Jones matrices of a tilted plate for Gaussian beams

Jones matrices of a tilted plate for Gaussian beams

J. C. Cotteverte, F. Bretenaker, and A. Le Floch

The Jones matrix of a tilted plate is theoretically and experimentally described, taking into account the

contribution of the successive internal reflections of the incident fundamental TEMOO Gaussian beam. This

induces variations of up to 10% of the transmission coefficients of the Jones matrix with angle of incidence.

The consequence of the influences on the walkoff and the internal interference effects of the characteristics of

the plate and of the incident beam, especially its mode size and its radius of curvature, leads to important

different variations of the Jones matrix. Surprisingly, however, we show that for a given Gaussian beam the

Jones matrix of the plate does not depend on its position along the beam propagation axis, in spite of the

variations of mode size and radius of curvature whose effects compensate mutually. The Jones matrix of such

a plate used in a laser cavity containing a diffracting aperture is also investigated. In every case, good

agreement is observed between theory and experiments.

1. Introduction

Tilted plates are commonly used in laser cavities formany different purposes. They can be used to intro-duce loss anisotropies in solid state lasers' and in Zee-man lasers,2 or as Brewster windows in conventionallasers. In multimode lasers, for example, dye lasers,tilted plates are used as etalons to provide mode selec-tion.3 Such plates are also used to change the opticallength of the cavity to tune the frequency of the laser.2

In solid state lasers with external mirrors, the laser rodcan also be considered a tilted plate. 4 Finally, to someextent, some kinds of internal mirror optical fiber cou-pler5 consist of two parallel interfaces and may behaveas tilted plates. In many of these cases, the faces of theplate can be coated to adjust the reflectivity at theinterfaces. The usual Jones matrix6 for a simple un-coated plate is

rt 01

Lo ty]

where t,, and ty are the field amplitude transmissioncoefficients given by Fresnel laws7 : x-polarization isparallel to the plane of incidence, and y-polarization isperpendicular to the plane of incidence. Such a de-

The authors are with Universit6 de Rennes I, U.A. CNRS 1202,

Laboratoire d'Electronique Quantique/Physique des Lasers, F-35042 Rennes CEDEX, France; F. Bretenaker is also with Socit6

d'Applications Gen6rales d'Electricite et de Mecanique, B.P. 72, F-

95101 Argenteuil CEDEX, France.Received 27 February 1990.

0003-6935/91/030305-07$05.00/0.© 1990 Optical Society of America.

scription does not take internal reflections into ac-count. However, with a coated or uncoated plate, suchreflections exist and can induce variations of t. and ty.Such variations have already been shown to modify thelosses in a laser operating in a single linearly polarizedmode. 8

Another interesting case is the quasi-isotropic lasercavity containing a tilted plate that introduces x-ytype loss anisotropies and behaves as a partial polariz-er.9 In that case, knowledge of the complete Jonesmatrix of the plate is necessary to predict the dynamicsof the eigenstates of the laser. This case is one of manytypes of quasi-isotropic laser, such as CO2 lasers, solidstate lasers, laser diodes, and sealed mirror lasers.Some authors3 10 studied the interferences in a plate,but introduced Fresnel formulas in the usual Airy for-mulal without taking the walkoff, i.e., the transversaldistance between successively reflected beams, intoaccount. If one considers the walkoff,12 the Gaussiancharacter of the beam must be taken into account.The aim of this paper is to investigate theoretically andexperimentally the influence of Fabry-Perot effects ont,, and ty, taking into account the optical characteris-tics of the plate and of the beam (i.e., its mode size andits radius of curvature), in order to write the completeJones matrix of a tilted plate in a laser for the TEMoofundamental Gaussian beam. The first part of thiswork is devoted to the theoretical calculation of theJones matrices of the plate inside and outside thecavity, taking into account the role played by theGaussian beam and its geometric characteristics. Theexperimental verification of this calculation is made inSec. III, leading us to study the evolution of Jonesmatrices with the different parameters of the incidentGaussian beam and of the plate.

20 January 1991 / Vol. 30, No. 3 / APPLIED OPTICS 305

II. Theoretical Calculation

Let us consider the general scheme shown in Fig. 1,where a Gaussian beam of size w and radius of curva-ture p is incident with an angle i on a plate of thicknesse and refractive index n. The transversal spatial de-pendence of the electric field associated with the inci-dent beam can be written4 as

Ei(x,y) = A exp( x 2+y

2 ) exp[-2 (x2 + y2) (1)

where X is the wavelength of the considered beam, thex- and y-directions are defined in Fig. 1, and

-A A= Ax

gives the polarization of that beam. We suppose thatthe plate is thin compared to the Rayleigh range (0.2 min our case) to allow us to neglect the influence of theGouy phase shift. Let us recall the well-known reflec-tion coefficients rx~ry and transmission coefficientstxsty for the two consecutive interfaces that are deter-mined according to Fresnel laws7:

r = [tan(i r)] 12a

y= [sin(i +r]'(2b)

= sin2i sin2rsin2(i + r) cos2(i - r) (3a)

= sin2i sin2r (3b)sin2(i + r)

where r is the refraction angle given by the Snell-Descartes law:

sini = n sinr. (4)

Our purpose is to calculate the exact transmissioncoefficients of the plate, taking the internal reflectionsinto account. Consequently, the electric field associ-ated with the successive output beams can then bewritten as

Et(x,y) Bp exp(ipO6) ex[-(x-pAx) 2 + 2 ]p0O W

X ex .ir ((X - pAX)2 + y2)], (5)

where

B. [t=r1PA (6)

contains the polarization and intensity of the pthtransmitted beam,

4rrne= Ay-- cosr (7)

is the phase difference for one round trip in the plate,and

Ax = 2e tanr cosi (8)

LENSPLATE

Fig. 1. Experimental setup without an aperture. Polarization ofthe incident Gaussian beam is arbitrary. All the intensity of thesuccessive beams is focused on a detector. The dotted lines repre-

sent wavefronts of the successively transmitted beams.

is the transversal walkoff between the successivelytransmitted beams.

Such an expression is valid as long as the plate is thinenough for w and p to be unchanged by the multiplepropagations in the plate. Here we consider materialssuch as uncoated glass (n 1.5) for which r and r are-4% so that one can neglect the terms withp > 1 in Eq.(5), leading to

tA] ( y2 + y2\ [u 21]E,(x,y) = t.AXexp~ x2 )Y exp[.'- (X2 + 2

+ [t:rA] exp(ip) ex[ (X AX)2 + ]

X exp{ [(x - AX)2 + y2]}* (9)

In a laser, one must use a diffracting aperture ofdiameter k centered on the main transmitted beam toselect the TEMoo mode (see Figs. 2 and 3). In thatcase, one can calculate intensity transmission Tj of theplate for j = x or y:

+0/2 102/4_y2T y = l¢2 1 Etj(X,y)12 (0__ /2 _12/4-y2,(0

f dy dx Eij(xy)12

which using Eqs. (1) and (9) becomes

2tj2 (/2 d 2y2 \ r, 2/4-y2 // 2X2 \TW=-/ dY exp 2 dxexp(\- _)+ 2rj expE x2+ ( - A) 2 CoSb+- [ 2 -(x - Ax)21V (11)L I JI P1),U1

where the term proportional to r? has been neglected.In the case where no aperture is introduced ( =

approximation), the calculation can be performed ana-lytically. Indeed, using the fact that

coS{, + [X2 - (x - Ax)2]} = cos cos{$ [X2 (x - Ax)21}

-sinip sin [X2 - (x -AX)2]}, (12)ApII

306 APPLIED OPTICS / Vol. 30, No. 3 / 20 January 1991

Consequently, the evolution of the Jones matrix

M(iwd) = T

Fig. 2. Tilted plate in a stable resonator with an aperture.

PLATE APERTURE

Fig. 3. Experimental setup with an aperture.

and'3

J dx exp(-ax2) cos(2bx) = . exp(- ) (13)

+ b

J dx exp(-ax 2 ) sin(2bx) = 0, (14)

the transmission coefficient becomes

= tj(1 + 2rj cos' ex{ - [-2 (p A)]}) (15)

Let us introduce the so-called interference maximumwalkoff L by

L =[ + (wA)2]r ' (16)

which gives the order of magnitude of the walkoffabove which interferences no longer occur. L seems todepend on the main characteristics w and p of thebeam. As4

p(z) = + (2)] (17b)[Z I XZ)2 (17b)

one can see that Eq. (16) reduces to

L = wo, (18)

leading to

Tj = tj2[1 + 2rj cosVI exp(- Au)]. (19)

The same calculation, taking all the internal reflectionorders into account, would lead to

j = t(2 I(k{1 + 2 E exp[- 2 w cos(2p)}

+ 2rj2+l Eexp[-(2p + 1)' 24_ cos[(2p + 1)4,]), (20)

of the plate with the angle of incidence i does notdepend on the position of the plate along the Gaussianbeam but only on its waist wo.

This striking phenomenon can be understood as theresult of two geometric effects related to the nature ofthe Gaussian beam itself. On the one hand, for a givenvalue of mode size w, a decrease of p leads to a destruc-tion of the interferences due to a quicker transversalvariation of the field phase. On the other hand, for agiven radius of curvature p, an increase of mode size wleads to two contrary effects: an increase of the over-lapping of the two beams that promotes the interfer-ences and an averaging of the interference term due tothe larger transversal variations of the field phase.The competition of these two effects, associated withthe usual evolutions of w and p along the beam, leads tothe invariance of the plate transmission along thebeam.

Ill. Comparison with the Experimental Results

Except for Fig. 9, the experimental evidence of thesephenomena was performed by single-pass transmis-sion measurements at X = 6328 A with a laser beamhaving a size w = 365 ,m, a radius of curvature p = 0.7m, and a waist size w0 = 260 Aim. We used polishedplates with X/10 surface figure tolerance. A first seriesof experiments was made without the diffracting aper-ture (0 = a) to investigate the role played by thedifferent experimental parameters (see Fig. 1). A sec-ond series of experiments was performed with a dif-fracting aperture centered on the main output beam(see Fig. 3) to compare the experimental results withthe theoretical calculation of the Jones matrix of theplate.

A. Role of the Experimental Parameters

1. Influence of Thickness e of the Plate onOverlapping Beams

The typical evolution of walkoff Ax vs angle of inci-dence i is shown in Fig. 4(a) for a 2.1-mm thick plate.One can see that this walkoff is maximum for i =arcsinJnI - n 95o in our case) and is (AX)max= 2e/ -1-2n/n 2-1 (1.65 mm in our case). Theangular range of oscillations in transmittivity de-pends on the ratio wo/e because of the exponentialfactor in Eq. (19). If wo < e, oscillations are only forsmall or large angles i. On the other hand, when wo >e, oscillations are for any angle i. This phenomenonis illustrated in Fig. 4. Figures 4(b) and (c) show theexperimental transmittivity Ty vs i for two differentplates. The first [Fig. 4(b)] is 2.1 mm thick with n =1.46, and the second [Fig. 4(c)] is 0.16 mm thick with n= 1.52. The corresponding theoretical results givenby Eq. (11) with 0 = or by Eq. (19) are displayed inFigs. 4(d) and (e), respectively. One can see that in thecase of Figs. 4(b) and (d) (e = 2.1 mm > wo), the


(21)

0 30 60ANGLE OF INCIDENCE (degree)

I -I

0.5 (C) N d

° 30 60 9ANGLE OF INCIDENCE (degree)

TY

90ANGLE OF INCIDENCE (degree)

Ty

ANGLE OF INCIDENCE (degree)

Ty 0.5

C0

(e)

theo30 60

ANGLE OF INCIDENCE (degree)TY

30 60ANGLE OF INCIDENCE (degree)

Fig. 4. (a) Evolution of walkoff Ax vs angle of incidence i for the 2.1-mm thick plate. (b),(c) Experimental evolution of Ty vs i, respectively,for 2.1- and 0.16-mm thick plates. (d),(e), Corresponding theoretical curves. (f) Same as (d) except that p = X (plane wave).

oscillations occur only for i < 15° and i > 80°, althoughin the case of Figs. 4(c) and (e) (e = 0.16 mm < w0), theyoccur for any angle i. Moreover, good agreement canbe observed between experimental and theoreticalcurves. The role of the radius of curvature of the beamis illustrated in Fig. 4(f), computed with the sameparameters as in Fig. 4(d), except that p = (planewave). In that case, the interference of the beams canbe observed for larger values of the walkoff, up to i200.

2. Influence of the Plate Thickness on the Periodof Oscillation

Equation (7) shows that the evolution of withrespect to i depends on thickness e of the plate. Thisdependence is illustrated in Fig. 5. This figure dis-plays the theoretical and experimental evolutions ofT. and Ty for two plates (e = 2.1 mm and e = 0.16 mm)for small angles i. One can see that the oscillations arequicker for a thick plate [Figs. 5(a)-(d)] than for a thinplate [Figs. 5(e)-(h)], with good agreement betweentheory and experiments.

3. Influence of the Polarization of Incident LightIn the case of the usual Jones matrix of a plate given

by Fresnel laws, the polarization plays an importantrole due to the existence of Brewster angle iB definedby taniB = n for which the light polarized in the planeof incidence is totally transmitted [see Figs. 6(a) and(b)]. In our case, the existence of such an angle forwhich partial reflections at the interfaces disappear forthe x-polarization induces another effect as shown inFigs. 6(c) and (d) (experimental) and 6(e) and (f) (the-oretical) for e = 0.16 mm. Indeed, the oscillationsdisappear at the Brewster angle for x-polarization al-though they remain for y-polarization, leading to twodifferent evolutions of the Jones matrix elements.

B. Jones Matrix of the Plate in a Laser CavityLet us now check experimentally our model used to

calculate the Jones matrix of the plate in a stableresonator. Such a situation is shown in Fig. 2. In thiscase, the cavity (mirrors and diffracting aperture) se-lects the main transmitted beam. To describe the


Ax(mm)

0.5

Tyo

30

L

0 NGLE OF NDENE10ANGLE OF INCIDENCE Idegreei

Ty

20

0


T 05- (C)

o _the___'A 10 2ANGI.E OF INCIDENCE (dcgree)

).5 (d)

o theo0 O E

ANGLE OF INCIDENCE (egrec


TY

20

T 0.

ANGLE OF INCIDENCE (degree)0 10

1

T 05

C


0 10 20ANGLE OF INCIDENCE (degree)

Fig. 5. T (left column) and T, (right column) vs angle of incidence i for the 2.1-mm thick plate (a)-(d) and for the 0.16-mm thick plate

(e)-(h), experimentally measured for (a),(b),(e),(f) and theoretically computed for (c),(d),(g),(h).

1…

A.5F (a)

0 030 6 0 9(ANGLE OF INCIDENCE (degree)

1,

.5- 7 0 O0 30 60 9'


Ty 0.5[

(

Ty

O 30 60 9ANGLE OF INCIDENCE (degree)

1

A.5 OFNCdENEd

0 30 60 9ANGLE OF INCIDENCE (degree)

30 60ANGLE OF INCIDENCE (degree)

TY 0.5

0

(f)

theo30 60


Fig.6. T2 (left column) and T, (right column) for the 0.16-mm thick plate: (a) and (b) are the usual Fresnel curves; (c) and (d) are the experi-

mental curves; and (e) and (f) are the corresponding theoretical curves. Note that the oscillations disappear for x-polarization at the Brewsterangle (56.7°).


T 0.' (a)

exP

.

12 (b)

, expo 10 2(

0.5'

0

(e)

exp

5 (g)

0 theo

\ V\JVVVIVV

(h)

)theo

(b)

0

0

l-.

vu

v

I

Ty (

3 ZO0

1

n1

I

C

10

30 30

1�

T 0 °' (a)

0 C xp0 10



I -Pi

0.5 - (b)

0 OX 0 10 2

ANGLE OF INCIDENCE (degree)0

Ty

20

1--

'.5 (e)

0 *XP ,0 10 9!


TY

0

0

10


10


20

20

Fig. 7. T (a)-(c) and Ty (d)-(f) vs the angle of incidence: (a),(d) experimental without aperture; (b),(e) experimental with aperture; (c),(f)theoretical with aperture. Note that the aperture decreases the mean value of the transmission coefficient.

behavior of the plate in such a system and to obtainquantitative experimental results, we performed theexperiment shown in Fig. 3 with the aperture centeredon the main transmitted beam. For a 6328-A He-Nelaser, diameter 4 of the aperture is usually chosen sothat _ 4.5w.14 In this case, the aperture plays anegligible role in matrix M and one can take = inEq. (11). However, for a high gain laser such as a 3.39-,um He-Ne laser, one must introduce important dif-fraction losses to restrain any mode but the TEMoofrom oscillating.'5 This is the reason we choose 0 2.4w to investigate the behavior of such a plate in ahigh gain laser.

The experimental setup is made so that the apertureis centered on the main beam and follows it as angle ofincidence i varies. Let us consider a 2.1-mm thickplate, which is the order of magnitude of conventionalplates. Figure 7 displays a comparison between thetheoretical and experimental results with and withoutan aperture for x- and y-polarization. The influenceof the aperture is to decrease the total transmittedintensity for any angle of incidence because the partsof the beams which interfere pass through the aper-ture. Here again, good agreement can be observedbetween experiment and theoretical results computedwith Eq. (11). In the case of the use of such a plate as apartial polarizer in a quasi-isotropic laser, the evolu-tion of the loss anisotropy tty vs i is shown in Fig. 8.Knowledge of this ratio is particularly important for aZeeman laser,2 for example. One can see that in ourcase the relative variations of t/ty - reach 20%.

1.005

tx

10 5 10

ANGLE OF INCIDENCE (degree)Fig. 8. Ratio t/ty of the Jones matrix elements computed from Eq.(11) (oscillating curve) and from Fresnel law (nonoscillating curve)

for the 2.1-mm thick plate.

If one wants to build an isotropic laser and avoid theFabry-Perot effects between a window and a mirrordue to the residual reflectivities of the windows of thecell, one can tilt those windows at a small angle chosenso that t/ty is close to 1 to minimize loss anisotropies.

It is worthwhile to note that, for a given Gaussianbeam, particularly in a laser cavity, the interferencemaximum walkoff given by Eq. (16) does not dependon the position of the plate along the beam but is equalto the waist mode size wo. A striking example of this


0.5 (C)

O theo .

T 0.5

0

ex:p (d)

6. Xp . . . .

1-

0.5 (f)

O theo . ,

I

o

j

� I 1191111M.-

5

1

TX 0.9

0.80

1

T 0.9

0.8 exp0


(b)


20

ferent parameters of the plate, the Gaussian beam, andby the eventual laser cavity have been studied, show-ing that the behavior of many conventional lasers canbe affected by these interference effects. Besides, it isshown that the calculated modified Jones matrix of theplate does not change along the propagation axis of theincident Gaussian beam. In every case, good agree-ment was observed between theory and experiment.Our study was restricted to an incident fundamentalTEMoo Gaussian mode. Jones matrices correspond-ing to higher-order Gaussian modes are different; how-ever, they can be calculated using the same method.

We wish to thank Y. Le Grand and J. P. Tach6 fortheir helpful remarks. This research was supportedby the Centre National de la Recherche Scientifique;the Direction des Recherches, Etudes, et Techniques;the pole optique et opto6lectronique de Formation desIngenieurs par la Recherche Technologique; and theEtablissement Public R6gional.

20

Tx 0 9 1}lll (C

0.8 theo0 10 20


Fig. 9. (a),(b) Experimental evolutions of TX vs i for two differentpositions z of the plate along the Gaussian beam measured from thewaist (z = 0.64 m and z = 1.40 m, respectively). (c) Corresponding

theoretical curve computed from Eq. (19).

phenomenon is shown in Fig. 9. The two curves inFigs. 9(a) and (b) are recorded for two different posi-tions of the plate along the Gaussian beam, at 0.64 and1.40 m from the beam waist. One can see that, asexpected, the two curves are identical. The corre-sponding theoretical curve is shown in Fig. 9(c). Thiscurve, computed from Eq. (19), is typical of this beamand independent of the position of the plate along thepropagation axis.

IV. Conclusion

The Jones matrices of a tilted plate outside or insidea laser cavity for an incident Gaussian beam have beentheoretically and experimentally investigated, takinginto account the Fabry-Perot effects due to the reflec-tions at the interfaces of the plate. Large variations ofthe transmission coefficients-up to 10% for uncoatedconventional silica plates-with respect to Fresnellaws have been isolated. The roles played by the dif-

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12. F. Hillenkamp, "Influence of Interference Fringes of Equal In-clination on the Reflection of Laser Beams from Plane ParallelPlates," Appl. Opt. 8, 351-354 (1969).

13. I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Seriesand Products (Academic, New York, 1968), p. 480.

14. D. C. Sinclair and W. E. Bell, Gas Laser Technology (Holt,Rinehart & Winston, New York, 1969).

15. J. Dembowski and H. Weber, "Optimal Pinhole Radius forFundamental Mode Operation," Opt. Commun. 42, 133-137(1982).

20 January 1991 / Vol. 30, No.3 / APPLIED OPTICS 311

(a)

exp I

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Jones matrices of a tilted plate for Gaussian beams

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