Chapter 5 Wave-Optics Analysis of Coherent Optical...

Chapter 5Wave-Optics Analysis of Coherent Optical

Systems

Prof. Hsuan-Ting Chang

January 5, 2016

Prof. Hsuan-Ting Chang Chapter 5 Wave-Optics Analysis of Coherent Optical Systems

Chapter 5

Wave-Optics Analysis of Coherent Optical Systems

Contents:

◮ 5.1 A thin lens as a phase transformation

◮ 5.2 Fourier transforming properties of lenses

◮ 5.3 Image formation: monochromatic illumination

◮ 5.4 Analysis of complex coherent optical systems


5.1 A thin lens as a phase transformation - 1

1. A lens is said to be a thin lens if a ray entering at coordinates(x , y ) on one face exits at approximately the same coordinateson the opposite face, i.e., if there is negligible translation of aray within the lens.

2. A thin lens simply delays an incident wavefront by an amountproportional to the thickness of the lens at each point.

3. Referring to Fig. 5.1, the total phase delay suffered by thewave at coordinates (x , y ) in passing through the lens may bewritten as

φ(x , y ) = kn∆(x , y )+k[∆0−∆(x , y )] = k∆0+k(n−1)∆(x , y ),

where n is the refractive index of the lens material, kn∆(x , y )is the phase delay introduced by the lens, and k[∆0 −∆(x , y )]is the phase delay introduced by the remaining region of freespace between two planes.


Fig 5.1


5.1 A thin lens as a phase transformation - 2

4. Equivalently the lens may be represented by a multiplicativephase transformation of the form

tl(x , y ) = exp[jk∆0] exp[jk(n − 1)∆(x , y )]. (5.1)

5. The complex field U ′l across a plane immediately behind thelens is then related to the complex field Ul(x , y ) incident on aplane immediately in front of the lens by

U ′l (x , y ) = tl(x , y )Ul(x , y ).


5.1.1 The thickness function - 1

1. We first adopt a sign convention: as ray travel from left toright, each convex surface encountered is taken to have apositive radius of curvature, while each concave surface istaken to have a negative radius of curvature.

2. Figure 5.2 shows the split three parts of a lens. The totalthickness function can be expressed as

∆(x , y ) = ∆1(x , y ) + ∆2(x , y ) + ∆3(x , y ).

3. The thickness function ∆1(x , y ) is given by

∆1(x , y ) = ∆01 − (R1 −√

R21 − x2 − y 2)

= ∆01 − R1(

1−√

1− x2 + y 2

R21

)

.

4. The second part comes from a region of glass of constantthickness ∆02.


Fig 5.2


5.1.1 The thickness function - 2

5. The third component is given by

∆3(x , y ) = ∆03 − (−R2 −√

R22 − x2 − y 2)

= ∆03 + R2(1−√

1− x2 + y 2

R22).

6. Combining the three expressions for thickness, the totalthickness is seen to be

∆(x , y ) = ∆0−R1(1−√

1− x2 + y 2

R21)+R2(1−

√

1− x2 + y 2

R22),

where ∆0 = ∆01 +∆02 +∆03.


5.1.2 The paraxial approximation

1. If only paraxial rays are considered, that is, only values of xand y sufficiently small to allow the following approximationsto be accurate:

√

1− x2 + y 2

R21≈ 1− x

2 + y 2

2R21√

1− x2 + y 2

R22≈ 1− x

2 + y 2

2R22.

2. With the help of these approximations, the thickness functionbecomes

∆(x , y ) = ∆0 −x2 + y 2

2(1

R1− 1

R2). (5.8)


5.1.3 The phase transformation and its physical meaning -

1

1. Substitution of Eq. (5-8) into Eq. (5-1) yields the followingapproximation to the lens transformation:

tl(x , y ) = exp[jkn∆0]

× exp[

−jk(n − 1)x2 + y 2

2(1

R1− 1

R2)

]

.

2. The physical properties of the lens (n,R1,R2) can becombined in a single number f called the focal length, which isdefined by

1

f≡ (n − 1)( 1

R1− 1

R2).



2

3. Neglecting the constant phase factor e jkn∆0 , the phasetransformation becomes

tl(x , y ) = exp[−jk

2f(x2 + y 2)].

Note that it neglects the finite extent of the lens.

4. Figure 5.3 shows several different types of lenses with variouscombinations of convex and concave surfaces. The focallength f of a double-convex plano-convex, or positive meniscuslens is positive, while that of a double-concave, plano-concave,or negative meniscus lens is negative.


Fig 5.3



2

5. Consider the effect of the lens on a normally incident,unit-amplitude plane wave. The field distribution Ul in front ofthe lens is unit, and U ′l behind the lens is

U ′l = exp[−jk

2f(x2 + y 2)].

This expression can be interpreted as a quadraticapproximation to a spherical wave.



3

6. If f > 0, then the spherical wave is converging towards a pointon the lens axis a distance f behind the lens. If f < 0, thespherical wave is diverging from a point on the lens axis adistance f in front of the lens. (See Fig. 5.4)

7. Under nonparaxial conditions, the emerging wavefront willexhibit departures from perfect sphericity (called aberrations),even if the surfaces of the lens are perfectly spherical.


Fig 5.4


5.2 Fourier Transforming Properties of Lenses –1

1. One of the most remarkable and useful properties of aconverging lens is its inherent ability to perform 2-D Fouriertransform.

2. Several different configurations for performing the transformoperation are described. In all cases the illumination isassumed to be monochromatic – coherent systems.

3. The information to be Fourier-transformed is introduced intothe optical system by a device with an amplitude transmittancethat is proportional to the input function of interests.

◮ The device may consist of a photographic transparency or anonphotographic spatial light modulator (SLM), capable ofcontrolling the amplitude transmittance in response toexternally supplied electrical or optical information.


5.2 Fourier Transforming Properties of Lenses –2

4. Figure 5.5 shows three arrangements that will be considered.


5.2.1 Input placed against the lens - 1

1. In this case, the disturbance on the lens is

Ul(x , y ) = AtA(x , y ).

2. The finite extent of the lens can be represented by a pupilfunction P(x , y ) defined by

P(x , y ) =

{

1, inside the lens aperture0, otherwise

3. The amplitude distribution behind the lens becomes

U ′l (x , y ) = Ul(x , y )P(x , y ) exp[−jk

2f(x2 + y 2)].



4. To find the distribution Uf (u, v ) in the back focal plane of thelens, the Fresnel diffraction formula (Eq. 4-17) is applied.Thus, putting z = f ,

Uf (u, v ) =exp[j k

2f(u2 + v 2)]

jλf

×∫ ∫ ∞

−∞

U ′l (x , y ) exp[jk

2f(x2 + y 2)]

× exp[−j 2πλf

(xu + yv )]dxdy , (5− 13)

where a constant phase factor e jkf has been dropped.



5. Substituting Eq. (5-12) in Eq. (5-13), the quadratic phasefactors within the integrand are seen to exactly cancel, leaving

Uf (u, v ) =exp[j k

2f(u2 + v 2)]

jλf

×∫ ∫ ∞

−∞

Ul(x , y )P(x , y ) exp[−j2π

λf(xu + yv )]dxdy .

(5− 14)6. When the physical extent of the input is smaller than the lens

aperture, the factor P(x , y ) may be neglected, yielding

Uf (u, v ) =exp[j k

2f(u2 + v 2)]

jλf

×∫ ∫ ∞

−∞

Ul(x , y ) exp[−j2π

λf(xu + yv )]dxdy .



7. The complex amplitude distribution of the field in the focalplane of the lens is the Fraunhofer diffraction pattern of thefield incident on the lens, even though the distance to theobservation plane is equal to the focal length of the lens,rather than satisfying the usual distance criterion for observingFraunhofer diffraction.

8. The Fourier transform relation between the inputtransmittance and the focal-plane amplitude distribution is nota complete one, due to the presence of the quadratic phasefactor.



9. In most cases only the intensity across the focal plane is ofreal interest, and the phase distribution is of no consequence.Measurement of the intensity distribution yields knowledge ofthe power spectrum (or the energy spectrum) of the input.Thus

If (u, v ) =A2

λ2f 2|∫ ∫ ∞

−∞

tA(x , y ) exp[−j2π

λf(xu + yv )]dxdy |2.


5.2.2 Input placed in front of the lens - 1

1. Consider the more general geometry of Fig. 5.5(b). The inputis located a distance d in front of the lens. Let F0(fX , fY )represent the Fourier spectrum of the light transmitted by theinput transparency, and Fl(fX , fY ) the Fourier spectrum of thelight incident on the lens; that is,

F0(fX , fY ) = F{AtA}, Fl(fX , fY ) = F{Ul}.

2. Assuming that the Fresnel or paraxial approximation is validfor propagation over distance d , then using Eq. (4-21), giving

Fl(fX , fY ) = F0(fX , fY ) exp[−jπλd(f 2X + f 2Y )],

where we have dropped a constant phase delay e jkd .



3. Letting P = 1, Eq. (5-14) can be rewritten

Uf (u, v ) =exp[j k

2f(u2 + v 2)]

jλfFl(

u

λf,v

λf)

=exp[j k

2f(1− d

f)(u2 + v 2)]

jλfF0(

u

λf,v

λf)

=A exp[j k

2f(1− d

f)(u2 + v 2)]

jλf

×∫ ∫ ∞

−∞

tA(ξ, η) exp[−j2π

λf(ξu + ηv )]dξdη.



4. A quadratic phase factor again precedes the transformintegral, but that it vanishes for the very special case d = f .Evidently when the input is placed in the front focal

plane of the lens, the phase curvature disappears,

leaving an exact Fourier transform relation!

5. Here we have entirely neglected the finite extent of the lensaperture. To include the effects of this aperture, we use ageometric optics approximation. That is, the distance d issufficiently small to place the input deep within the region ofFresnel diffraction of the lens aperture, if the light werepropagating backwards from the focal plane to the plane ofinput transparency.

6. Figure 5.6, the light amplitude at coordinates (u1, v1) is asummation of all the rays traveling with direction cosines(ξ ≈ u1/f , η ≈ v1/f ). However, only a finite set of these raysis passed by the lens aperture.


Figure 5.6



7. The projected lens aperture limits the effective extent of theinput, but the particular portion of tA that contributes to thefield Uf depends on the particular coordinates (u1, v1)being considered in the back focal plane.

8. The value of Uf at (u, v ) can be found from the Fouriertransform of that portion of the input subtended by theprojected pupil function P, centered at coordinates[ξ = −(d/f )u, η = −(d/f )v ].

Uf (u, v ) =A exp[j k

2f(1− d

f)(u2 + v 2)]

jλf

×∫ ∫ ∞

−∞

tA(ξ, η)P(ξ +d

fu, η +

d

fv )e−j

2πλf

(ξu+ηv)dξdη.



9. The limitation of the effective input by the finite lens apertureis known as a vignetting effect. For a simple Fouriertransform system, vignetting of the input space is minimizedwhen the input is placed close to the lens and when the lensaperture is much larger than the input transparency.

10. In practice, it is often preferred to place the input directlyagainst the lens in order to minimize vignetting, although inanalysis it is generally convenient to place the input in thefront focal plane, where the transform relation isunencumbered with quadratic phase factors.


5.2.3 Input placed behind the lens - 1

1. Consider Fig. 5.5(c). The input is now located a distance d infront of the rear focal plane of the lens.

2. In the geometric optics approximation, the amplitude of thespherical wave impinging on the object is Af /d .

3. The particular region of the input that is illuminated isdetermined by the intersection of the converging cone of rayswith the input plane. If the lens is circular and of diameter l ,then a circular region of diameter ld/f is illuminated on theinput. This effective region can be described by the pupilfunction P[ξ(f /d), η(f /d)].

4. Using a paraxial approximation to the spherical wave thatilluminates the input, the amplitude of the wave transmittedby the input

U0(ξ, η) =

{

Af

dP(ξ

f

d, η

f

d) exp[−j k

2d(ξ2 + η2)]

}

tA(ξ, η).


5.2.3 Input placed behind the lens - 2

5. Assuming Fresnel diffraction from the input plane to the focalplane, Eq. (4.17) can be applied to the field transmitted bythe input.

Uf (u, v ) =A exp[j k

2d(u2 + v 2)]

jλd

f

d

×∫ ∫ ∞

−∞

tA(ξ, η)P(ξf

d, η

f

d) exp[−j 2π

λf(ξu + ηv )]dξdη.

6. Up to a quadratic phase factor, the focal-plane amplitudedistribution is the Fourier transform of that portion of theinput subtended by the projected lens aperture.

7. The scale of the Fourier transform is under the control of theexperimenter. As d increases, larger transform size. As ddecreases, smaller transform size.


5.2.4 Example of an optical Fourier transform

1. Figure 5.7 shows a transparent character 3 and thecorresponding energy spectrum.


5.3 Image Formation: Monochromatic Illumination

1. The most familiar property of lenses is their ability to formimages, which are the distributions of light intensity thatclosely resembles the objects.

2. The image may be real in the sense that an actual distributionof intensity appears across a plane behind lens, or it may bevirtual in the sense that the light behind the lens appears tooriginate from an intensity distribution across a new plane infront of the lens.

3. Here we consider image formation in only a limited context:(1) we restrict attention to a positive, aberration-free thin lensthat forms a real image.(2) we consider only monochromatic illumination, arestriction implying that the imaging system is linear incomplex field amplitude.


5.3.1 The impulse response of a positive lens - 1

1. Referring to the geometry of Fig. 5.8, our purpose is to findthe conditions under which the field distribution Ui canreasonably be said to be an “image” of the object distributionUo .

2. To express the field Ui by the superposition integral:

Ui(u, v ) =

∫ ∫ ∞

−∞

h(u, v ; ξ, η)Uo(ξ, η)dξdη,

where h(u, v ; ξ, η) is the field amplitude produced atcoordinates (u, v ) by a unit-amplitude point source applied atobject coordinates (ξ, η).

3. The properties of the imaging system will be completelydescribed if the impulse response h can be specified.


Figure 5.8



4. To produce high quality images, Ui must be as similar aspossible to U0. Equivalently, the impulse response shouldclosely approximate a Dirac delta function,

h(u, v ; ξ, η) ≈ Kδ(u ±Mξ, v ±Mη),

where K is a complex constant, M represents the systemmagnification.



5. To find the impulse response h, let the object be a δ function(point source) at coordinates (ξ, η). Then incident on the lenswill appear a spherical wave diverging from the point (ξ, η).The paraxial approximation to that wave is

Ul(x , y ) =1

jλz1exp

{

jk

2z1[(x − ξ)2 + (y − η)2]

}

. (5− 25)

6. After passing through a lens (focal length f ),

U ′l (x , y ) = Ul(x , y )P(x , y ) exp[−jk

2f(x2 + y 2)]. (5− 26)



7. Finally, using the Fresnel diffraction equation (4-14) toaccount for propagation over distance z2,

h(u, v ; ξ, η) =1

jλz2

∫ ∫ ∞

−∞

U ′l (x , y )

× exp{j k2z2

[(u − x)2 + (v − y )2]}dxdy ,

(5− 27)where constant phase factors have been dropped.



8. Combining Eqs. (5-25)–(5-27) and neglecting a pure phasefactor,

h(u, v ; ξ, η) =1

λ2z1z2exp[j

k

2z2(u2 + v 2)] exp[j

k

2z1(ξ2 + η2)]

×∫ ∫ ∞

−∞

P(x , y ) exp[jk

2(1

z 1+

1

z 2− 1

f)(x2 + y 2)]

× exp{−jk[(ξz 1

+u

z 2)x + (

η

z 1+

v

z 2)y ]}dxdy .

This is the relation between the object Uo and the image Ui .


5.3.2 Eliminating Quadratic Phase Factor: The Lens Law -

1

1. Two quadratic phase terms are independent of the lenscoordinates (x , y ):

exp[jk

2z2(u2 + v 2)] and exp[j

k

2z1(ξ2 + η2)],

while one term depends on the lens coordinates:

exp[jk

2(1

z 1+

1

z 2− 1

f)(x2 + y 2)]

2. We first choose the distance z2 to the image plane so that theterm in the last term above will vanish. That is,

1

z 1+

1

z 2− 1

f= 0

This is the classical lens law .Prof. Hsuan-Ting Chang Chapter 5 Wave-Optics Analysis of Coherent Optical Systems


2

3. Consider the quadratic phase factor that depends only on thecoordinates (u, v ). This term can be ignored under either oftwo conditions:

3.1 It is the intensity distribution in the image plane that is ofinterest, in which case the phase distribution associated withthe image is of no consequence. =⇒ A very usual case.

3.2 The image field distribution is of interest, but the image ismeasured on a spherical surface, centered at the point wherethe optical axis pierces the thin lens, and of radius z2.



3

4. Finally, consider the quadratic phase factor in the objectcoordinates (ξ, η). It has the potential to affect the result ofthat integration significantly. There are three differentconditions under which this term can be neglected:

4.1 The object exists on the surface of a sphere of radius z1centered on the point where the optical axis pierces the thinlens.

4.2 The object is illuminated by a spherical wave that isconverging towards the point where the optical axis pierces thelens (Fig. 5.9).

4.3 The phase of the quadratic phase factor changes by an amountthat is only a small fraction of a radian within the region ofthe object that contributes significantly to the field at theparticular image point (u, v) (Fig. 5.10).


Figure 5.9


Figure 5.10



4

5. The end result of these arguments is a simplified expression forthe impulse response of the imaging system.

h(u, v ; ξ, η) =1

λ2z1z2

∫ ∫ ∞

−∞

P(x , y )

× exp{−jk[( ξz1

+u

z2)x + (

η

z1+

v

z2)y ]}dxdy .



5

6. Defining the magnification of the system by M = − z2z1, the

minus sign being included to remove the effects of imageinversion, we find a final simplified form for the impulseresponse,

h(u, v ; ξ, η) =1

λ2z1z2

∫ ∫ ∞

−∞

P(x , y )

× exp{−j 2πλz2

[(u −Mξ)x + (v −Mη)y ]}dxdy .

(5− 33)



6

7. Thus, if the lens law is satisfied, the impulse response is seento be given (up to an extra scaling factor 1/λz1) by theFraunhofer diffraction pattern of the lens aperture, centeredon image coordinates (u = Mξ, v = Mη).


5.3.3 The Relation Between Object and Image - 1

1. If the imaging system is perfect, then the image is simply aninverted and scaled replication of the object. Thus accordingto geometric optics, the image and object would be related by

Ui(u, v ) =1

|M |Uo(u

M,v

M).

2. Under this case, we have

h(u, v ; ξ, η) −→ 1|M |δ(ξ −u

M, η − v

M).

3. To include the effects of diffraction, we return to theexpression (5-33) for the impulse response of the imagingsystem. The impulse response is that of a linearspace-variant system, so the object and image are related bya superposition integral but not by a convolution integral.



4. To reduce the object-image relation to a convolution equation,we must normalize the object coordinates to remove inversionand magnification. Let ξ̂ = Mξ and η̂ = Mη. Equation (5-33)can be reduced to

h(u, v ; ξ̂, η̂) =1

λ2z1z2

∫ ∫ ∞

−∞

P(x , y )

× exp{−j 2πλz2

[(u − ξ̂)x + (v − η̂)y ]}dxdy , (5− 36)

which only depends on the differences of coordinates(u − ξ̂, v − η̂).



5. Let x̂ = xλz2

, ŷ = yλz2

, ĥ = 1|M|

h. Then the object-image

relationship becomes

Ui(u, v ) =

∫ ∫ ∞

−∞

ĥ(u − ξ̂, v − η̂)[ 1|M |Uo(ξ̂

M,η̂

M)]d ξ̂d η̂,

orUi(u, v ) = ĥ(u, v )⊗ Ug (u, v )

where

Ug (u, v ) =1

|M |Uo(u

M,v

M)

is the geometric optics prediction of the image, and

ĥ(u, v ) =

∫ ∫ ∞

−∞

P(λz2x̂ , λz2ŷ ) exp[−j2π(ux̂ + v ŷ)]dx̂d ŷ

is the point-spread function introduced by diffraction.Prof. Hsuan-Ting Chang Chapter 5 Wave-Optics Analysis of Coherent Optical Systems


6. Two main conclusions obtained:(1) The ideal image produced by a diffraction-limited opticalsystem is a scaled and inverted version of the object.(2) The effect of diffraction is to convolve that ideal imagewith the Fraunhofer diffraction pattern of the lens pupil.

7. The smoothing operation associated with the convolution canstrongly attenuate the fine details of the object, with acorresponding loss of image fidelity resulting.


5.4 Analysis of Complex Coherent Optical Systems

◮ The number of integrations grows as the number offree-space regions grows, and the complexity of thecalculations increases as the number of lenses includedgrows.

◮ The introduction of a certain “operator” notation that isuseful in analyzing complex systems.


5.4.1 An Operator Notation - 1

1. Several simplifying assumptions are used here:◮ We restrict attention to monochromatic light, that is, limit

consideration to what we call “coherent” systems.◮ Only paraxial conditions will be considered.◮ We will treat the problems in this section as 1-D problems

rather than 2-D problems.

2. Most operators have parameters that depend on the geometryof the optical system being analyzed.

3. Parameters are included within square brackets [ ] followingthe operator. The operators act on the quantities contained incurly brackets { }.



4. Basic operators are given as follows:◮ Multiplication by a quadratic-phase exponential. The

operator Q is defined as

Q[c]{U(x)} = e j k2 cx2U(x),

where k = 2π/λ and c is an inverse length. The inverse ofQ[c] is Q[−c].

◮ Scaling by a constant. Symbol: V ,

V [b]{U(x)} = b1/2U(bx),

where b is dimensionless. The inverse of V [b] is V [1/b].



◮ Fourier transformation. Symbol: F

F{U(x)} =∫

∞

−∞

U(x)e−j2πfxdx .

◮ Free-space propagation. Symbol: R

R[d ]{U(x1)} =1√jλd

∫

∞

−∞

U(x1)ej k2d (x2−x1)

2

dx1,

where d is the distance of propagation and x2 is the coordinatethat applies after propagation . The inverse of R[d ] is R[−d ].



5. Some simple and useful properties are listed below:5.1 V [t2]V [t1] = V [t2t1]5.2 FV[t] = V [ 1

t]F ⇒ A statement of the similarity theorem of

Fourier analysis.5.3 FF = V [−1] ⇒ Follows from the Fourier inversion theorem,

slightly modified to account for the fact that both transformsare in the forward direction.

5.4 Q[c2]Q[c1] = Q[c2 + c1]5.5 R[d ] = F−1Q[−λ2d ]F ⇒ A statement that free-space

propagation over distance d can be analyzed either by aFresnel diffraction equation or by a sequence of Fouriertransformation, multiplication by the transfer function of freespace, and inverse Fourier transformation.

5.6 Q[c]V [t] = V [t]Q[ ct2]



6. Two slightly more sophisticated relations are

R[d ] = Q[ 1d]V[ 1

λd]FQ[ 1

d], (5.51)

which is a statement that the Fresnel diffraction operation isequivalent to premultiplication by a quadratic-phaseexponential, a properly scaled Fourier transform, andpostmultiplication by a quadratic-phase exponential, and

V[ 1λf

]F = R[f ]Q[−1f]R[f ],

which is a statement that the fields across the front and backfocal planes of a positive lens are related by a properly scaledFourier transform, with no quadratic-phase exponentialmultiplier.



7. Table 5.1 summarizes many useful relations between operators.


5.4.2 Application of the operator approach to some optical

systems - 1

1. Figure 5.11 shows the first example: The goal is to determinethe relationship between the complex field across a plane S1just to the left of lens L1, and the complex field across a planeS2 just to the right of the lens L2.

2. The first operation on the wave takes place as it passesthrough L1. → Q[−1f ]

3. The second operation is propagation through space overdistance f . → R[f ]

4. The third operation is passage through the lens L2. → Q[−1f ]


Figure 5.11



systems - 2

5. The entire sequence of operations can be represented by asystem operation S,

S = Q[−1f]R[f ]Q[−1

f]

By means of Eq. (5.51),

S = Q[−1f]Q[1

f]V[ 1

λf]FQ[1

f]Q[−1

f]

= V[ 1λf

]F ,

where the relations Q[−1f]Q[1

f] = Q[1

f]Q[−1

f] = 1 have been

used to simplify the equation.



systems - 2

6. This system of two lenses separated by their common focallength f performs a scaled optical Fourier transform, withoutquadratic-phase exponentials in the result, similar to thefocal-plane-to-focal-plane relationship derived earlier.

7. The result explicitly in terms of the input and output fields,

Uf (u) =1√λf

∫ ∞

−∞

U0(x)e−j k

fxudx ,

where U0 is the field just to the left of L1 and Uf is the fieldjust to the right of L2.



systems - 3

8. The second example that contains only a single lens is shownin Fig. 5.12.

9. Here the object or the input to the system, located distance dto the left of the lens, is illuminated by a diverging sphericalwave, emanating from a point that is distance z1 > d to theleft of the lens.

10. The output of interest here will be in the plane where thepoint source is imaged, at distance z2 to the right of the lens,where z1, z2, and the focal length f of the lens satisfy the lenslaw, z−11 + z

−12 − f −1 = 0.


Figure 5.12



systems - 4

11. The system of operators describing this system is

S = R[z2]Q[−1

f]R[d ]Q[ 1

z1 − d]

◮ Q[ 1z1−d

] represents the fact that the input is illuminated by adiverging spherical wave.

◮ R[d ] represents propagation over distance d to the lens.◮ Q[− 1

f] represents the effect of the positive lens.

◮ R[z2] represents the final propagation over distance z2.12. Apply the lens law immediately, replacing Q[−1

f] by

Q[−z−11 − z−12 ].



systems - 5

13. There are several different ways to simplify this sequence ofoperators.

14. Use the relationship in the 4th row and 3rd column of Table5.1.

R[z2]Q[−1

z 1− 1

z 2] = Q[z1 + z2

z22]V[−z1

z2]R[−z1].

15. The two remaining adjacent R operators can now becombined using the relation given in 4th row and 4th columnof Table 5.1.

S = Q[z1 + z2z22

]V[−z1z2]R[d − z1]Q[

1

z1 − d].



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16. Next Eq. (5.51) is applied to write

R[d − z1] = Q[1

d − z1]V[ 1

λ(d − z1)]FQ[ 1

d − z1].

17. Substitution of this result yields an operation system

S = Q[z1 + z2z22

]V[−z1z2]Q[ 1

d − z1]V[ 1

λ(d − z1)]F .

18. The last steps are to apply the relation (5.50) to invert theorder of V and Q operators in the middle of the chain,following which the two adjacent V operators and the twoadjacent Q operators can be combined. The final resultbecomes

S = Q[

(z1 + z2)d − z1z2z22 (d − z1)

]

V[

z1

λz2(z1 − d)

]

F .



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19. A more conventional statement of the relationship between inthe input field U1(ξ) and the output field U2(u) is

U2(u) =exp[j k

2(z1+z2)d−z1z2

z22 (d−z1)u2]

√

λz2(z1−d)z1

∫ ∞

−∞

U1(ξ) exp[−j2πz1

λz2(z1 − d)uξ]dξ.

(5.57)The field U2(u) is again seen to be a Fourier transform of theinput amplitude distribution.



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20. The results reveal some important general facts not explicitlyevident in our earlier analyses: The Fourier transform planeneed not be the focal plane of the lens performing the

transform! Rather, the Fourier transform always appears in the

plane where the source is imaged.

21. The quadratic-phase factor preceding the Fourier transformoperation is always the quadratic-phase factor that wouldresult at the transform plane from a point source of lightlocated on the optical axis in the plane of the inputtransparency.



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22. A few general comments about the operator method ofanalysis:Advantage – it allows a methodical approach to complexcalculations that might otherwise be difficult to treat by theconventional methods.Drawbacks –(1) Being one step more abstract than the diffraction integralsit replaces, the operator method is one step further from thephysics of the experiment under analysis.(2) To save time with the operator approach, it is necessarythat one be rather familiar with the operator relations of Table5.1. Good intuition about which operation relations to use ona given problem comes only after experience with the method.


Chapter 5 Wave-Optics Analysis of Coherent Optical...

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