A Field Compensated Michelson Spectrometer by Allan ... · 1.6 Recent Developments in Field...
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A Field Compensated Michelson Spectrometer for the Visible Region by Allan William Philip Kerr April, 1975 A thesis submitted for the degree of Doctor of Philosophy of the University of London and for the Diploma of Imperial College Physics Department, Imperial College, London, S.W.7.
Transcript of A Field Compensated Michelson Spectrometer by Allan ... · 1.6 Recent Developments in Field...
A Field Compensated Michelson Spectrometer
for the Visible Region
by
Allan William Philip Kerr
April, 1975
A thesis submitted for the degree of Doctor of
Philosophy of the University of London and for
the Diploma of Imperial College
Physics Department,
Imperial College,
London, S.W.7.
Abstract
The field compensation principle, first stated by Hansen, and
applied to interferometric spectroscopy by P.Connes and others, allows
the resolution-luminosity product of an interferometer to be substantially
increased. The gain, of such a system is limited by the aberrations of
the optics. This thesis concerns the application of the field compensation
principle to two beam interferometry. A Michelson spectrometer has been
constructed for use in the wavelenght region 400nm - 1000nm, with a
resolution of 105 at 500nm and a resolution-luminosity product of
2.5 m2steradians. The theory, construction and practical limitations
of the system are discussed.
(
Preface
The concept of the Four-Prism Field-Compensated spectrometer
was proposed by Professor James Ring and Mr. John Schofield who
began work on the instrument in 1970. I took over the project in
October,1971 and completed the mechanical and optical design of
the instrument by the end of my research studentship.
I would like to take the opportunity'of expressing my gratitude
to Professor Ring for his advice and encouragement throughout this
project and to Mr. William Stannard for his assistance in the
designing and construction of the mechanical components. I am also
indebted to many of my colleagues for their advice and help. Finally
I wish to thank the Science Research Council for providing me with
a research studentship.
iv
Contents
Chapter 1
Introduction Page
1.1 Interference Spectroscopy 1
1.2 The Resolution-Luminosity Product 1
1.3 1 The Field Compensation Principle 3
1.4 Aberrations 5
1.5 Systems of Field Compensation 7
1.6 Recent Developments in Field Compensated Spectrometry 17
The Four-Prism
Chapter 2
Field-Compensated Michelson Spectrometer
2.1 Path Difference 22
2.2 The Field Compensating Condition 23
2.3 Spherical Aberration 25
2.4 Chromatic Aberration 27
2.5 Zero Path DifferenCe 29
2.6 The Optical Parameters 32
.Chapter 3
Numerical Ray Trace
3.1 Introduction 38
3.2 Theory 38
3.3 The Size and Shape of the Fringes 48
3.4 The Gain'and Resolution-Luminosity Product 52
Chapter 4
Page Anti-Vibration Mounting
4.1 Introduction 55
4.2 Theory of the Harinyx System 55
4.3 Design Parameters for the Harinyx System 61
4.4 Spring Parameters for the Harinyx System 63
4.5 Damping for the Harinyx System 66
4.6 Conclusion 67
4.7 The Suspension System 69
Chapter 5
Mechanical Design
5.1 Introduction 70
5.2 Mounting of the Optical Components 73
5.3 Airbearings 76
5.4 Design Theory of Airbearings 78
5.5 Design Parameters for Airbearings 82
5.6 The Resistance to Flow in the Plastic Feed Pipes 84
Electrical Design
5.7 Control of Path Difference 87
5.8 Linear Motor Design 89
5.9 To Find the Flux Density in the Air Gap 90
5.10 Damping 91
Chapter 6
Conclusion
6.1 Adjustment of the Spectrometer 94
6.2 Experimental Verification 101
6.3 Application to Raman Spectroscopy 102
6.4 Conclusion 108
Appendix A 109
Appendix B 112
Appendix C 113
118 References
*vi
Page
• •
1
Chapter 1
Introduction
1.1 Interference Spectroscopy
Common to all spectroscopic instruments is the phenomena of
interference. This is obvious in the case of the grating, which is simply
a multi-beam interferometer, but it is also true for the prism. In the
latter case the dispersion of the light is due to the optical retardation
of different wavelengths in the prism material resulting in zero order
constructive interference. Historically the classification, interference
spectroscopy(1), has been applied to techniques of spectroscopy that
involve high orders of interference such as the Fabry-Perot and Michelson
interferometers.
Jacquinot(2) was the first to show that classical interferometers
like the Michelson possess a fundamental advantage over the prism or
grating instrument. They have a higher luminosity for an equivalent
resolving power, giving them an advantage in experiments that are limited
by source brightness.
1.2 The Resolution-Luminosity Product
The luminosity of different spectrometerst can be compared by
measuring the resolution-luminosity product, RLP. This is given by the
expression
RLP = (XAw)/(dX) (1.1)
where X is the wavelength of the incident radiation, A is the area of
Footnote to further classification of spectroscopic instruments may be
made according to the type of detector that is used. The term "spectrograph"
is applied to instruments that employ an image receiver and "spectrometer"
to those employing a total flux detector.
Mirror
Beamsplitter
Mirror
l.
fig.2
2
Logio(BL)
m2 steradians
0 2 4 6 8 Log10R
Resolution-luminosity products of spectrometers N
with common aperture 020cm2. (0.Grating
(B).Classical Michelson. (C).Field-compensated Michelson
fig.1
3
some pupil or stop in the system, w is the solid angle of radiation
passing through the exit pupil and dA is the resolvable spectral element.
The resolution-luminosity product of a classical Michelson interferometer
is a constant for a given aperture and approximately one hundred times
greater than that for a prism or grating spectrometer with an equivalent
aperture (fig.1).
It is possible to change the resolution-luminosity product of an
interferometer in such a way that it is no longer a constant. The result
of this conversion is to increase the useful field of the spectrometer
by making the path difference a more slowly varying function of the angle
of incidence i. The spectrometer is said to be "compensated" or "field
widened".
1.3 The Field Com eusation Principle
In a classical Michelson spectrometer set at zero path difference
the image of the reflector in one arm is superimposed on the reflector
in the other arm and vice-versa. The path difference between any two rays
entering the spectrometer is zero and independent of the angle of
incidence (fig.2).
If one of the reflectors is displaced a distance d, the Condition
of mutual imaging no longer applies. The path difference of a ray at an
angle i to the axis is given by
A = 2d cosi (1.2)
and the path difference of the axial ray is given by
Ao = 2d (1.3)
It is clear that the path difference of the off-axis ray is smaller than
the axial path difference by an amount di2 plus higher order terms. This
dependence on cosi causes the resolution-luminosity product of the
interferometer to be limited.
Field compensation(3) involves the introduction of optical systems
into the interfering beams in order to maintain approximately mutual
imaging of the reflectors at all values of path difference.
The concept of the variation of path difference with off-axis angle
can be expressed formally(4) by writing A - Ao in a power series of i as
follows:
A - Ao = Ai +
3 Di4 + Ci + Di + higher order terms
(1.4)
Thus for the classifical Michelson spectrometer
A - Ao = Bi2 + higher order terms (1.5a)
Similarly the equation for the field compensated spectrometer is
A - Ao = Di4 + higher order terms
(1.5b)
A comparison between equations 1.5a and 1.5b indicates that the advantage
of the field compensated system over the classical Michelson spectrometer
arises from the elimination of the second order term in Bi2. The result
is a gain in luminosity without any reduction in resolution, the gain
being limited by the aberrations of the optics.
It is possible to increase the field compensation still further(13) .
This is done by introducing a small axial displacement between the virtual
images of the reflector. The term in i2 is re-introduced into the
expression A - Ao, which becomes
A - Ao = ei2 + Di4 + higher order terms
(1.5n)
By choosing negative values of e it is possible to partially cancel the
terms in i2 and i4 making possible the use of off-axis angles greater
than that indicated by equation 1.5b.
1.4 Aberrations
Field compensated systems are liable to spherical aberration,
chromatic aberration and astigmatism.
(a) Spherical aberration
As explained in the previous section, the variation of path
difference with angle is modified in a field compensated system such that
A - Ao = Di4 + El6 + • • • +
(1.6)
where D is a function of the refractive index of the glass in the system.
. The term in l is the spherical aberration, the higher order terms becoming
the higher order symmetric aberrations.
■
(b) Chromatic Aberration
since. the refractive index is a function of wavelength for
all optical media other than a vacuum, two effects *are observed in a
field compensated system ():
(1) The optical path length for all rays including the axial
rays will be a function of wavelength and so distort the computed spectrum.
(2) The dispersion in the system will re-introduce a variation •
of path difference with off-axis angle for all waverengths except the
compensated wavelength. For example, in the Mertz(6) first system (see
equation 1.10) the variation of path difference with i2 is re-introduced.
(c) Astigmatism
Here the path difference of similar off-axis rays will vary
in different planes. As a result the fringes at the exit aperture of the
instrument are hyperbolic in shape when the central fringe is made as
symmetric as possible (). If one plane is compensated at the expense of
the other the fringes are elliptical in shape, giving rise to a reduction
in...the efficiency of the system.
I
t / 1 Mirror
Plane parallel
slab \.
Beamsplitter
Mirror
Wedges.
4
Mertz's First System
- 6
fig. 3
7
1.5 tyjsofFielLCoo2Sstei nensation
Ring and Schofield(5) have proposed that field compensated systems
may be divided into two categories:
A. Those in which the optics consist of plane surfaces(4,6,7,8)
B. Those which consist of curved surfaces(9,10)
A further classification may be made according to the number of
translations that are involved when introducing a path difference in the
spectrometer. Interferometers that involve a change in dielectric
thickness(6) as the mirror in that arm is displaced invariably require a
complex servo-control system linking the two translations.
Systems involving only one translation are,simpler to construct
and operate(11)
All the systems proposed are to some degree liable to spherical
aberration and chromatic aberration although these may be partially removed
in some systems. Those that are classified under A are liable to
astigmatism.
The field compensated interferometers classified in categories
A and B are described in the following sections. Details are given of
the gain and aberrations of the different systems.
A.(i) The Mertz First System(6)
This system, that was proposed by Mertz and analysed by
Bouchareine and Connes(7), consists of a classical Michelson interferometer
in which a plane parallel block of glass, refractive index n, is inserted
into one arm. A similar glass block but of variable thickness is
introduced into the other arm (fig.3). The path difference of a general
off-axis ray is given by
A = 2 [e(n cosr - cosi) + d cosi] (1.7)
8
where e is the thickness of the excess glass and d the amount the
mirror is displaced in that arm. To find the field compensation condition
it is necessary to find the relationship betWeen e and d. Expand cosr
and cosi as a power series in i, equation 1.7, collect the terms in i2
and equate them to zero. The relationship is given b7
d - erci (1.8)
The. expressions for the spherical aberration and the chromatic aberration
can be found in reference 7 and are given by:
Spherical aberration:
. i4 = o
1+12
JTh where A
o is the path difference for the axial ray.
Chromatic aberration:
.2 = Li-an an1 At A +
o n-
2n(n2-1)I
(1.9)
(1.10)
where (Sn is the change in the refractive index of the glass over the
spectral range.
Since the field of the spectrometer is defined by the criterion (7)
that the path difference of an off-axis ray and a paraxial ray must not
differ by more than Athe maximum field compensated angle imc is
a i2
8n2
o me =
.2 1 = 2n — mc R
where R is the resolution and is equal to o . For a classical
Beamsplitter.
Direction of
motion
Fixed prism
Moving
prism
Bouchareine and Connes's System
fig.4
10
Michelson interferometer at the same resolution
.2 2 1 = — m R (1.12)
Thus the gain in luminosity of the field compensated system over the
classical Michelson spectrometer is
.2
G = flmc 1 mc
nmm
2R (1.13)
For an instrument with a resolution of 104 at 0.5pm and refractive indices
1.5 the gain is 210.
(ii) Bouchareine and Connes' Prism Interferometer (7)
Bouchareine and Connes have developed a prism interferometer
that involves only one translation instead of two as in the Mertz First
System. In this system two identical prisms replace the plane mirrors
of a Michelson interferometer (fig.4). The back surfaces of the prisms
are silvered. If one prism is moved in the direction of the apparent
position of its reflecting surface (the dotted line, fig.4), the apparent
distance remains fixed but the optical path difference changes.
The path difference is given by:
A = 2x n sina + tanY.n2
I
sin2 a-1 cosy (1.14)
cos
where n is the refractive index, a is the prism angle, 3 = sin-1(n sina
= tan-1 tan2a and x is the translation.
tans
The expression for the coefficients of spherical, chromatic and
astigmatic aberrations, derived in reference 7, are given by
Spherical Aberration
LA oi4
;TT?: (1.15)
•11
-7/7"/"/7 //•///,,
System 1.
V
System 2.
Shepherd's Systems
fig.5
12
where Ao is the path difference of the axial ray
Chromatic Aberration
Ao6ni.2 — '
2n(n2 -1)
(1.16)
where do is the change in the refractive indices of the glass over the
spectral range.
Astigmatism
The astigmatism of the prisms is not equal for the two arms, except
at zero optical path difference, and it is this that limits the size of
the field. The astigmatic aberration limit is given by
Aotan2 a.1
2 (1.17)
4
Thus the gain in luminosity of this system over the classical interferometer
is
G = 2 cot2a
(1.18)
For a = 8° the gain is approximately 100 at a resolution of 4
(iii) Shepherd's Systems(8)
Shepherd has proposed two systems which can be scanned
by only one translation (fig.5).
In the first system the path difference is given by
A = 2d cosi cosa - 2dn cosr sing (1.19)
where d is the translation and i and r are the off-axis angles in
air and glass. The field compensation condition is
tang = n (1.20)
.13
Mirror
Hilliard: and Shepherd's System
fig.6 •
14
The spherical and chromatic aberrations are the same as for the Mertz
System. Consequently the gain is of the same order of magnitude.
In the second system the path difference is
A = 2d cos cosi + 2d sine (n cosr - cosi)
(1.21)
which is compensated when
tans 3= n
n-1 (1.22)
Again this system is similar to the Mertz System.
(iv) Hilliard and Shepherd's System(4)
This instrument has been constructed specifically for the
purpose of measuring Doppler line widths of emission lines of the upper
atmosphere. It has a fixed path difference in glass, the path differenci.
as a whole being oscillated over a few hundred fringes (fig.6). The
large field at zero path difference is transferred to any desired path
difference by this method. The path diffefence is given by
A = 2 [in cosi - do cosio] (1.23)
where d and i are the glass thickness and ray angle with the normal
for the path in glass, do and 10 are the analogous quantities for the
path in air. The field compensation condition is given by
d tani = dotanio (1.24)
Using the limiting conditions for small angles, equation 1.24 reduces to
do = din (1.25)
and the path difference for the axial ray is given by
Ao = 2d(n-1in) (1.26)
.15
• 11.1, *MOO S INN* IMMO OMNI 1111. 4111110111 ONO -.lib
Telescope
Telescope and
mirror move
together
Beamsplitter
I Fixed 1 mirror
■
TIM(
I ■„ ■ ■ ■
I,1 1 1 -L.__ ... •■ 411111■ OM. ■•■■■• NNW MEW .1■11, .1.■F ...No- ,...... ■Nol
Connes's Afocal System
fig.7
16
The spherical and chromatic aberrations are the same as for the Mertz
System.
B. (i) The Afocal System(9)
Hansen(12) originally proposed the concept of field
compensation and it was Connes who constructed the first instrument.
In each arm of a Michelson interferometer a telescope is placed,
consisting of a positive and negative lens of equal power (fig.7).
If the telescope in one arm and the mirror in the other arm are displaced
simultaneously it is possible to keep the image of each mirror superimposed
on the other in the beamsplitter to a Gaussian approximation. It is
necessary to use a telescope with a magnification 2-1. The telescope
and mirror are mounted on a common carriage; the system is subject to
all the aberzations associated with lenses. The one advantage of this
system is that it is not limited by considerations of dielectric thickness.
(ii) Cuisenier and Pinard's System(10)
The authors have proposed a field compensated interferometer
in which the secondary mirror of a cat's eye retro-reflector is deformed
as the path difference is changed. The advantage of a cat's eye retro-
. reflector is that off-axis rays incident on the secondary mirror are
spatially separated and the necessary compensation can be introduced by
varying the radius of this mirror. This presents two problems: To
maintain the spherical shape of the secondary mirror as its radius varies
and the synchronisation of the change of radius with change in path
difference.
Input
Output 2
Output 1
stop
fig.7a
17
1.6 !Est. 2Eflopr..._,,22.151 in Field-Compensated Spectrometry
Work on the Four-Prism spectrometer and the two systems of
field compensation described in this section have been carried out
concurrently.
Elsworth, James and Sternberg's System(29)
The design is based on a system proposed by Mertz(11)
and
modified by James and Sternberg(30). The path difference is introduced
by moving one of the glass blocks so as to introduce extra glass into
one arm and extra air into the other. See fig. 7a for the optical
layout. The field compensation condition is
sins = psin(a-0) ' (1.27)
where p is the refractive index and the angles a and 8 are shown in
fig.7a. For a translation x of the carriage the path difference, A, is
= 2x(psina-sin(a-8))
(1.28)
To avoid the possibility of total internal reflection of the marginal
rays and guard against the effects of polarization a must not exceed 27°. r_
1 1
Field stop
_J Sliding carriage
N
Direction "NI of motion
oft diaphragm
Beam splitter
Gas B
18
Spherical aberration, caused by the introduction of glass at path
differences other than zero, limits the size of the field. In the
original design an oblique air gap was present in one arm and the
authors found that the astigmatism and coma of this gap caused them
considerable problems. If the gap was not kept to less than 2um it
limited the size of the maximum field compensated angle. This was
overcome by introducing a compensating oblique gap into the other
arm. Now the two gaps must not differ in thickness by more than 2um.
- Chromatic aberration restricts the spectral range to a few hundred
wavelenghts. The instrument has a resolution-luminosity product
of 0.1 cm2sr.
B. Field-Widened Michelson Spectrometer with nolizainuLEL52
Hirschberg(31)proposed this system and Steel(32)showed how
it could be made field compensated. In this method two spherical mirroc-
A and B, of radius r, are adjusted with their centres of curvature,
cA and cB' at the surface of the beamsplitter, fig.7b.
Mirror A
Gas handling section
Gas A
D
•••■•••■•••
Window
Window
Mirror B
- fir .7b
'19
The surfaces of the beamsplitter are also made spherical of radii r,
fig.7c.
Beam splitter
Mirror
fig.7c
When the tr,a arms of the spectrometer are filled with different gases
of refractive indices n1-and,n2' the path difference for the rays
through the centres of curvature cA and cB is
P = 2(nl-n2)r (1.29)
The variations of path difference between a parallel ray and an off-axis
ray are of the fourth order. For a point at a distance re from the axis
at the beamsplitter and a point at a distance r4/n at the mirror in
planes an azimuth * apart, the extra path difference is
1 pe202c0s4 '4n/n ' 2
(1.30)
where n is the refractive index of the gas. The spectrometer is
field compensated. The resolving power is given by
4 1-n
2)
(1.31)
20
where D is the distance between the spherical mirror and the beamsplitter,
fig.7b. Assuming that one chamber is filled with sulphur hexafluoride
and the other with helium, then n2-n1=102. D is one metre, then
R=105 at 0.4pm. The chromatic aberration is small because of the low
dispersion of gases. A one metre spectrometer with 16cm diameter optics
has a resolution-luminosity product of 40m2steradians.
Upper arm of spectrometer
1.
13
A
V Bemnsplitter Mirror
Y
21
Lower arm of spectrometer
/ /
Direction of
motion
Four Prism System
fig.8
.22
Chapter 2
The Four-Prism Field-Compensated Michelson Spectrometer
Theory
The optical layout of the spectrometer is illustrated in fig.8.
It consists of two prisms and one plane parallel block of glass. in each
arm of the interferometer. Prisms 1 and 2 have the same refractive index,
na, as the glass block 6, nb, being the refractive index of the other
optical components. Three possible variations of the system exist
depending on which pair of prisms are translated to produce the path
difference between the interfering beams. Only the translation of prisms
1 and 4 will be considered.
2.1 Path Difference
The path difference is introduced by a common translation x of
the two large prisms 1 and 4 in a direction parallel to their common faces
with the small prisms 2 and 5. Consider the path difference of a general
off-axis ray in a glass prism whose thickness has been increased by Xsina
fig.9
.23
By Fermat's principle EF = naGH
where na is the refractive index of the glass.
the path difference = (AB + BC)na
AB = Xsina Cosr
a
where ra is the angle of refraction
BC = AB Cos 2ra
= Xsina cos 2ra Cos ra
naXsina the path difference is: A = cosy
(1 + cos(2ra)) a
but cos2ra = 2cos
2ra-1
A = 2Xnsina cost a a
Similar calculations apply to the beam in the other arm of the
spectrometer. Thus the total path difference is given by:-
A = 2X (aacosr
a sina - nbcosrb sin3)
(2.1)
2.2 The Field Compensating Condition
To derive the field compensating condition it is necessary to
expand cosra and cosrb as a function of i.
From Snell's Law sin i = n sin r a,b a,b
24
. 2 cos2ra,b+sin r a,b = 1
• cosra,b = (17- sin2ra,b) i
2 = (1 - sin2. a,b)
1 .3
cos ra,b 2
(1 - • (i - i3)2) na,b 6
Expanding the root binomially, neglecting terms which will give rise
to powers greater than the fourth in the final expression
- .3 1 .3 1 1
cos rab = (1 - 2 (i - )2 4 (1 )4) , 8n
2na,b 6 a,b
= (1 - 1
2n2 (1
2 _ .1. 36 )
a,b a,b
.4 4 6
8n 4 1 (i4 +...))
.2 .4 1 1 , cos ra,b = 1 - 2 2 (1/3 -14 na,
2b )
2na,b 2n a,b (2.2)
substituting the expression into 2.1
.2 .4 .2 .4 . 1-1 1/ 1/4n2„ . + 1 (1/3 _1/44))) A = 2x Eiasina( —2 + --2- ( 3 - ail- nbsinok 2• 2 2n 2n 2n2na b
(2.3) 2 By equating the terms in . to zero, the field compensation condition
is given by
na sing sins -b
(2 .4)
Consequently,
= =
and for
substituting
A -
the
nasina(11.---
the principal
2.6 into
Ao = x
path difference of an off-axis ray is given
.4
1/ 1/. 2„ _ i4 1/ ( 3 - 4na)) 2
nbsina(14---f ( /3 - 2na
2nb
ray the path difference is
Ao = 2x [aasina nbsin
2.5
2 — sina(1/3 - 1/4ni) sina(1/3 - 414nb)
by
1/. l/
.4 1
2„-
25
(2.5)(2.5)
(2.6)
(2.7) a nb
The advantage of a field-compensated system arises from the cancellation
2 . of the i2 term, and is best illustrated by comparing equations 2.7 and
I.5a the expression for the clasgical Michelson.
2.3 Spherical Aberration
To determine the coefficient of spherical aberration, which is
responsible for limiting the gain of the system, it is necessary to derive
4 the term in i4._ This is given by 2.7
A Ao -sing 1/ 1/, 2, sins (1/3
- 1/443)1 i a
x n ( 3 - 4na) - nb (2.8)
substituting 2.4 and 2.6 into equation 2.8 gives
A - to
2° 2 nb a
° [/4 2 - 1/4n1i4 2(na-nb)
(2.9)
A - Ao Ao i4 2 2
8nanb (2.10)
26
A
V Beamsplitter
V
le1 and MI are the virtual images of the mirrors 4
M1
and M4
fig.10
27
2 . Alternatively, by re-introducing an 1 term into equation 2.5, it is
possible to eliminate the spherical aberration for a particular off-axis
angle 1 3)c . Re-arranging equation 2.5 gives
A Do x [7i2 ( (1/3_ Viin:)s!.na (1/3 -1/4n2)sinIB ) a nb
Thus for a particular off-axis angle is
the spherical aberration will
be eliminated when
[
1/ a n
a
(1/3 _1/4112\ sin$ 2 b/ n ,
1c . e ....
. - ( ,3 _,4o2) sina I/
u
In practice this is achieved by displacing the virtual images of the mirrors
a distance e, as illustrated in fig.10. The effect of this displacement
is to make possible the use of off-axis angles even greater than indicated
by equation 2.10. The choice of is is most conveniently found by
numerical ray tracing.
2.4 The Coefficient of Chromatic Aberration
The spectrometer is designed to operate in the visible region of the
spectrum and so it is necessary to take into account the effect of
chromatic aberration. Consider changes (SA in A, the path difference of
an off-axis ray, due to changes of Sna and Snb' which are the changes in
refractive index of the glasses over the spectral range. na and nb are
mid-range values. A is given by equation 2.1:
= 2x(nasina cos ra - nb
cos rbsinS)
Substituting for cos ra and cosx.1) from equation 2.2
28
.2 .2 A = 2X(na(1 2) sing - nb (1 1 ) sine)
2na 2nb
'
.sina sinnb , = 2xnasina - 2Xnbsina - LX1
2, 2 2na
but Ao = 2x(nasina - nbsine)
(sing sine) i2 A = Ao x
a) 1 na nb
2 (2.11)
The variation in path difference may be written
6A ._EL 6n 4. g A 6 ria a 3nb nb
Consequently, equation 2.11 becomes
SA = a 2 un - `?
a a b dnb - x [
n2 a
. a . dnb
'511 n2 Ao sing On
sine
n . = 2k (sins On - sine Onb) -
sina Sna si2a drib)].2
na nb
sina, , 2
sing sine 2 = 2x sing (Ona - dn.b sina )
ona 2 dnb)i
na nb
Using the field compensation condition 2.4
nb SA = 2x sina -Sn. . + 2x sin
a b nb )
On - .2 a 1
n2 2 a
1.2
but Ao 2xsina a2-n2)
a b • 4
611b Sn
a) (la
7nb6nb) o i2 nb
n a
SA = 2 (na
2 -nb) 2(na-nb)
(2.12)
29
Connes and Bouchareine have pointed out that the introduction of this
variation in path difference has two effects. The first term in equation
2.12 will serve to distort the spectrum and the second term re-introduces
2 a dependence on the
. term, resulting in a reduction of the gain. The
influence of the second term in equation 2.12 can be reduced if
Sna
6nb (2.13)
na nb
As a result the system will be achromatic for two wavelengths and nominally
achromatized between these wavelengths.
2.5 Zero Path Difference
For a wide field at zero path difference it is necessary that the
total thicknesses of each pair of prisms and the path difference in air
Should be in certain ratios. There are two ways in which this can be
achieved.
In the first method it is necessary to find the ratios of
pathlengths in the prisms and air which give a wide field at zero path
difference. The condition that there is zero path difference for the
principal ray is
Lana Ihnb =
(2.14)
where I.a is the minimum thickness of the prism pair 1 and 2, see fig.11,
kb
is the minimum thickness of the other prism pair 3 and 4, and y is
the extra path length in air, assumed to be in the first arm of the
interferometer.
fig. 11
3.
to a• 2n
2 4. 2 -2bnb' 2
a 2
.2.2 .2 .y 0
• 30
' • i2 Equating the . term of the path difference equation 2.3
to zero gives
ta tb
y •'"""' = 0
.na nb (2.15)
From equations 2.14 and 2.15
/13 R = - y.] n
a 1113-37
= a nb na
n2
y(1-n) _fib Oa_ - a b
tb b a (n2 -1)
(2.16) y 2 2
na-nb
Combining equations 2.16 and 2.17 gives
nb(na2 -1)
b = .a
na(nb-1) (2.18)
31
And
= [ y na
ta n
b
ftana+y
b na
2 2 nb
Y(nb -1) = ta(n a na
2 ta
na(nb-1)
Y n2 n2 a' - b
(2.17)
This gives the ratios
2 2 2 2 na-n2 \/ na-n2
y : ta : t
b = 1
2 2 )1- na(nb-1) nb(na-1)
The values depending on the minimum thickness of the thinnest prism.
These ratios provide a wide field for only one wavelength. If different
glass types are chosen to minimize the variation of zero path difference
with wavelength, the achromatism for change in path difference is lost.
The second method is to make path length in each arm identical by
introducing a plane parallel block of glass into each arm of the spectrometer,
see fig.8, the glass block in the arm containing prisms 1 and 2 having
refractive index nb and thickness equal to the minimum thickness of the
prism pair 3 and 4 in the other arm. Similarly, the block in the arm
containing prisms 4. and 5 has refractive index na and thickness equal to
the minimum thickness of the prisms 1 and 2. This will give a wide field
32
at zero path difference. In practice the blocks are made slightly
thicker to allow the production of short interferograms that are
symmetric about zero path difference. It is this method that is used
here.
2.6 The Optical Parameters
The choice of glass types will depend on the following design
criteria:
A. ApertUre,size and resolution.
The instrument will have an aperture of 10 cm diameter and a
0 resolving power of 105 at 5000A.
B. Achromatism
The system will be achromatic for two wavelengths and approximately
achromatized between. these wavelengths. Equation 2.13 can be used to
define the value of the refractive indices that will satisfy this design
specification.
C. Size of the large prisms
The size of these components will be related to the resolving
power and the field compensation condition, equation 2.4. To determine
the miniman:prism volume required consider the prism illustrated in fig.12.
It will be assumed that both prisms are rectangular. The correction for
circular prisms is Small as it occurs in both terms (see fig.12 for prism
volumes V1'V2 and V3).
33
L
fig. 12
V1
= (2L +'A tana).A.A
V2
= x cosa.x sina.A/2
= x cosa (L + Ltana).A
where A is the aperture diameter
L is the edge thickness necessary for rigidity
a is the prism angle
x is the translation distance
Total volume V = V1 + V
2 + V
3
V =A((2L+A tana)A + x cosa (L+L tana + 2 --sina)) (2.19)
but x = 2(na
sina-nbcosa)
A 0
R=105
34
Volume of High
dex Prism (Litres)
10 20 30 40 Prism Angle (Degrees)
fig.13
35
Using the field compensation condition equation 2.4
AoN x
2sina
where na 2 2 na-nb
substituting into equation 2.19 gives
A2m2 V = AP AAL + A2tana + 2NA + cota (AtiEL +
2
The minimum volume will be given by dV/dx = 0
a .min = sin 2 2
A + An 2 8
(2.20)
0 Fig.13 illustrates the variation of V with a for R = 105 at 5000A.
The best compromise was found to be the two glasses
Schott LAK8 nd
= 1.713003
main dispersion ni-ne = 0.013245
and
Schott LLF1 nd = 1.548140
main dispersion nF-ne = 0.011980
The angle a for the large prism was fixed at 200, see fig.13, as
this was easiest for the glassmakers to produce. After polishing the
36
final value of the angles was
a = 19° 58' 50"
and
8 = 17° 59' 15"
Considerable difficulty was encountered when polishing the
- reflecting surfaces of the large prisms due to the effects of thermal
expansion. The original specification was to polish the reflecting
surface to an overall tolerance of X/20 but this was found to be
impossible - consequently the specifications were relaxed. Instead of
polishing the whole reflecting surface to X/20, 3" sections along the
whole length of the prism were polished to within X/20 such that a
/ variation of
A /4 could exist over the whole surface, as shown la fig.14.
/4.
4 • or ■
fig. 14
X 37
Translations
A1A2=.812
A2A3=d23
A3A4=834
A4A5=645
A5A6=c356
Direction Cosines
A :a a y-oct i a t y' 111 11 1,
A„ccialyi4.a282Y2 . 4
A3 a 3 ' 2 2y 2 2 2 2 A01$2y2-,a3$3y3
A5'a3'B3Y3-aP3Y;
N'aP3i34(14134Y4
Refractive Indices
n=1; ni =na; n"=nb.
Plan of one arm of the spectrometer illustrating the path
of a skew ray
fig. 15
38
Chapter 3
Numerical Ray Trace
3.1 Introduction
In deriving the path difference, equation 2.1, it was assumed that
there were no air gaps between the prisms in the system. However, as
Ring and Schofield(3) have pointed out, translation of the prisms is
not possible unless gaps are present, because of friction and the
possibility of optical contacting of the prism face; as a consequence
air gaps were introduced between the components in the system. The
influence of these gaps on fringe shape was determined by a numerical
ray trace. In addition, the effect of drive inaccuracies on the gaps
was computed. In the next section the ray tracing equations that form
the basis of the computer program are derived for one arm of the system
and the results of the numerical ray trace are summarised and discussed
in the final section on the size and shape of the fringes.
3.2 Theory
Consider the first surface of the compensating glass block, fig.15,
to te parallel to the xy plane and to intercept the z axis at z=0. The
equation of the surface is z=0 and the normal to the surface is a unit
vector pointing in the direction given by
v =K
with components
v(x,y,z) = (0,0,1)
The light ray, incident on the surface at the point Al, is a unit
vector e specified by direction co-ordinates a1,81 and 11. The
39
refracted ray is a unit vector e', with direction co-ordinates a1,81,y1,
the law of refraction relating the two vectors by the equation
1 1 n e = ne + bv- (3 )
The constant b is determined by taking the scalar product of both sides
of 1, see fig.16.
v.v = 1
fig.16
f e.v = cos 8 e .v = cos e
t (3.1)
b = n cos - n cos 8 (3.2)
/ v The next step is to determine the direction co-ordinates al'
vY1
by using the law of refraction in the form
fw f
n (a.i+01 J+Y1 K) = n(a1 i+aL j+Y K)+(n case) -n cos 8 )K
-- --
Comparing coefficients
1
a1 = n a1 a.
1
B1 = n 1 (3.3) b. --
v I Y1 = n Yi + (n cos e
I-n cos e ) c.
—1, r n n
.40
where nf is the refractive index of the glass block.
To evaluate cos 0 use 3.1
e.v = cos e = Y — — 1 (3.4)
Taking the vector cross product of 3.2 to determine cos 6'
n sin 6 = n sin 6
n'2(1-cost 6') = n2(1-cos26)
n cos 6 = (n12 -n2 +n2 cos2 0)
(3.5)
This completes the ray trace through the first surface, by means of
the refraction equations 3.3, 3.4 and 3.5. The refracted ray now becomes
the incident ray for the second surface and the co-ordinates (x2,y2'z2)
of the point A2 in which the refracted ray meets the second surface
must be determined. If (512 denotes the distance between the points Al
and A2 then the co-ordinates are given by
. A = A + e a lt —1 — 12 (3.6)
Comparing coefficients
x2 = x1 + ad12
f
Y2 = Yi yi 2
z2 = z
1 + Y161 2
(3.7)
r
but the second surface is parallel to the first and intercepts the
41
z Axis at z = el
z2
= zl + e
l = e
l
The distance dl2
is given by
612 = (x2-xl)2+(572-571)2+(z2-z1
)2
substituting for x2,y2 and z2 gives
e1 612 Y
1
(3.8)
This completes the calculation of the distance between the surfaces;
similar calculations at the second surface of the compensating block
give
n 1 a a2
n
02 = -n 1 ntat
b
cos-8 = Y' (3.9) 1
cos Eit = (n2-n,2+n,2cos20)
d
n Y = 1 + (n cos 0' -n' cos 0)
2
The'co-ordinates at (x3'y3,z3) of A
3 in which the refracted ray meets the
.42
third surface are given by
x3 = x2 + 82 3 a2 a ,
and the distance 2
Y3 =
z3
is
62 3
-
Y2 + 62 3 $2
z2 + 62 3 i2
el+e2-z
2
4 (3.10)
(3.11)
(1-a2-) 1 26,2
2
where e2 is the distance between the first surface of the small prism
and the second surface of the compensating block.
At the first surface of the sma..1 prism, fig.15, application of
the law of refraction gives
= a —
2 nil a2
a, n 2 n" 2
(3.12)
cos 0= Y2
cos 8'= (n"2 -n
2 +n2 cos
2 6)
n"
(n"cos -n cos ) Y 2 n 2
net
a
b
d
e
where n" is the refractive index of the prism. The incident ray meets
43
the surface at the point A4 where co-ordinates .are given by
x4 = x
•
3 + `'3 4 a2 a
Y4 = Y
•
3 + 63 4 82
(3.13)
z = z + 6
4 3 3 4 2 C
° Before the translation 63 4
can be computed the equation of the second
surface of the prism must be derived. This equation is found in terms
of the intercepts on the x and z, axes.
The intercept on the z axis is z = Q, and that on the x axis
is Q/tan a where Q.= e1+e2+e3. The equation of the surface is then
given by
= 1 Q/tan a
• • x sin a+ z cos a= Q
(3.14)
where .a is the angle between the first and second surfaces of the
small prism. To derive 63 4
is necessary to find the co-ordinates
of the point of intercept, A4, of the incident ray and the surface.
Since the ray passes through the points A3 and A4 it should satisfy
the equation(14)
x4 x3
a' 2
7.4-Y3 z4z3 Y'
2 2
a
a • att (z -z ) 4 3
x4 _ -
Y,4.,x3 b (3.15) 2
44
(x4-x3) = 2 z
a2 3
'substituting 3.15b into 3.14 gives the co-ordinate
.c
QYcos a +z3 a' sin a - x3Y3sin a (3.16) z4 - a3sin a + Y3cos a
-Similarly substituting 3.15c into 3.14 will give
Qa t cos a +x3Y2cos a -z3 a' cos a
(3.17) x4 &2sin a +Y3cos a
Substituting 3.16 and 3.17 into
634 = [(x4-x3' '')"4-3731‘2
4.'z4-z3)2 ] 1
where y4 = Y3 + r32' 634
will give
634 Qcos a -x3sin a -z3cos a
a2 sin a +Y2 cos a (3.18)
Application of the law of refraction at the second surface of
the prism will give the direction cosines
cos 8 = a' sin a +Y2 cos a 2
cos 8 = (n2-n"2+n"2cos28)4y n
a3 n" a' (n cos 8' -n"cos e) sin a c (3.19) —
R3 - eat 3 2
a
.45
Y3 = n" YI 4. (n cose'-n"cos e )cos a z
To evaluate the effects of an oblique air gap on the path
difference, the large prism is rotated through an angle 0 in the
horizontal plane and through an angle (I) in the vertical plane, as
shown in fig.17
X
z
fig. 17
The translation 645 will be derived by the same method that was
employed to compute 634
The equation of the first large prism surface is
x
y z - 6sin(a+0)/tana tan. 6tan(a+0)/tana 1 6/tan a
where 6 = e1+e2+e+e
4
46
which becomes, after reduction
xsinasin(a+e) cos cp -y sin a sin (I) +z sin a cos (a+0) cos cp
= cScos a cos 4) sin(a+0) (3.20)
Since the incident ray passes through the points A4 and A5 it should
satisfy the equation
x5-x4 y5-y4 z5-z4 a
b
c
a3
Y5
5
a3
$3(x5-x4)
Y3
+Y4
+ z4
a3
13(x5-x4) a3
(3.21)
Substituting 3
'
.21.b and c into 3.20 gives
x5 = cSa3cos a cos cp sin (a+8)-63x4sin a sin cp +y4sin a sin go a3
1 +y3x4sin a cos (a+0) cos cp -z4a3sin a cos (a+13)cos cb
[a3sin a cos (I) sin (a+8)-63sin a sin tp +Y3sin a cos cp cos (a+8)]
(3.22)
The procedure is repeated to obtain y5 and z5.
The co-ordinatEs x5'y5 and z5 are substituted into
1 = [(x5 -x4 )
2+(y5 -y .4)2+(z5-z4)2] 2
4 5
rS4 5
= 6cosacoscPsin (a+0)-x4s inacos cpsin (a+8) +y4s inasincP-z4s inacos cpcos (a+81
a3sinacosOin (a+0)-63sinasincp+Y3sinacos ybcos (a+e)
(3.23)
47
(545 is the translation across the oblique air gap between the prisms.
The direction cosines at this surface are
rose = a3sin(a+e)coscp-a3sinc1)+Y3cos(a+e)coscP a
cosO' (n"2-n2+n2cos201
n
a3 na3
•
(n"cose'-ncose)sin(a+0) cos (op) c .n„
-
n" n
(3.24)
a' 3 na3
•
(n"cose'-ncos )sin4 - n" nfi
d
Y' nY3
•
(n"cose'-ncose)cos(a+e) cos (4) e 3 • nft nfi
The final translation, 85 6 • , is to the rear surface of the prism
as shown in fig.15. The equation of this surface is
y tan(I) R Rsincl) Rtan4 1 (3.25)
(8+e5) 8 sine tana
which after reduction becomes
x sinecoscp-y sincp+z cosecoscp ='R sinecoscp
The method used to derive 15 34 and 45 is employed to compute 856
56 [y.3sinecoscP-a3sin4)+Y3cosecos]
[Rsinecos(1)-x5sinecosq)+y5sincf)-z5cosecosil (3.26)
.48
The light ray reflected off the rear surface obeys the law
of reflection given by
e' = e - 2vcos$ •■■• ••••• (3.27)
where v is the normal to the surface, e the incident ray and e'
the reflected ray.
The direction cosines are found by comparing coefficients of
3.27 where v = sinei-sing+cosK
cos$ = e.v
f 1
= a3sinecos0-$3 sintpl-Y3cosecos$cosch a - .
a4 = a3-2.sinecosOcoscp b
f34 = a3+2,sincpcos$ c
(3.28)
Y4 Y3-2,cosecos$coscl)
The ray returns through the system. The translations and
direction cosines are calculated in a similar manner to those described
earlier. These equations form the basis of the computer program, shown
in appendix A, that was used on the IBM CDC 6400 computer to determine
the influence of the air gaps and drive inaccuracies on the shape of
the fringes.
3.3 The size and shape of the fringes
The field of the spectrometer is defined by the criterion that
the path difference of an off-axis ray and a paraxial ray must not differ
by more than X(7). The size is limited by the aberrations of the optics
and is found by using equation 2.10, the/coefficient of spherical
•
49
aberration, which is
A i4 A —
Do =
8n2n2
a
Consequently, using this equation and the criterions, for field size
it is possible to derive the formula for the maximum field
compensated angle ic. It is given by
where na
and nb are given by equation. 2.16. If the resolution of
the instrument is 105 at 0.5pm th‘n
1. = 90
As explained at the beginning of the chapter, it is necessary
to introduce air gaps between the optical components in the system.
The influence of these gaps can best be determined by measuring the
change in value of the maximum field compensated angle for different
gap widths as shown in Table 1.
Air gap (um)
5
Maximum field compensated angle (degrees)
9
10 8
15 8
20 7
25 7
50 3
75-• 2
100 1
Table 1
50
The gap widths between the prisms in each arm are the same and
the field compensated angle is measured in the horizontal plane xg,
fig.15. Up to a gap width of 8 microns the spherical aberration will
limit the size of the fringes. Beyond this point the fringe size is
governed by the width of the gap. The dramatic reduction in the value
of the field compensated angle and, as a result, the gain of the system,
is due to refraction at the glass-air interface between the prisms in
each arm. The prism faces are at different angles in the horizontal
plane. Consequently the light is refracted at a different angle in each
arm. And as a result the path length travelled by the light is different
in each air gap.
When the field compensated angle is measured in the plane
perpendicular to the horizontal plane, the yz plane fig.15, the only
limit to die fringe size is set by the spherical aberration as shown
in Table 2.
Air gap Maximum field compensated angle (inn) (degrees)
25' 9
50 9 •
75 9
100 9
Table 2
Theoretically it is possible to increase the size of the field
by displacing the virtual images of the mirrors as discussed in
chapter 2. If this is done the effects due to refraction are eliminated
51
in the horizontal plane only to be re-introduced into the vertical
plane as shown in Table 3.
Distance between virtual images (microns)
Max. Field Comp. Angle (degrees)
Horizontal Plane Vertical Plane
55 10 6
50 9 6
45 8 7
40 8 7
35 7 7
30 6 8
Table 3
If the virtual images of the mirrors are kept superimposed
the system is astigmatic because complete compensation can only be
obtained for one plane containing off-axis rays. The result of this
is to make the fringes at the exit aperture elliptical to a first
approximation. Photographs of the fringes at zero and maximum path
difference are in chapter 6.
Complete compensation can be obtained by displacing the virtual
images of the mirrors, although the maximum value of the field
compensated angle is reduced, see table 3.
In order to introduce a path difference into the system it is
necessary to mount the prisms on a common carriage as explained in
chapter 5. The carriage is located in the vertical and horizontal
directions by air bearings. As a consequence it is advisable to calculate
52
the effects of drive inaccuracies on the shape and size of the fringes.
It is possible for the carriage to tilt in the vertical plane yz
introducing gaps of different widths into each arm. These gaps can
differ in width by a maximum of 5 microns without any change in the size
of the field. Similarly if the carriage rotates in the horizontal plane
xz the gaps may differ in width by a maximum of 21 microns or 6 arcsec.
In addition to investigating the effects of gaps in the system
it is important to determine the tolerances on the prism angles because
these decide the extent of the field widening. Calculations show that
a 10 arcsec change in the smaller of the two prism angles gives aft degree
change in the field compensated angle ic. Therefore the maximum allowable
tolerance on the prism angle is *10 arcsec.
3.4 The Cain and Resolution-Luminosit Product
For a simple Michelson spectrometer the size of the field is
X 2
.2. 2X 2 =
M Ao (3.30)
As in the previous section the field size is defined by the criterion
that the path difference of an off-axis ray and a paraxial ray must not
differ by more than X. R is the resolving power.
For this field compensated spectrometer the field size is given
by equation 3.29
.2 'MC 2nanb ,/ (3.31)
53
Therefore the gain of the field compensated spectrometer over the simple
Michelson spectrometer is
w
.2 wc 1M
i C
G = = = neb [21 (3.32)
For a resolution of 105 at 5000A the gain is 1185.
The resolution-luminosity product, RLP, is given by equation 1.1
RLP = RAw
= RAni MC (3.33)
Substituting equation 3.31 into equation-3.33 gives
RLP = 2nAnanb (3.34)
The spectrometer, which has an aperture of 0.1m will have a RLP of
58.5m2 steradians. Both the gain and the RLP has been calculated on the
assumption that the size of the field is only limited by the spherical
aberration of the optics. However, it is in fact the astigmatism which
limits the size of the field. Therefore assuming air gaps are introduced
between the prisms a more realistic value for the maximum field compensated
angle is 31° which corresponds to air gaps of 0.00175", see table 1.
This gives a gain of 120 and a RLP of 2.36m2steradians for an aperture
of 0.05m.
54
casting
Steel girder framework
Haringx Anti-vibration Mounting
fig.18
55
Chapter 4
Anti-Vibration Mountings
4.1 Introduction
The isolation of vibrations from such causes as running machinery
or traffic is of extreme importance in the case of an optical interferometer.
Consequently the function of an anti-vibration mounting used in an optical
experiment must be to:
(i) Reduce the coupling between the instrument and its surroundings.
(ii) Damp out any sudden disturbance.
(iii) Prevent any coupling between the various modes of vibration.
One of the anti-vibration mountings tried in this project was
proposed by Harinyx(15'16) and consists of two equal masses, one of which
was the spectrometer itself, the other an auxiliary mass. The spectrometer
was supported, by weak springs placed at each of the four corners, on a
steel girder framework with the auxiliary mass suspended underneath it
and attached to it by four large springa as shown in fig.18.
4.2 Theory for the Harinyx System(15)
Consider a spectrometer of mass m supported by a spring of
rigidity c, as shown in fig.19a, and suppose that the foundation on which
the spring stands vibrates with an angular frequency w and amplitude a0.
Then in the absence of damping the spectrometer will oscillate with the
same angular frequency w as the foundation but with an amplitude a which
is frequency-dependent. From the differential equation of motion of the
.56
spectrometer it can be shown that
a 1 ao c c mu)2
(4.1)
and that the resonance frequency is
w m
The frequency characteristics are shown in fig.19b and by careful choice
of the parameters c and m the resonant frequency can be made much
lower than the lowest frequency occurring in the interfering vibrations.
a.
fig.19
(Philips Technical Review)
In addition to reducing forced vibrations it is necessary to
limit the effects of sudden disturbances which could set the spectrometer
vibrating. This can be done by introducing extra damping. The ideal
system of damping is a dash pot containing oil carried by a fixed point
• 57
in space with the plunger attached to the spectrometer as shown in fig.20a.
The amplitude ratio of this system is given by
(4.2) [aa011: (w2-1)2i.112T2
where w = , q = -h- and w2 = wo wo o m
with frequency characteristics that are illustrated in fig.20b
a.
b. fig.20
(Philips Technical Review)
As the damping is increased so the resonance peak is reduced and the
duration of the free vibrations is shortened. In practice an arrangement
like this cannot be achieved and is only of theoretical interest.
Harinyx has shown that it is possible to imitate this system of
"absolute damping" by introducing the damping element between the
spectrometer and an auxiliary mass M attached to the spectrometer by a
spring of rigidity C. The amplitude ratio of such a system is
a -2 2 -2
= -4 (w -P) +qw ao [um -(1+p)24r il2Z20-1 ) (4.3)
m m+M and u =
58
where
w = -41) = m+M o
C . m+M p = c m
q - k m+MI
The frequency characteristics are more complicated as shown in fig.21b.
a
M
a.
fig.21
(Philips Technical Review)
The figure has been drawn for the special case u = 0.5 and
p = 0.5 where the points of intersection A and B of all the frequency
characteristics lie at the same height.
59
When the damping factor q equals infinity the spectrometer
and auxiliary mass are rigidly connected and have frequency characteristics
identical to those of the undamped system in fig.19. Conversely for
q = 0 the system has two degrees of freedom in resonance at two different
frequencies as shown in fig.21b. For intermediate values of q the
frequency characteristics always pass through the two points A and B.
At high frequencies, the amplitude ratio is given by
aI 12
c ao - - - 2
Pto mw
and is identical to that for the undampedsystem. Here the auxiliary
mass plays no part when the frequency of the interfering vibrations is
much higher than the resonant frequency and that the damping is effective
in reducing the vibrations caused by sudden impulses or disturbances.
By considering the rate of decay of the free vibrations Harinyx
has shown that the optimum choice of parameters will occur when the -
points of intersection A and B occur at the same height in fig.21b. His
criterion is that the highest peak in the frequency characteristic should
be as low as possible. This leads to the relation
p ,(4.4) optimum
and that the optimum value of the damping parameter is
qoptimum = J1.5p(1-a
As a result the choice of parameters is narrowed down to the selection
of a suitable mass parameter p. If the frequency characteristic is to
be as low as possible a large auxiliary mass should be used. In practice,
60
it is never larger than the main mass, and in general they are made equal.
Comparison of the frequency characteristics of the various systems
discussed here shows that the system with an auxiliary mass is the most
efficient.
So far the analysis has been restricted to a body that has only
one degree of freedom. Any rigid body suspended in space by a set of
springs as described earlier will posses six degrees of freedom: three
of translation and three of rotation. These degrees of freedom are in
practice coupled, that is to say, a translation in one direction will
give rise to translations and rotations in other directions. To prevent
coupling the body, in this case the spectrometer, should have three
mutually perpendicular axes of elasticity passing through a point and
coinciding in direction with three principal axes of inertia. A principal
axis of elasticity is a line along which si applied force will produce a
displacement in the same direction.
The necessary conditions exist if the mounting is symmetrical
with respect to two mutually perpendicular planes, as shown in fig.22,
and if, at the same time, the springs are of equal rigidity in directions
perpendicular to these planes
X
fig.22
•6].
The height of the centre of elasticity can be calculated from the rigidity
of the springs. If an auxiliary mass is attached, again the principal
axis of elasticity of the spring attachments must coincide in direction
with the principal axis of inertia of this mass. Care must be taken to
ensure that the introduction of damping does not give rise to coupling.
4.3 Design Parameters for the Harinyx System
It is desirable that the mounting should react the same way in the
vertical as in the horizontal direction. This can be achieved by using
helical springs, whose axial and lateral rigidities are equal, as the
resilient elements. Under this constraint Harinyx(16) has shown how the
relative spring compression, E, and the slenderness ratio, 10/D, can be
represented graphically, fig.2,..3 and used to find the spring parameters.
1.0
0.8
0.6
0.4
0.2
0 2,o D
1.0 1.2 1.4 1.6 1.8 2.0
fig.23
(Philips Technical Review)
to is the free length of the spring and D the diameter.
62
The relative spring compression is related to the resonant
frequency wo of the system by the equation
wo = g 1
Eto (4.5)
and it follows from the graph that wo can be defined by the diameter
of the springs.
Before deriving the parameters of the helical springs it is
necessary to deduce the relationship between the spring rigidities of
the primary and secondary springs. From equation 4.3 the rigidity
parameter p was defined as
p
and the mass parameter u as
C m+14- = c M (4.6)
m m+M (4.7)
but ,from equation 4.4
pop = u , (4.8)
substituting equations4.7 and 4.8 into 4.6 gives
C m u2
= — . c M
but m = M as explained previously
2 u = —
and from equation 4.7 u =
C =
(4.9)
63
This is the relationship between the spring rigidities.
4.4 Spring Parameters for the Harinyx System
Let the diameter of the secondary springs, that support the
auxiliary mass, be 8", the compression ratio of the springs to be 0.6.
From Harinyx's graph
-2. = 1.77 when E = 0.6
The free fen,-;'.h of the spring is
to = 1.77 x 8 = 14.16"
and the compression. is
= 0.6 x 14.16 = 8.496"
• Consequently the resonant frequency is
Wa = 1.073 hertz
The auxiliary mass M is 480Kg
. 480
' the mass carried by each spring is - 120Kg •
= 264. lbs
64
Thus the rigidity C of these springs is
C 264 8.496
= 31.1 lb/in
The ridigity of the primary springs c is given by equation 4.9
c = 4C = 124.4 lb/in
Each spring carries 528 lb.
• the compression is • • 528
= 4.24" 124.4
Again if the compression ratio of the primary springs is 0.6, the
. free length to is 24 - 7.08" and the compressed length is 0.6
Pt -
=' Tog-4.24 = 2.84"
Ito Since E and — are related, the diameter of the primary springs is
D 7.08 1.77
- 4"
The resonant frequency is given by
981 too
(2.54) (2.84)
= 9.54 rads-1
= 1.857 hertz
65
Thus 1.857 Hz is the dominant frequency of the system, which is
acceptably low.
The centre of elasticity
The height of the centre of elasticity(16)
, which is of
fundamental importance, is given by the equation
h = 2(1-is2)
wh..re £2 is a correction factor depending upon the relative axial
compression g and can be found from the graph fig.24,Since -°= 1'0
E _- 2.5
fig.24
(Philips Technical Review)
Thus the position of the centre of elasticity for the secondary springs is
h = 2(1-1.25)
= -0.252
.66
where h is 1.416" below the bottom of the springs. For the primary
springs h is -0.71".
The table below is a summary of the parameters of the helical
springs.
Free length Compressed Rigidity Centre of Dia. Resonant length lb/in Elasticity Frequency
Secondary 14.16" 5.664" 31.1 1.416" 8" 1.073 Hz
Primary 7.08" 2.84" 124.4 0.71" 4" 1.857 Hz
Table 4
An anti-vibration mounting was constructed according to the theory given
here.
To verify the resonant frequency of the primary springs the
spectrometer and auxiliary mass were clamped together and observations
of the oscillations of the apparatus against time were taken. By applying
the principle of least squares the resonant frequency was found to be
1.754 t 0.038 Aertz
This' is in good agreement with the theoretical value.
4.5 Damping for the Harinyx System
The damping mechanism consists of four square cups fixed to the
auxiliary mass in which four others, fixed to the base of the spectrometer,
can move freely in all directions. The cups are filled with oil of a
given viscosity to a height H = 1.45B where B is the side of the basal
plane of the inner cup. Harinyx(15) has shown that when the baths are
filled to this height the damping coefficient is the same in the
67
horizontal and vertical directions. The damping coefficient can be
varied by changing the viscosity of the damping liquid but this is not
critical as far as optimum damping is concerned. This is advantageous
because the viscosity of the oil and hence its damping ability depends
upon temperature, so that optimum damping would ont be achieved at
one temperature. This damping mechanism was found to satisfactorily
reduce free vibrations caused by a sudden disturbance.
4.6 Conclusion
Initially a simple anti-vibration mounting employing the principle
of the Julius suspension mounting, described in the next section, was
tried. Vibrations in the horizontal plane are isolated by this system
but those in the vertical plane are not. Although this is satisfactory
in the laboratory it may not prove so outside, for example, when used
at the Coudd focus of a telescope. In an attempt to improve on this
design the more complicated Harinyx mounting was constructed and tried.
However, this system has one serious disadvantage which is the
tilting of the main mass when a heavy body, in this instance the carriage
that supports the two large prisms, moves across it. The only feasible
method of overcoming this problem would be to have an equivalent mass
moving in the opposite direction, the motion controlled by a servo loop.
Due to the awkward shape of the casting and the weight of the equivalent
mass, the introduction of a servo-control system would add unnecessary
complications. Consequently under these circumstances it was considered
simpler to return to the pendulum suspension system.
Suspension System
fig.25
.6 8
.69
4.7 The Suspension System(17)
The mounting is the same in principle as the Julius suspension
system, but on a much larger scale.. The spectrometer is hung by steel
wire with the points of attachment in the same plane as the centre of
gravity of the spectrometer in order that any translational vibrations
which are transmitted by the pendulum support are not converted into
rotational vibrations that are more serious. The mounting behaves in
the same way as a pendulum and is illustrated in fig.25. Vibrations
in the vertical direction are not isolated by this arrangement but
vibrations in the horizontal plane are, provided that the period of the
pendulum system is greater than that of the external disturbances. In
this arrangement the steel wire is one metre long, which gives a pendulum
period of 2 sec. It was found unnecessary to introduce damping, although
it would have been relatively simple V., do.
70
Chapter 5
Mechanical Design
5.1 Introduction
Thel main structural member of the spectrometer is a casting
made from pig iron. Unfortunately meehanite, a more stable grade of
cart iron, was unobtainable. A casting was given preference over a
prefabricated assembly for several reasons. Greater rigidity is more
readily obtainable in a casting than in prefabricated structures by
adding flanges or other stiffening pieced. By suitably designing the
casting time can be saved in the machinery operations, e.g. the surface
of the area to be machined can be raised above the surface of the
remainder of the work. Generally the machined surfaces will only
constitute a small fraction of the whole surface. In addition iron
machines better than other metals.
A further advantage which is sometimes overlooked is the difference
in the elastic properties of cast iron and steel. The internal damping
of vibrations in cast iron is much greater than in mild steel and, in
• the case of an optical instrument, it is an advantage to have a material
that will not readily transmit vibrations.
Nevertheless there are disadvantages in employing a_casting.
The obvious one is the tendency of'a casting to 'creep' or change shape
over a period of months after it has been made. This is overcome by
weathering, a process that involves leaving the casting in the open air
for up to eighteen months. Alternatively, the casting can be subjected
to a process of heat treatment which is very expensive.
• • •
View of the upper arm of the spectrometer showing the mirror,compensating
glass block,prisms and the common prism carriage
View of the lower arm of the spectrometer showing the compensating glass block,prisms
and the lower half of the common prism carriage
fig. 26
Rotating plate
73
The only prefabricated assembly of the spectrometer is the
common carriage supporting the two large prisms. It is made from
aluminium to save weight (see photograph).
5.2 Mounting of the Optical Components
The optical mountings, which incorporate the principles of
kinematic design, are made from mild steel because weight is not an
important factor. Adjustments for tilt and position are made using
screws with a Model Engineer's thread. The small prisms, whici, are made
from circular glass blocks, have rotating as well as tilt and position
adjustments. Without this it is impossible to align the inclined face
of the small prism with the equivalent face of the large prism.
To design the mountings for the two large prisms it is necessary
to take into consideration the gap tolerances derived in chapter 3.
Each mounting has adjustment for height and rotation. In addition they
have one degree of translational freedom.
A mounting consists of three 1H mild steel plates cut in the shape
of the prism, as shown in fig.26.
Brass pivot
• 74
The bottom plate has three adjustable jack screws that locate in vee
grooves on the common carriage. They provide the height adjustment.
The next element of the mounting is the rotating plate which is
located in the centre of the bottom plate by a brass pivot. The plate
rests on three ball bearings, see fig.27.
P.V.C. collar
Rotating plate
Base plate
fig. 27
A steel pillar is fixed'to one edge of the plate about fifteen
centimetres from the centre against which a micrometer, with a steel ball
bearing mounted on the end, is brought to bear. At the other end a spring—
loaded plunger is fixed to the bottom plate and brought to bear on the
edge of the rotating plate, as shown in fig.28.
Spring plunger
75
Steel pillar
fig. 28
A 5 arcsec rotation is achieved for a 0.0015-inch translation of the
micrometer. Greater accuracy has been obtained by attaching a 10 to 1
gear ratio component to the micrometer.
The top plate is part of the kinematic slide which provides the
one degree of translational freedom. This last adjustment is supplied
to give control over the width of the gap between the small and large
prism. However, the weight of the large prism and the space available
for the whole mounting combine to make it difficult to design a smooth
and accurate slide. The slide is illustrated in fig.29. The rotating
plate acts as the base of the slide and the milling of the vee grooves
is carried out in one operation.
Vee
Rotating plate
Spring plunger
76
Spring plunger and steel pillar attached to rotating plate
not shown
fig. 29
The translational motion is controlled by a screw dnd spring plunger
arrangement, shown in fig.29. The screw thread is a in M.E. with 40 T.P.I.
The sliding surfaces are lubricated with a carbon-impregnated grease to
combat the 'stick slip' action of the mounting caused by the excessive
weight of the prisms.
5.3 Airbearings
A common translation of the large prisms, in a direction parallel
to their common faces with the small prisms, introduces the path difference.
The ratio of the carriage translation to path difference is 5:1. Since
the motion of the carriage must be free of 'stick slip' and accuracy
of the order of '/20 20 is required for the carriage servo-control loop,
the choice of bearing becomes extremely important. A is the wavelength
of a helium-neon laser.
77
The choice of bearings lies between air bearings, oil bearings
and ball bearings. Air bearings were chosen to support and control
the motion of the carriage for the following reasons:
(i) Zero starting friction and only a small running friction.
(ii) Absence of wear since the bearing surfaces are not in contact.
(iii) Negligible heat generation.
(iv) Film thicknesses of the order of 12-371pm resulting in an
averaging effect on the motion, so that machinery errors in the components
of the bearings become less significant.
They were given preference over other types of bearings for the
following reasons:
. (i) Freedom from vibration compared with ball bearings.
(ii) Zero starting and negligible running friction compared with
oil bearings that have inherent damping which varies with temperature.
Against this the disadvantages can be summarised as follows:
(i) The likelihood of air flowing around in the arms of the
spectrometer causing a random variation of path difference. This cam
be avoided'if the air flow is laminar and kept to a minimum.
(ii) Certain designs of bearings can lead to air hammer instability.
(iii) The necessity of having a clean and oil-free air supply
which is accomplished by introducing filters into the air line.
Most of the work on airbearing in this country has been carried
out by H.L. Wansch and his colleagues(18-24) at the National Engineering
Laboratory. They have formulated a design theory for a system of flat
airbearing on the basis of their experimental observations. It is
necessary to summarise this theory which is the basis of a computer
program, appendix B, used to derive the basic design parameters for the
• 78
•
air bearing system.
5.4 Design Theory of Airbearing
The steady state solution for flat-pad air bearings is based
on equating the mass flow through the restrictor and that through the
bearing clearance. The flow through an orifice type restrictor is
given by(25)
[ M = CHAP 2y (y.-1)RT • 2/Y Y
(5.1)
while for a circular pad the flow through the bearing gap is
M - 3 2 -P a
2 ffh (p )
12uRTItn= ri
where the symbols are defined as
Effective orifice area
CD Orifice coefficient coefficient of discharge
h Air gap
M .Air flow mass
P• Air supply pressure
p Pocket pressure
Pa Atmospheric pressure
R Universal gas constant
Y Ratio of specific heats
u Viscosity of air
r1 Pocket radius for pocket bearings
• 79
r2 Radius of pad, i.e. distance of orifice to edge of pad
Number of unit bearing pads joined in a slideway
Combining these equations gives the air film thickness
r 12uRTtn yiCDAP
n(132 -Pa2 )
h3 2/ Y+1
2Y [f] [1)P-1 (Y-1)RT •
(5.3)
On the basis of experimental results the load bearing relationship
is given by
W = 1.3nK1(p-Pa)1.1
a0.8
(5.4)
where K is a constant depending on the presence or absence of an
orifice pocket. The value of KLwhich gives the best agreement with
experimental results was found to be
K1 = 0.206 for orifice compensated bearings
with 1" diameter pocket.
Rearranging equation 5.4
0.91
1.3nK 0.8
J a 1
p
+ pa (5.5)
and simplifying equation 5.3
r2
h3 C1knr1
AP
2 (1) -Pa
2 )
2/
[PP I
■■•
where 1 2
12uRTC C1
2y (y-1) RT
0.91 -
1.3n:la!"
+Pa 1.71- i0.91 -
P
r2 C 1 rl
[ w 1.41
1.3nKia0.8 +Pa P
80
Also assuming adiabatic flow through the orifice, y = 1.41.
Substituting equation 5.5 into equation 5.10 gives
W
[1.3nK1a0.8
1.82 + 2pa
0.91
.8 1.3nK alp
(5.7)
The air flow equation is found by substituting equation 5.5 into equation
5.1.
W
0.91 -
[ 0.8 • +pa 1.3nK a 1
P
10.91 -L.71
1.3nK1alln +Pa
P M =
1.41
5.8
where C2 = CD
The constants C1 and C2 are obtained from experimental data. For
pocketed bearings they are
C1 = 1.41 x 10 -4
C2 = 44
In equation 5.7 r1 = 1/16 for pocketed bearings and for square
shaped bearings r2 is taken as the shortest distance between the
orifice and the bearing edge, i.e. r2 = . In using the design
equations for a single-pad bearing, where the air flow is allowed to
2y
Bearing surface
Horizontal Bearings
fig.30
Air feed pipes
1- 1 Distribution
Boxes
Bearing attached to carriage wall
.81
82
exhaust at the four edges of the bearing, experimental results indicate
that n, the number of unit pad bearings joined in a slideway, becomes
1 and the 1.3 value associated with n in equations 5.7 and 5.8 also
becomes 1.
The stiffness of the bearings is found by computing h, the air
gap, for two load values W+614 and W-6W, and is given by
Stiffness - h -h W-614 W+614
Equations 5.7, 5.8 and 5.9 are used in the computer program to find the
orifice area, air flow and stiffness of the bearing.
5.5 Design Parameters for Airbearings
The bcarings supporting the carriage consist of three square pads
made from stainless steel. Each pad has a single central inlet orifice f,
and an orifice pocket of I" diameter by 0.005" deep. The area of each
pad is 3 square inches. To ensure that they all lie in the same plane
the pads were ground after they were mounted on the carriage.
The bearings controlling' motion in the longitudinal direction
consist of four square pads similar in design to those supporting the
carriage. Each pad is 1 square inch in area. To take into account any
fluctuations in pressure these bearings are set up as illustrated in
fig.30. Adjustments for tilt and position can be made to ensure that the
bearing pads are in the same plane as the bearing surface. Also the gap
between the bearing surfaces can be varied.
26W 5.9
The design parameters for the bearings supporting the carriage
are shown in Table 4. These parameters are calculated on the basis that
.83
each bearing pad carries a load of 30 lb, since the total weight of the
carriage and prisms is 90 lb.
Supply Pressure Gap Thickness Orifice Area Air Flow Stiffness
lb/in2 inch in 2 ft3/min tons/in
100 0.0008 0.0009 0.84 1314
100 0.0009 0.0012 1.19 985
100 0.0010 0.0017 1.64 761
100 0.0011 0.0022 2.18 602
Table 4
It is essential to keep the air flow to a minimum.' This is done with
a small orifice area, although in practice it is difficult to drill
small holes in stainless steel because it is a very hard material.
The smallest diameter obtainable was 0.0014". It is not necessary to
operate the system at a supply pressure of 100 lb/in. Experimental
observations show that the carriage will float provided the supply
pressure is above 75 lb/in2. Equation 5.8 indicates that a lower supply
pressure iS an advantage because the air flow is reduced.
The bearings controlling motion in the longitudinal direction
need only provide a force of a few pounds since any force acting in the
horizontal plane will be small. Experimental observations indicate that
supply pressure in the range 10 to 25 lb/in2 is adequate. Alignment
of these bearings with the bearing surface is critical if linear motion
is to be achieved. Misalignment will result in oblique gaps appearing
between the prisms, leading to a reduction in the size of the field if
the misalignment is greater than 5 microns.
84
All the bearing surfaces on the casting are ground flat to an
accuracy of 0.0002". The air to the system is fed through plastic tubes.
These are hung from the ceiling to minimise interference with the
.motion of the carriage.
5.6 The Resistance to Flow in the Plastic Feed Pipes
When air flows along a straight length of pipe of constant diameter
it loses energy owing to the effects of viscosity. If the system includes
bends and valves there will be further energy losses. To maintain flow
against these energy losses there is a pressure difference between the
two ends of the pipe. Before deciding on the length and internal diameter
of the feed pipes this pressure difference was calculated for plastic
tubing with a k" internal diameter.
It is assumed that the effects of viscosity are equivalent to the
frictional force at the.pipe walls. Let% 0 be the value of this-force
per unit area of the internal surface of the feed pipe, whose diameter
is d, and consider the forces acting on the cylinder of air within the
length 2, of the pipe. The air flow experiences a retarding force
1:01rdft , and to maintain the flow there must be an equal and opposite
force due to the difference of'pressure between the ends of the. pipe.
ird2 This force is (perpB 4 )— where p
A is the pressure at the inlet to the
pipe, and pB is the pressure at the outlet. Equating it to the
frictional force gives
d 0 = (PA –P8'42, (5.10)
This pressure difference is a measure of the resistance to flow. Ower
and Pankhurst(26)
point out that it is customary to express the resistance
85
in terms of a non-dimensional coefficient Y which is equal to
.t o/iev2 where iev2 is the kinematic energy of air. e is the mass
density of air and v is the mean velocity of flow. Thus, rewriting
equation 5.10 gives
PA-PB d ev2 2t (5.11)
If y is known, it is possible to calculate the pressure drop.
The resistance coefficient, y, can be found by using the equation
y = 4 log 3.7.-
(5.12)
given by Colebrook and White(27). Here d is the diameter of the pipe
and e is the average height of the roughness on the internal surface
of the pipe. For plastic tubing Colebrook and White give c = 0.00006".
For plastic tubing with a 1" internal diameter the resistance-
coefficient is
4 log. (3.7)(0.25)
•= (0.00006)
= 6.72.10-4
Rearranging equation 5.11 to find the pressure difference gives
Ap = 2yev
2t
86
The compressor supplies air at the rate of 0.35 cubic feet per minute.
The mean velocity of flow is given by
V
!rr2
where V is the volume of air flowing in unit time and 7r2 is the
cross-sectional area of the pipe.
0.35.96.96. 60n
= 17.12 ft/sec
The pressure difference is given by
Ap = 2.672.10-4.(17.12)
2(0.08).(48)
where the mass density of air, e, is 0.08 lb/ft3
A = 1.51 lb/ft2 per .unit length
Clearly this indicates that the pressure losses due to the flow
resistance may be neglected. As a result of this analysis the plastic
tubing is used for the feed pipes.
,87
Electrical Design
5.7 Control of Path Difference
To determine the exact path difference continuously it is
necessary to use a reference fringe system generated by a beam passing
through the interferometer in parallel with the radiation to be analysed.
A Helium-Neon laser produces an ideal reference beam. The carriage can
be displaced continuously or in a stepping motion when it moves from one
sampling point to the next.
The stepping method offers several advantages. The equi-distant
spacing of the sampling points can be achieved with a high degree of
accuracy, represented by a variation of not more than about 1°A over
the whole path difference. Another advantage is that with sources of
variable intensity the duration of eaca measurement can be controlled by
the flux.
The simplest method of producing motion is with a direct current
linear motor that has only one moving part. It consists of a permanent
magnetic field, B, that interacts with a tangential current I flowing in
a coil of N turns which is connected to the carriage. This interaction
produces an axial force given by
F = IBLN (5.13)
where NL is the length of the wire in the magnetic field. By changing
the magnitude and direction of the current the force and direction of
motion change.
Attached to carriage
Linear Motor
fig.31
Air gaps B Ceramic magnets
.4,_Steel bar
.89
5.8 Linear Motor Design
The coil of the linear motor is wound round a hollow aluminium
former which has a square cross-section. The former is joined to the
common prism carriage. Above and below the coil is a row of ceramic
bar magnets. To complete the magnetic circuit a square steel bar passes
through the former as illustrated in fig.31.
Ceramic permanent magnetic material, rather than alnico, is used
because it has a higher magnetic coercive force. Further, the B-H
demagnetization curve is nearly linear; consequently the magnetized
pieces can be taken apart and reassembled without requiring remagnetization.
To provide the same force as tha present coil, a smaller coil can
be used if the magnetic field interacts with'all fOur sides of the coil.
This has the advantage of reducing the self inductance of the coil which
is important because the carriage is displaced in a stepping motion from
one sampling point to the next by applying a series of regularly recurring
pulses with a period of milliseconds. Such an improvement can be achieved
by using a coil of circular cross-section with circular ceramic magnets
to provide the radial magnetic field. Unfortunately it was not found
possible to find a manufacturer who could make the magnets to the
dimensions and tolerances required.
The linear motor has the following design parameters:
Length 0.1 metre
Cross-Section Area 1.56.10-4metre2
No. of turns 80
The remanent induction of the ceramic magnet. 0.22 weber/metre2
Resistance of gauge 18 manganin wire. 2.4 ohms
Self Inductatite of the coil 25mH
• 90
The self inductance of the square coil is calculated by using equation
1.8 in appendix C.
The motor produces a maximum force of four newtons. This force
has been derived using the assumption that the magnetic flux density
in the air gap in which the coil moves is the same as the remanent
induction of the ceramic magnets. This is true if the air gap is less
than 0.05". Due to design problems the gap is set at 0.12".
5.9 To find the flux density in the air gap
Assuming that there is no leakage
BAa
= Bm m A (5.18)
where Aa,Am are the cross-sections of the air gap and the magnet
respectively, and Bm is the magnetic induction in the magnet. The total
magnetomotive force in the magnetic circuit is zero, hence
Hda+Hmdm =
•
(5.19)
where da,dm are the path lengths in the air gap and the magnet respectively
and Hm is the magnetic field in the magnet. Combining equations 5.18 and
5.19 gives
BHAada = -Bm Hm Am dm (5.20)
In a permanent magnet(28) the field inside is the,' demagnetization field'
due to the 'free magnetic poles' near to the ends of the magnet. The
negative sign arises since this field is in the opposite direction to Bm.
M 91
Reducing equation 5.20 gives
B2 = uBHd om mm da
(5.21)
where Am
= AA
Here uo = 4n.10henry/metre
Bm = 0.115 weber/metre
2
Hm = 8.9.105 ampere/metre
dm = 0.635.10
-2metre
da = 0.031.10
-2metre
B = 0.1625 weber/metre2
which is the magnetic flux density in the air gap.
Consequently the maximum force produced by the linear motor is
3.25 newtons.
5.10 Damping
To provide stability to the servo-control loop, active or passive
damping must be introduced. The simplest method to implement is passive
damping, which is used here. It consists of damping baths, equal in
length to the total carriage displacement, filled with a viscous liquid,
in this instance silicone oil. Vanes which form an integral part of
the carriage run between vanes in the damping baths, the latter extending
along the length of the damping baths.
92
The frictional force between the vanes of the carriage and the
silicone oil depends upon the velocity of the carriage and is equal to
the force necessary to shear the oil. It is given by
p = by (5..22)
where b is a constant. According to Newton's law of viscous flow for
streamline motion the force necessary to shear the silicone oil is given
by
dv p =
dx (5.23)
dx i where is the velocity gradient set up in the liquid between the dx
moving and stationary vanes. A is the area of the damping vanes over
which this force acts and n is the coefficient of internal friction,
more commonly known as the coefficient of viscosity.
Equating equations 5.22 and 5.23 gives
dv by = nk-- dx (5.24)
By assuming that the velocity of the silicone oil in contact with
the stationary vanes is zero, equation 5.24 reduces to
by = riAv dx
(5.25)
where v is the velocity of the silicone oil in contact with the moving
vane. Equation 5.25 reduces to
b riA dx
(5.26)
• 93
Consequently, the total resistance to motion is
pA
= ritldx (5.27)
Equation 5.27 is the damping term in the second order differential
equation for the servo-control loop.
The total surface area of the carriage vanes is 0.3 metre2 and
the gap between the moving and stationary vane is 0.038.10-2metre
2.
94
CHAPTER 6
CONCLUSION
6.1 Adjustment of the Spectrometer
The alignment procedure involves equalising the path lengths of
both arms in air and glass. To begin with the path lengths in air are
made equal by setting up the beamsplitter, mirror and compensating glass
blocks in their approximate positions, as shown in fig.32, and observing
the images of the source caused by reflections from the surfaces of the
compensating glass blocks. The source is a mercury discharge lamp with
a green filter.
Mirror Upper Arm
Compensating glass block
Source
V
Observer
Beam splitter Lower Arm
ompensating glass block
fig. 32
The images are brought into coincidence by adjusting the screws of the
compensating glass block mountings. At this point a ground glass screen
is placed in front of the mercury discharge lamp and in the focal plane
.95
of the input lens to provide an extended source of light. Fringes should
be seen crossing the field of view. Further adjustment will result in
two circular fringe patterns. One set of fringes is caused by reflection
from the front surfaces of the compensating glass blocks and the other is
due to reflections from the rear surfaces. To ascertain which surfaces
are responsible for the fringe patterns methyl alcohol is poured over the
rear surface of one of the compensating blocks and the effect on the
fringe patterns observed. One set of fringes are distorted as the methyl
alcohol flows down the surface of the compensating block and it is these
that are formed by reflections from the rear surfaces.
To equalise the path lengths in air only the fringes caused by
reflections from the front surfaces of the compensating blocks need be
observed. The mirror and beamsplitter positions are adjusted until three
or four fringes fill the field of view at which point the path difference
in air is within a few dozen wavelengths of zero. Finally the mercury
discharge lamp is replaced with a white light source. A movement of not
more than one tenth of a millimetre in the position of the mirror should
be sufficient to produce white light fringes indicating that the path
difference is only a few wavelengths.
Next mount one of the small prisms and isolate that arm of the
spectrometer by inserting a sheet of black paper into the other arm as
illustrated in fig. 33. Several images of the source, which is the
mercury discharge lamp, may be seen. These images are brought into
coincidence by adjusting the screws of the small prism mounting.
I .96
Upper Arm
Black paper Lower Arm
Beam splitter Small prism
Source
Compensating glass block
Observer
fig. 33
A ground glass screen is inserted in front of the mercury source to
give an extended source. Fringes should be seen across the whole field
of view and are formed in the gap between the compensating glass block
and the small prism. It has been assumed that the plane faces of the
compensating glass block are parallel.-: The position of the prism is
adjusted until four or five circular fringes fill the field of view.
At this point the plane face of the prism is parallel to the faces of
the compensating glass block. A similar alignment procedure is adopted
for the prism in the other arm.
The final stage of the alignment procedure involves equalising the
path lengths in glass. This is done by setting the common prism carriage
in the position that corresponds approximately to zero path difference.
Both large prisms are set up on the carriage with their hypotenuse faces
parallel to the carriage edge. The carriage moves at an angle of 20° to
r- -
Direction of
motion
S .97
the faces of the compensating glass blocks as shown in fig.34.
Carr'age
•r-- Casting
fig.34
Consequently the hypotenuse face of the prism whose angle is
approximately 18° will not be parallel to the equivalent face of the
small prism in that arm. Therefore the position of the small prism
must be adjusted until the hypotenuse faces of both prisms are parallel.
There is no optical method of checking that they are parallel; as a result
it may be necessary to readjust the.position of the small prism at a
later stage.
It follows that having changed the position of the small prism
the compensating block must be re-aligned as described previously.
Similarly the images of the front surfaces of the compensating blocks are
superimposed again by adjusting the angle of the mirror in the upper arm
(see fig.32). Having carried out this re-aligned procedure two bright
images of the source will be seen. Minor adjustments to the prism whose
S
Direction of motion -cc
Large prism
Small prism
98
angle is 18° will bring both images into coincidence. By replacing the
ground glass screen fringes should be seen across the field of view.
Further adjustments will give circular fringes at infinity.
Field compensated fringes occur when the virtual images of the
reflectors are approximately superimposed, as explained in chapter 1,
and are most easily found at zero path difference. Replace the mercury
discharge lamp and filter by a white light source such as the ordinary
mains voltage lamp. Use the translational control on one of the large
prism mountings to move that prism at right angles to its reflecting
surface until white light fringes appear. The path difference for the
spectrometer is then within a few wavelengths of zero, fig.35. To
verify that the system is field compensating move the carriage and observe
how much the fringes contract, figs. 36 and 37.
As explained in chapter 3, the degree of field compensation will
vary with the size of the air gaps between the prisms in each arm.
Consequently it may be necessary to vary the size of the air gaps until
the maximum field compensation is achieved.
Compensating glass block
fig.35
99 •
Fringes near zero path difference
fig.36a
Fringes at 5.0 cm path difference
fig.36b
Source : Mercury discharge lamp
.100
Fringes near zero path difference
fig.37a
Fringes at 5.0 cm path difference
fig.37b
Source : Helium- Neon laser
0 101
6.2 Experimental Verification
An ordinary mercury discharge lamp is employed as the light source.
The 546nm line is isolated by using a filter with a bandwidth of 13nm
Fringes at approximately zero path difference and at maximum path
difference are shown in figures 36a and 36b respeC#vely. At an optical
path difference of 5 cm the time delay of the spectrometer is much
greater than the coherence time of the discharge lamp - consequently the
visibility of the fringes is very poor. This is clearly illustrated in
figure 36b.
To obtain fringes of good visibility at a resolution of 105 it is
necessary to find a source with a narrow line width. A helium-neon laser
is such a source. This has a bandwidth of approximately 0.018 giving it
a coherence time that is very much greater than the time delay of the
spectrometer. Fringes at approXimately zero path difference and maximum
path difference are shown in figures 37a and 37b.
In both cases the size of the fringes is consistent with the values
previously derived for the system with air gaps of-,0.002" between the
prisms. In addition the elliptical shape and size of the fringes
confirms that the system is astigmatic and that it is this aberration
which governs the size of the fringes.
In order to verify that the spectrometer is achromatic in the
visible region of the spectrum it is necessary to measure the diameter
of the central fringe at different wavelengths. Line spectrum sources
at 486nm, 546nm, 589nm and 638nm were chosen. The spectrometer was set
up with an optical path difference of 1 cm and the diameter of the
central fringe was measured with the aid of a telescope. Only a small
variation in fringe size was detected over the spectral range 486nm to
638nm confirming that the system is nominally achromatic between the
102
design wavelengths.
6.3 Application to Raman Spectroscopy
One of the possible uses of this instrument is in the field of
Raman scattering. Unfortunately a lack of time ha6 prevented completion
of the project. Asa result only the theory behind this application can
be discussed here.
The design of an instrument capable of resolving the various
components of Raman scattering is governed by the extremely feeble effect
of such scattering. For a liquid sample with light from the visible part
of the spectrum incident upon it, the intensity of the molecular sratter'ng
is 105
of the incident intensity. It must be emphasized that this is
the total molecular scattering (including Rayleigh scattering) and of
this perhaps only 1% may contribute to the, Raman spectrum. For gases
where the density is lower, the intensity is correspondingly,lower.
If a spectrograph is employed to detect the weak Raman lines fast
photographic emulsions will be necessary(33)
. These tend to be heavily
fogged and consequently exhibit poor contrast which is fatal if the
presence of the anti-Stokes line is to be detected. Any quantitative
interpretation on the bases of photographic density measurements of
spectral lines often proves as hazy as the lines measured. The graininess
of the emulsion will often limit the resolving power to a value below the
intrinsic resolving power of the spectrograph.
Some commercial spectrometers developed primarily for Raman
spectroscopy such ss the Cary Model 81 Raman spectrophotometer(34)
and
the Spex 1400-11 Double spectrometer employ diffraction grating as the
dispersive element(35). The important feature common to both these
instruments is the use of two plane diffraction gratings in their optical
• 103
system which allows the luminosity to be increased by a factor of two
for a given resolution. One of the disadvantages of using a diffraction
grating arises from the scattered light in'the system.
This scattered light arises from the grating surfaces(33)
Superimposed on a diffracted light beam at any particular angle is
background resulting from spurious reflections caused by an imperfect
grating. Modern gratings are ruled in evaporated aluminium and replicated
in plastic. Both have surfaces which exhibit irregularities that cause
light to be scattered and reflected at random. Further imperfections
to the grating surface will be introduced by any surface faults on the
diamond that produced the original grating and also by any chatter from
the diamond as it rules the grating. Each of these imperfections will
result in the scattering of light. This stray light forms a continuous
background that may mask some or all of the Raman lines depending upon
their intensity.
Spectrographs are more efficient than spectrometers if
consideration is given to the energy emitted by a source since all this
energy is received simultaneously by a spectrograph whereas in a
spectrometer the elements of the spectrum are recorded sequentially.
There is a gain in signal to noise ratio of J-N-I when N spectral elements
are measured simultaneously instead of sequentially06)
This gain in signal to noise is realizable, when the measurements
of the N spectral elements are independent of each other so that the
signal to noise ratio of any given element is not affected by any of the
surrounding elements. This condition is fulfilled if separate detectors
are used or different areas of the same detector are used as in the
spectrograph. Multiple detectors will only become feasible if the
number of elements is small.
104
To overcome the problem of thousands of elements, a technique
that has been used in telecommunications is adopted. It is the
principle of multiplexing which allows a number of communication channels
to be carried by a single line. To differentiate between the separate
signals it is only necessary to modulate each radiation in a sinusoidal
mode at a different frequency before it reaches the detector. The output
signal is thus composed of N superimposed sinusoidal signals, each
having an amplitude proportional to the flux representing the intensity
of its own spectral element. The spectrum of the source is derived by
carrying out a Fourier analysis of the output signal.
A Michelson spectrometer is suitable for modulating the incident
spectral radiation. The resolution-luminosity produtt of the Michelson
spectrometer is much greater than for either the prism or grating
instrument of the same aperture (see chapter 1) and it has this multiplex
advantage as well(36). It must be emphasized that the multiplex advantage
only applies in the infrared region of the spectrum and that the detectors
are limited by photon noise in the visible.
Consider monochromatic radiation of wavenumber o to be received
by a Michelson spectrameter(37). If the incident flux is B(a) the emergent
flux is
B(6) = B(a)cos2(wad) = IB(a)(1+cos(271-a6))
(6.1)
where 6 is the path difference. If the path difference is varied
linearly with time at a rate v, the flux becomes
B(6) = 1B(a)(1+cos(2wolit)) (6.2)
105
The output flux is now the sum of a constant term and of a term varying
sinusoidally in time at a frequency v = av proportional to the frequency
of the incident radiation.
If the incident radiation consists of a number of different
wavenumbers, in the interval a to a + da the intensity is given by
B(a)da, the total emergent flux is
B(S) = B(c)(1+cos(2Trovt))dc (6.3)
The variable part of the expression is simply the cosine Fourier transform
of the function B(a). The inverse of the Fourier integral, ignoring the
numerical factor of is given by
B(a) = 2r B(o)cOs(27ta6)d6 —m
= 4 r B(6)cos(27rac9d6 (6.4)
Thus the correct spectrum for positive frequencies is obtained through
a cosine transform of the.interferogram., equation
To make an accurate calculation of the function B(S) it is
necessary to evaluate it for all path differences between 0 and =. In
practice cS can only be increased to some maximum value this this will
mean that, instead of having the function B(S), only its product with
the rectangular function of width 26m and unit height can be known.
If the spectrum is calculated, the Fourier transform of this product
is obtained instead of the required result B(a). Now, the Fourier
transform of the product of two functions is equal to the convolution
of their transforms. Hence instead of B(a) the 'smeared spectrum' is
found.
106
B' (o)= B(o) * A(o) (6.5)
where A(o) is the Fourier transform of the rectangular function 6 f/ 2Sm
and is given by
A(o) = SINC [211-od] (6.6)
The situation is analogous to the spectrum found using conventional
spectroscopic instruments. A disadvantages of the Michelson spectrometer
is the undesirable shape of the instrument profile due to the presence
of comparatively large secondary maxima, see fig.38 that would mask
both the Stokes and anti-Stokes lines of the Raman spectrum. The height
of these secondary ma=xima can be reduced by a mathematical technique
called rpodzation.
fig.38
A suitable function H(S) is chosen to replace the function
rect.(1-0 as the multiplying factor of the interferogram between the
limits -6m and +6m. This is equivalent to choosing the ideal
• 107
instrument profile. Apodization can be carried out physically by
attenuating the signal from the detector to an extent that is variable
with path difference or it maybe performed mathematically by multiplying
. the ordinates of the interferogram by suitable factors and employing a
computer to evaluate the spectrum. The result of apodization is to rz
S reduce the resolving power.
Since certain regions of the spectrum are of interest to the Raman
spectroscopist it would be useful to isolate these regions. By convolving
the interferogram with the Fourier transform of an appropriate filter
function it is possible to obtain this isolation. The spectrum obtained
is the product of the spectral distribution and the filter function in
the particular region of interest. This operation is carried out before
apodization.
To sum up the Michelson spectrometerlas the advantage of a
higher resolution-luminosity product than either the prism or grating
spectrometer. The noise level is set by he photon noise of the detector
rather than scattered light as in the case of a diffraction grating.
In addition the field compensated Michelson spectrometer has the
advantage of an even greater resolution luminosity product without loss
in resolving power. These properties should make it easier to detect the
very weak Raman lines which in other systems are masked by the high noise
level. Furthermore the spectral region 400nm to 900nm(38) is of most
interest to the laser Raman investigators. Consequently it is an
additional advantage if the Michelson spectrometer is nominally achromatic
for at least part of that region.
108
6.4 Conclusion
The Four Prism system cannot be said to be better than other
similar instruments for all possible applications. The mechanical
'simplicity of a single translation has been accompanied by much
greater optical complexity and cost. Elimination of „the chromatic
aberration is an advantage when a large spectral range is required.
However,the achievement of a large resolution-luminosity product
has been the most important factor in the design of this instrument.
The experimental value of the RLP found is 2.5m2steradians which is
very close to the theoretical value of 2.36m2steradians for a gap of If
0.00175 derived in chapter 3. The increase in luminosity at a given
resolution will undoubtedly improve the signal to noise ratio by
making a larger signal available at the detector. Accordingly it will
allow the study of low luminosity sources at resolving powers up
to 105.
• Appendix A '109
Computer program for numerical ray trace
PiRIAL CCLL.G7. FCRTRAN CCPIFILER KRONOS 2.1SX PSR2+ 74/12/i . ,
C FIFO !-IMPENSATE0 MICHELSON INTERFEROMETER • i C RAY -RACE FOR FOUR PRISM SYSTEM
is ;'7'.1.. T I LshOyKsJ
2s Vt!'”7:Inti x (15) 0 (15) ,Z (1.5) IL (1.5) 010.5) ,N (15) pl? (3) pS (1)) IR(2), 1 Fi519K(7),Ji6),I(12)
3. WTI(6113O) 4. inn F)1Nit(tHit 15HTHE RCSULTS ARE) ,. 5s , 0909u4t.,431
6. 123 . Cz.".4-1.100004o431
7, WRTT7(6,135) C
8. 175 r1"A-(tH0167X121"C=1F12411-11) C L(1),In[10(i) mRE TILL INCIDTNT OIRaCTION CUSMES
go . . L(1)=Cla.Or.k0.017453,?g)
10, 1(1.)=(10.0)*(0.01745329)
11. . IO -1 120
124 1?1. L(OrtA1.0)(0,11745321) 13s - 129 4(1)=.(10.0).4=(01101745329)-Lti/ 14. 1(1)=1.0 _ 15. )112'%23300003 16,
17o 74A'.1005 47ITT(Ss 107) L(1)0(1)0(1)
19. 107 ;FCP,MAT(1H0,10X1 5h1.(1)= 1F7.50X,M(1i.=oF7.51.5X,544(1)=IF7.5)
.'d nrTA FOR FIRST ARM OF SYST;LM C .
20. .M2)=1.713003 21. • R(')=1.543140 22. %:(11,9) JEJ1745329)
' 23s 7(1)=6.64030025
24. l(2)=0.01000025
. program continued on the next page
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Y(1.)--tt15e .
▪
0.1 12?
ot TA r:DP THE Sa.01tD ARV OF SYSTEM
124 lt".)=1.51+'1,140 RU")=14,71391 3
({c). 01745329) (20.0)) 7(1) 726.10:))0 air)
F• (1) =501419j 942 (51?-(14.5030'23134-C"SIN(A))
Z11)-=0.0 122 L f")::AnOS (1)4COS (L (1))) /(2(2)})
'1(1)=V7COS ((r:(1)•dCOS(H(1)))/(Z(2))) CO'I (N (1)
I▪ (?) =4 7:r.;_ (C...-:Th-ZI(?;(2)4+2...(P(1)":2)* (1e0•.(COS (I (1)) )*=-2.) )) / (•R(2) ) ) 417)=ACO.J C--;(1)4CCS C1(1) ) tk (2) ) + (R(2)*COS (I (2)) -R(1)4.COS (1(1)) )./
1 CM 2))) S(1.) T3 T'IE TH THRCUCH THE COMPEnAIING GLASS 1LOCK LAK 8
511.)=(7(2)/LI:J3 (N (2)) ) ) X(2)=X (1) +S(1) 4CCS (L (2)) Y.( '") ,7'•=Y (1) +S( 1) *COS( t•f( 2)) Z(?)=Z (1) -1-:-“1)'CC:(N(2)) L (1)=4°C,OS (2) (1.. (2) ) (R(1) )) %;(7)=A7,CC.)S R (2) 'q.;CS (0(2) )(R(1)) ) I(3)=APCUS(CuS(N(2) ) ) • IC4 )=ARGO:,t(:‘,T7I (1; (1)."2-(R(2)"2)*(1.0- (COSII (3)) )442) )) (^(1) ))
(7)=4 Et..0:, (R(?)4=-. (11(2) )/ (R(1))+ (R(i) 006 (I (4))-R(2)4C,OS(I(3)) )/ 1. (?t{)))
St 2) 13 Tm- AIR GAP 3(2)=(F:(1)+E(2)-7(2))/COS(N(3)) )C( -4 1r.X(2)+:.(2)a`CCS(1.(3)) f
(2) +3(2)'1.COS ('4(3)) C3) •=Z (2) <3(2) J'COSttl( 3)) •
L(4 )=AR130;:“R(1).k.CCt. (L. (3) )/ (RUM ) CO =aE•Att. CI (3) ) / (R(3)) ) • I(5)=AE130 .̀.)(CO3(N (3))) (6) =1,1.?COSUSORT(;'( 3)**2-(K(1)"2)*(1,07(COS(I(5) ) ) *4.2) 'i)/ (R(3) Di (V)=AiRCO-3(r: :1) 4CCS (N ( -3) )/ (R (3) ) + (k(3)4C1X-: (I (6))-R(1)+C.:OS (1,(5))
f").(7)) 5(3) IS THE PATH THI;GLIGH THE g)RISM LLF1(1)
3f T)=( ( (E(1) +£(2)+E(3) )a'COS(4) -X (3)*SIN(A) -Z (3)*COS( A)) r (Ile'(L (4) )—..:IN(A) +COS (11(4))*C01:(A))) - X (4)=X (3) 4-'z;(3)4-.CGS (L (4)4
14)=Y (3) +S(3)4CCS. (t'' (4)) Z(4)
•
(3) +:.3(3)*.COS (t1(4)) I (7)=ARGO (CV:AL (4) )''.S.IN(A)+COE (N(4) )*00S(A) ) I (i)=Y-:,;0:., ((:Ar-a(iit1).'s4-2- (R (3) 4-'''2):̀ (1.0- COOS (I (7)) ) 4'2) )) (7 (1) ) ) L(5)=ARCOSC(r.(3)'COS(L (4) ))/ (r<a)) +P.(1)"“:#03 (1(1)) -R (3) 'COSii(7) ) ) f !..1.11( / (r(1)) )
fc )z.:COS (1-1; (1) MOO ) (R (70 =0, Tr":0S (CR (3 )'`COS (N (4) ) ) OM.) ) “ R(1) 4"COS (1 (0))''R (3) !COS(' (7) )
1.) :;ns ( / (IA )) .-4., (1)+E(2)+E(3)4-(GAP)/(COS(A))
S(43 LS TH7 AIR GAP ;7 Ci kEEN THE PRILtiF 3 (4)= (r)';;CY...;(•-•,) 4COS(C,PSIN(t 1.$) -X ( (+)*SIA )*SIN(B)
1. fY /&) (A) (C).-Z ( * 31:1 (A) *COS (Cr COS tA +5)) / 2 (7,'n--“L(5WSINCP.)-, t,OSc-A 4SIIICA+1)—CUStr1(5))*SiliA) *SIN (C) 3 +COS (Nt5) )*C0S 3)4.ri(A))
X ( -3)=X (4) +S(4) -*CGS (L (5)) Y (51=Y (4) +:;(4) (!•4 (9)) 1(71=Z (4) +S(4)+GCS(M(5)) I (1)=-44COS (COS (L (5) (A+R)+COS (L) -COS CI 15) )"SIN(C) +7:". (N(5) (r+3) - I(1r)=:-R:OS( (SORT (E (3)4" 2-(R(1)4'2)*(1•0-(COS (I(9)) ) "2) ) i(;-;(3)
i )) L (5) =r;kr:•00S(R(1)*CCS (L(5) ) /(a( 3)) +{K(3) "COS (I t10).)-R(1)*COS CT CO)) )-'7.I.1(A+3)-1COS(C)/ (R(3))) I (5)=A RC,X3 (R (1)41..C17 ) / (3) ) tR(3)1( COS /I (1C) )-P. (1) 'COS 17.(E13 WSI'I(C)/ (Q (3)))
NIF(C) --.--1..F.C3SC=.(1) CC!: ttl( 5) ) / (m( 3) ) (k( 3) *COS CI( 10) )-R(1)*COS 1. ri)))+OUSt.k+r.14C0".-i(r;)/ (R(3)))
$15) PATH PrZI'3N LLF1(2) r=1 47 (5) + (,14-COS (A)r SINtr.)) )/ tSIN(A))
j) GO TO 130 S(5)=-: tT4COS (C)-X(5)*SINI(r3)*633(C) 4-Y (5)4SIN(C)
2 '01 0) '''COS (EA 'f...; (6) )/ (70`3 CL (6) )4SiN tb) 4-0O3;(t.;)-;,OS (N (6) )4' SItl((;) 2 4-t0IIN(6))--COSF-3)."COS(C))
110
• ■■•■■•■•
1.3 fr 'iii)
O 131 e:.(5)=(T-X(5)'':iIt4(3)-Z(5)*COS(B))/(CCS t6)) 4Sit4(3)+
triF.(10 (6wr., of.:( -A ) 131 X (6)=X (5) +S(5) .̀ 'CO.S(L()
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1 4- riC(11(6) ) -- CrISCIPTOS(C) 1.1"r)r- ,,..nr.L1`“:(:05(t. (CA )-(2.0) 4 SPI(C..)*COS(I (/1)) 4 COS(C))
ti'lrzA"-:007; (CV; (X tE) + (2,93 (C)*-00f;tI (11) ( 7 )=-.)°,'S kW (N% (6) t2*0) CuS (ElkeUS (I til) V- COS(C))
S(5) __
t3ATH 11 PR1S.4. LL.F1(2) 3(5) =(0-z-CoSt ,I)#COS(C) 4SIN(A+',) -X(6) 4.SItItA) *COS(0) iiSIN(A +3) +Y(A1 ,••••-u.il (C) t6rc:7,Ill (A)-7-00'.3(C) 4-- 00F.: 'A+E.;) )
2 (7.rirl (LC) cOS G)=SiN(m+6)-DUStH (7)) 4SIN (A) v-sit4 (c) 3 4-1r.,,,(N(7))=.C1SiCr, COS (A+3)*SIN(A)) •
X( 7 ) =X(6) 6) -4- CCS(L( 7)) Y1T)--,1 (6) -1-'..;(5)"Cc• -a (m(7)) • T(T)=Z (6) (t•1(7)) L(4)74,%?1_3(.):3(;:.(,) -'CCS. (L(7) ) (1(1)3+ tR() 4.GOS (I (1'1) )-F;(1)LOS
i (T(n)))k SI14 (A+ t:IjS(.";),f(2,(1))) tti(7) )/ CR (1) ) (R(3)*COS LI (103 )-P.( 1) .1 COS
'CI ielnt :1 u:)/tc(I)l) (8 )•=-Ar!Cr)S (7) ) /(C.(1) ) M(3)*COS (I (10) ) -R (1)*COS T( ) -CeStC)/(2,(1)))
5(7) r3 THE PP,TH THE AID. GAP 3 (7)=( (1) (2) 4•:- (3) C.OF. (6) -X (7)t SIN (A) -2(-7)*COS (A)) / (f.1/F (-L(8) ) A) +CuS (N(5) )*GOS (A))) (A)=X(7) 4-7_)(7) .1- C- CS(L(5))
f (11=1, (7) +S (7) --- CtiZ ( 11( 3)3 . Z (1)=Z (7) +:.(7)=4 CCS (N (8))
• L(g)=A'coS C-“1.)• COS (L (I)) )/ (Z (3) ) i(P(1)*COS (I (61) -R(3) 4 COS(I(7)) 1
3)))
4(9)=AcCOS tR(1)*CCF (M(8) ) / (P(3)) ) Ott ))=ikrrOS((ri"(i) ..:CIS(N( -3)))/ (R(3))+IR(1) 2'COS C3.(3))—(3)*P OOS(i. (7) ) ) 1, ;(v--:(A)/C.( -)))
St S) rs TIE Prl Tri 1 i-LCk.JC-11 PRISM ILF1(1) S (8)r: (7(1) (2) -2 (E) )/CoS(N(9))
• ( -3)=x (1_,) 4.:.;(5) (L(P)) (9)=y (8) +S(5)''- CU5(?':(9))
7 ( .. :r.Z(3)+:-A3)•44 COSiN(g)) L(ir)=ARCVAR(3) 4 CUE (L(9)) f(R(1) )) Mr1r1= PAR:r..)!TAR(3PCCS(1(0 )),/ (R(1))) 4(17)=AiCOS(fc13)'COSO-((9))/tk(i))+(R(3)*COI(6))-R(1) 4'
( 5)) / (7.(i) )) SES) 13 THE PATH br,ROS::, THE AIR GAP
(1) -: F(1)-2 ( )./CCS (N (10)) ,t(in)=Y.(9) (9) *CCS U. (10)) f(t1)=Y(9) +5; J'COS (1(10)) .7.(tr)=7(9) (9) 4 CC: (N(10)) L(1.1):-•::, ;-:70:=(P (L (10))/ (R(2))) 1(1.1)=AP.C.:OSP (19) ) / ('Z(2))) U(11)=APCOS(R(1)'CGS(N(10))/iR(2))t(R(1)*COS(I(4))-P (2)''COS(I(3) )
1..)11(...(2)f) S(10) 13 ''.ETU ;14 PATH IN 3LOtAc LAKE
1(1.P)=-.7 (1.3)/C,OS IN (11) ) Kitt) =X(10)+S. (10) 4C OS(1(11) ) Y(11.)=Y(10)4:S(10) 4'COS(;1(1.1)) !.(1.1)="(10)+:..(10)"(.WAN(11)) .1.1121=; P.7.',OSCR(2) .'CC'..3 0_ (11) )) • 14 (1?)=;LDtr,-,(a(2) ,- cos
• Nr i,) - AP,r;us(9.(2)*cosciqui»+c(2)*cosci (23 ) -R (1)*COS (I (1) ) )) P-- (r(3).GF_-- .1.71306.3) GO 10 123 'CI )=S(1)= .::(2)+S (2)+S(3) 4 R(3)+S(4)+S(5)'M +
) +S(7) +S (d) '.R(3) +3 (9) +S (10) 4-2,(2) 20 Tr) 124
123 (!)=1 (1 )-"--.(2)-+.. (2) +;;(3) 4'R(3)+::;(4)+S (5)".+Z(3) + 1 3 Vir- R(3)+S(7)+5(,)) .°.k(3)+S(9)+St10')'“2(2)
p 1714:-.P (2) -2 (1) wzr-'rc(6114U) GAP
1.41) ciFTA'(1113) 65X1 41-GAP=,F8.6) 14 .:IT- (c)1 127) F
1?7 F) -",, AT (111277X151-cATii=„Fizel) TFCt Cil C) 4 ( 0.01745329) GO TO 121
r.LT.u.9003442405) GO TO 128
ElD •
S REFED,ENr7 MAP (/N= N REFERENCES IN STATEMENT)
AMTS S32-r") 3Y AOORT,SS •• (3=RELAT11E ADDRESS) (C= P<ELATI4E TO 1/)
102ft?' 3 0020609 K 002067e L 10213?-1 Y 0021513 Z 002110C. M
0021'..16=3 N 002n73
.112 Appendix B
Computer program for airbearings
PROGRAM KERR (OUTPUT, TA PE6=OUTPUT)
-C VERTICAL REAPING SYSTEM C TO CALCULATE CRIFICE FACTOR,STIFFNESS AND AIRFLOW FOR CARRIAGE
C POCKETED 9EARING - - - - - - - REAL K DIMENSION A (2) IW12) tli (3) sIK (2) ,P (3) ,C (2) IR (2) Vi(1) =30.0 A(1)=3.0 Kt 1)=106
- 2)=K(1)*W(1) F(1)=114.1 F( 2)=10000 K( 2)=0.20E 1=1,J C( 1)=1.41E-4 - C(2)=44.0 R( 1)=0.0625 R( 2)=0;-866 EE.TA=0.0 GAMMA=1( OI(2))/( (1.0)*T*K(2)*(A(1)"Ute,)))"(0.91)+P(1))/(P(2)) 1NR ITE(6,120) GAMMA
- 120 FORMAT (1H1, 6HGAMMA=IF6.4) M(1)=0003C4
102 11(1)=M(1)40.00li1 A( 2)=(((M(1))"3)*( (W( 2)/ ( (140) *T -liK(2)4 (A(1) " (0.13) )) ) 4*(1.82)4.
1 (2.0)*P(1) ((W(2)./ ((1.0)*T*K(2) 4" (A (1)" (0.8))) )" (0.91) ) ) ) 2 (C )*ALCG10 (12(2)/R (1) )*P (2)*( (GAMMA) " (1.41)- (GAMMA)" (1,71) )
- 344 (0.5) ) F=tCt 4 T*At2)*P(2) 4(11(1)**(BETA))*MGAMPA)) "(1.41)- (GAMMAi -1-4̀
.1(1.71)) 4.*(0.5)) CELIA= (((t, (2)+1.00)/ C(1.(l)*T*K( 2)*(AC1)"11.8)) )"(0•91)+P(1))
1/(F(2)) ° OMEGA=c(t# (2)-1.00)/ ((1.0)*T*K( 2)*(A(1) "C•8)) ) "(0.91) +P(1))
1/(P(2)) 2)= (C(1)*ALCC10 (P,(2)/R(1))*A (2)*P(i)*( ((DELTA)" (1.41)-
1 (DELTA)" (1.71 ))**(0.5))/ ((W(2)4.1eJC)/( (1* 0) 4 T 4-K(2)*(A(2)" 2 (0•8))W4 (1.82) (200)*(P(1))*( tki (2) -1-1.00/( (1.0)*T*K(2)*(Ai2) 3" (0•8) ) ) ) 4" (0•91) ))** (0.333) H3)=. (C )'ALCG10 (R (2)/ R (1) ) 4̀ A (2)*F 12) 4 ( ( (OMEGA)" (1. 111 )-
1 (OVEGA)" (1.71 )) ."(0.5))/( (14(2)-1.01) /( (1•0) 4-T*K(2) .?(A(2)*4- 2(t1.8))))**(i,82)4-(2.0)*(('11))*f (W(2) -1.3C)/ ((1.0) *T*K(21 4 (11,(2) 3 4*(0.8))) )4*(0.91) ))" (0.333) - - -
)(= ( (2.0) 4 (1.00)1 (H(3) -1-1(2))) WRITE(6 1 104) T I P(2) 1 A (2) t F,X ) H(1)
104 FORMATC1F1015Y 3 21-1T= I F6,2,5X,5HP(2)= I F6.2,5)( t5HA (2)= I F6.4 15X 1 - - 12HF=IF4,25.5X12HX=IF13065X 15f1H(1)=,F7,4)
- IF Ili (1).L.T.O. 001) Gt) TO 102 ST CP ENC
COMPILER SPACE
- • •
I.dL.sinG
(a2+42)
I.dL.sin3
d2
4
.113
APPENDIX C
The Inductance of a Square Coil
fig.1
The magnetic field at P, using Laplace's theorem, is given by
I.dL.SINO dH - r2
(1.
fig.2
114
In fig.2 dL is a small element of AB.
dL.sine = (a2+d2)SO = diSe
dL.sin20 =
Substituting into equation 1.
dH IsinedSO
d2
= I — sine60
Only the component along the axis need be considered
PH = PKsinKPV = PK.a/d
the horizontal component of equation 1.1 is given by
1a dH = — sinOSO
d
The magnetic field intensity due to one side AB of the square is
H 1= 12,) 'Tr - (3 sined8 d e
2Ia cosO
d2
2a2I 1 2 (a2 +x ) (2a2+x2)1
sinO
115
. The magnetic field due to all the sides of the square is
H 8a2I (1.2 (a2+x2)(2a2+x2)
So far the magnetic field intensity has bac-A derived for only
1 turn. Consider a coil of n turns per unit length.
fig.3
The field intensity at P due to a thin band of the coil of width dx
at A is
dH -
8a2nIdx (1.3 2 2 2 Aa +x )(2a +x2 )
Now AP=L and Ld0=dxsinO. Also sine = 7. a - and x2+a2=L2.
Substituting into equation 1.3 for dx and a2 +x2 gives
dH - 1 (2a2 +x2 )2
but (2a2+x2) =
( 2 2 /1 2a +x )
2 • a )a +x+ 1 (a2 +x2 ) +x
(sin28+1)1 sine (1.5
8nlade (1.4
116
Substituting equation 1.5 into equation 1.4 gives
dH - 8nIsined0
(sin20+1)1
If 01 and 02 are the values of 0 for A at the ends of the coil, the
total field H at P is
H = 8n1 %..01
(sing +1)
Integrating by substitution gives
-1 coshll 02 H = 8nI sin 2
01
where 0.2 = 7r- 0 and 01 = 6.
The magnetic field varies along the axis of the coil and is at a maximum
at the centre. Therefoi.e to find the maximum value of the self-inductance
substitute cos° into equation 1.6
At the centre of the coil cos0 - (a
2+L2)
equation 1.6 gives
[I
H = 16nI sin1 [ L
(2(L2+a2)) (1.7
In rationalised MKS units equation 1.7 reduces to
. _ H s in 4ra [ -1
(2 (L2 +a2 )) ya
r.0 2 sin8d0
(1.6
• L2
117
The magnetic flux through the coil is
klotatipi- L
L 2 L2 a2 1,21 01) = BA -
When the current I is varied the rate of change ofjlux linkage per
turn of the winding is
4.41.10nA.d1 sin 1 [ L [ dt ' dt. (2(a2+L2))4
The induced emf around N turns in the coil of lengthJ, is
4110112A . dI [ . -1 [ L dI = - sin Ln dt (2(a2+1,2))1d .1= dt
the inductance is
L - -4u0NZA [-sin-1 [ L
21 I (1.8
Lit (2(a
2+I.2))
.118
References
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119
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120
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