Chapter 5 Wave-Optics Analysis of Coherent Optical Systems

Transcription

1 Chapter 5 Wave-Optics Analysis of Coherent Optical Systems January 5, 2016

2 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

3 5.1 A thin lens as a phase transformation 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 coordinates on the opposite face, i.e., if there is negligible translation of a ray within the lens. 2. A thin lens simply delays an incident wavefront by an amount proportional to the thickness of the lens at each point. 3. Referring to Fig. 5.1, the total phase delay suffered by the wave at coordinates (x,y) in passing through the lens may be written 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 free space between two planes.

4 Fig 5.1

5 5.1 A thin lens as a phase transformation Equivalently the lens may be represented by a multiplicative phase transformation of the form t l (x,y) = exp[jk 0 ]exp[jk(n 1) (x,y)]. (5.1) 5. The complex field U l across a plane immediately behind the lens is then related to the complex field U l (x,y) incident on a plane immediately in front of the lens by U l (x,y) = t l(x,y)u l (x,y).

6 5.1.1 The thickness function We first adopt a sign convention: as ray travel from left to right, each convex surface encountered is taken to have a positive radius of curvature, while each concave surface is taken to have a negative radius of curvature. 2. Figure 5.2 shows the split three parts of a lens. The total thickness 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 (R 1 R1 2 x2 y 2 ) ) = 01 R 1 (1 1 x2 +y 2. R The second part comes from a region of glass of constant thickness 02.

7 Fig 5.2

8 5.1.1 The thickness function The third component is given by 3 (x,y) = 03 ( R 2 R2 2 x2 y 2 ) = 03 +R 2 (1 1 x2 +y 2 ). R Combining the three expressions for thickness, the total thickness is seen to be (x,y) = 0 R 1 (1 1 x2 +y 2 )+R R1 2 2 (1 1 x2 +y 2 ), R2 2 where 0 =

9 5.1.2 The paraxial approximation 1. If only paraxial rays are considered, that is, only values of x and y sufficiently small to allow the following approximations to be accurate: 1 x2 +y 2 1 x2 +y 2 R1 2 2R1 2 1 x2 +y 2 R x2 +y 2. 2R With the help of these approximations, the thickness function becomes (x,y) = 0 x2 +y 2 ( 1 1 ). (5.8) 2 R 1 R 2

10 5.1.3 The phase transformation and its physical meaning Substitution of Eq. (5-8) into Eq. (5-1) yields the following approximation to the lens transformation: t l (x,y) = exp[jkn 0 ] exp [ jk(n 1) x2 +y 2 ( 1 1 ] ). 2 R 1 R 2 2. The physical properties of the lens (n,r 1,R 2 ) can be combined in a single number f called the focal length, which is defined by 1 f (n 1)( 1 R 1 1 R 2 ).

11 5.1.3 The phase transformation and its physical meaning Neglecting the constant phase factor e jkn 0, the phase transformation becomes t l (x,y) = exp[ j k 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 various combinations of convex and concave surfaces. The focal length f of a double-convex plano-convex, or positive meniscus lens is positive, while that of a double-concave, plano-concave, or negative meniscus lens is negative.

12 Fig 5.3

13 5.1.3 The phase transformation and its physical meaning Consider the effect of the lens on a normally incident, unit-amplitude plane wave. The field distribution U l in front of the lens is unit, and U l behind the lens is U l = exp[ j k 2f (x2 +y 2 )]. This expression can be interpreted as a quadratic approximation to a spherical wave.

14 5.1.3 The phase transformation and its physical meaning If f > 0, then the spherical wave is converging towards a point on the lens axis a distance f behind the lens. If f < 0, the spherical wave is diverging from a point on the lens axis a distance f in front of the lens. (See Fig. 5.4) 7. Under nonparaxial conditions, the emerging wavefront will exhibit departures from perfect sphericity (called aberrations), even if the surfaces of the lens are perfectly spherical.

15 Fig 5.4

16 5.2 Fourier Transforming Properties of Lenses 1 1. One of the most remarkable and useful properties of a converging lens is its inherent ability to perform 2-D Fourier transform. 2. Several different configurations for performing the transform operation are described. In all cases the illumination is assumed to be monochromatic coherent systems. 3. The information to be Fourier-transformed is introduced into the optical system by a device with an amplitude transmittance that is proportional to the input function of interests. The device may consist of a photographic transparency or a nonphotographic spatial light modulator (SLM), capable of controlling the amplitude transmittance in response to externally supplied electrical or optical information.

17 5.2 Fourier Transforming Properties of Lenses 2 4. Figure 5.5 shows three arrangements that will be considered.

18 5.2.1 Input placed against the lens In this case, the disturbance on the lens is U l (x,y) = At A (x,y). 2. The finite extent of the lens can be represented by a pupil function P(x,y) defined by { 1, inside the lens aperture P(x,y) = 0, otherwise 3. The amplitude distribution behind the lens becomes U l(x,y) = U l (x,y)p(x,y)exp[ j k 2f (x2 +y 2 )].

19 5.2.1 Input placed against the lens To find the distribution U f (u,v) in the back focal plane of the lens, the Fresnel diffraction formula (Eq. 4-17) is applied. Thus, putting z = f, U f (u,v) = exp[j k 2f (u2 +v 2 )] jλf U l (x,y)exp[j k 2f (x2 +y 2 )] exp[ j 2π (xu +yv)]dxdy, (5 13) λf where a constant phase factor e jkf has been dropped.

20 5.2.1 Input placed against the lens Substituting Eq. (5-12) in Eq. (5-13), the quadratic phase factors within the integrand are seen to exactly cancel, leaving U f (u,v) = exp[j k 2f (u2 +v 2 )] jλf U l (x,y)p(x,y)exp[ j 2π (xu +yv)]dxdy. λf (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 U f (u,v) = exp[j k 2f (u2 +v 2 )] jλf U l (x,y)exp[ j 2π (xu +yv)]dxdy. λf

21 5.2.1 Input placed against the lens The complex amplitude distribution of the field in the focal plane of the lens is the Fraunhofer diffraction pattern of the field incident on the lens, even though the distance to the observation plane is equal to the focal length of the lens, rather than satisfying the usual distance criterion for observing Fraunhofer diffraction. 8. The Fourier transform relation between the input transmittance and the focal-plane amplitude distribution is not a complete one, due to the presence of the quadratic phase factor.

22 5.2.1 Input placed against the lens In most cases only the intensity across the focal plane is of real interest, and the phase distribution is of no consequence. Measurement of the intensity distribution yields knowledge of the power spectrum (or the energy spectrum) of the input. Thus I f (u,v) = A2 λ 2 f 2 t A (x,y)exp[ j 2π λf (xu +yv)]dxdy 2.

23 5.2.2 Input placed in front of the lens Consider the more general geometry of Fig. 5.5(b). The input is located a distance d in front of the lens. Let F 0 (f X,f Y ) represent the Fourier spectrum of the light transmitted by the input transparency, and F l (f X,f Y ) the Fourier spectrum of the light incident on the lens; that is, F 0 (f X,f Y ) = F{At A }, F l (f X,f Y ) = F{U l }. 2. Assuming that the Fresnel or paraxial approximation is valid for propagation over distance d, then using Eq. (4-21), giving F l (f X,f Y ) = F 0 (f X,f Y )exp[ jπλd(f 2 X +f2 Y )], where we have dropped a constant phase delay e jkd.

24 5.2.2 Input placed in front of the lens Letting P = 1, Eq. (5-14) can be rewritten U f (u,v) = exp[j k 2f (u2 +v 2 )] F l ( u jλf λf, v λf ) = exp[j k (1 d 2f f )(u2 +v 2 )] F 0 ( u jλf λf, v λf ) = Aexp[j k (1 d 2f f )(u2 +v 2 )] jλf t A (ξ,η)exp[ j 2π (ξu +ηv)]dξdη. λf

25 5.2.2 Input placed in front of the lens A quadratic phase factor again precedes the transform integral, 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 lens aperture. To include the effects of this aperture, we use a geometric optics approximation. That is, the distance d is sufficiently small to place the input deep within the region of Fresnel diffraction of the lens aperture, if the light were propagating backwards from the focal plane to the plane of input transparency. 6. Figure 5.6, the light amplitude at coordinates (u 1,v 1 ) is a summation of all the rays traveling with direction cosines (ξ u 1 /f,η v 1 /f). However, only a finite set of these rays is passed by the lens aperture.

26 Figure 5.6

27 5.2.2 Input placed in front of the lens The projected lens aperture limits the effective extent of the input, but the particular portion of t A that contributes to the field U f depends on the particular coordinates (u 1,v 1 ) being considered in the back focal plane. 8. The value of U f at (u,v) can be found from the Fourier transform of that portion of the input subtended by the projected pupil function P, centered at coordinates [ξ = (d/f)u,η = (d/f)v]. U f (u,v) = Aexp[j k (1 d 2f f )(u2 +v 2 )] jλf t A (ξ,η)p(ξ + d f u,η + d f v)e j 2π λf (ξu+ηv) dξdη.

28 5.2.2 Input placed in front of the lens The limitation of the effective input by the finite lens aperture is known as a vignetting effect. For a simple Fourier transform system, vignetting of the input space is minimized when the input is placed close to the lens and when the lens aperture is much larger than the input transparency. 10. In practice, it is often preferred to place the input directly against the lens in order to minimize vignetting, although in analysis it is generally convenient to place the input in the front focal plane, where the transform relation is unencumbered with quadratic phase factors.

29 5.2.3 Input placed behind the lens Consider Fig. 5.5(c). The input is now located a distance d in front of the rear focal plane of the lens. 2. In the geometric optics approximation, the amplitude of the spherical wave impinging on the object is Af/d. 3. The particular region of the input that is illuminated is determined by the intersection of the converging cone of rays with the input plane. If the lens is circular and of diameter l, then a circular region of diameter ld/f is illuminated on the input. This effective region can be described by the pupil function P[ξ(f/d),η(f/d)]. 4. Using a paraxial approximation to the spherical wave that illuminates the input, the amplitude of the wave transmitted by the input { Af U 0 (ξ,η) = d P(ξf d,ηf d )exp[ j k } 2d (ξ2 +η 2 )] t A (ξ,η).

30 5.2.3 Input placed behind the lens Assuming Fresnel diffraction from the input plane to the focal plane, Eq. (4.17) can be applied to the field transmitted by the input. U f (u,v) = Aexp[j k 2d (u2 +v 2 )] f jλd d t A (ξ,η)p(ξ f d,ηf d )exp[ j2π (ξu +ηv)]dξdη. λf 6. Up to a quadratic phase factor, the focal-plane amplitude distribution is the Fourier transform of that portion of the input subtended by the projected lens aperture. 7. The scale of the Fourier transform is under the control of the experimenter. As d increases, larger transform size. As d decreases, smaller transform size.

31 5.2.4 Example of an optical Fourier transform 1. Figure 5.7 shows a transparent character 3 and the corresponding energy spectrum.

32 5.3 Image Formation: Monochromatic Illumination 1. The most familiar property of lenses is their ability to form images, which are the distributions of light intensity that closely resembles the objects. 2. The image may be real in the sense that an actual distribution of intensity appears across a plane behind lens, or it may be virtual in the sense that the light behind the lens appears to originate from an intensity distribution across a new plane in front of the lens. 3. Here we consider image formation in only a limited context: (1) we restrict attention to a positive, aberration-free thin lens that forms a real image. (2) we consider only monochromatic illumination, a restriction implying that the imaging system is linear in complex field amplitude.

33 5.3.1 The impulse response of a positive lens Referring to the geometry of Fig. 5.8, our purpose is to find the conditions under which the field distribution U i can reasonably be said to be an image of the object distribution U o. 2. To express the field U i by the superposition integral: U i (u,v) = h(u,v;ξ,η)u o (ξ,η)dξdη, where h(u,v;ξ,η) is the field amplitude produced at coordinates (u, v) by a unit-amplitude point source applied at object coordinates (ξ, η). 3. The properties of the imaging system will be completely described if the impulse response h can be specified.

34 Figure 5.8

35 5.3.1 The impulse response of a positive lens To produce high quality images, U i must be as similar as possible to U 0. Equivalently, the impulse response should closely approximate a Dirac delta function, h(u,v;ξ,η) Kδ(u ±Mξ,v ±Mη), where K is a complex constant, M represents the system magnification.

36 5.3.1 The impulse response of a positive lens To find the impulse response h, let the object be a δ function (point source) at coordinates (ξ, η). Then incident on the lens will appear a spherical wave diverging from the point (ξ, η). The paraxial approximation to that wave is U l (x,y) = 1 jλz 1 exp { } k j [(x ξ) 2 +(y η) 2 ]. (5 25) 2z 1 6. After passing through a lens (focal length f), U l (x,y) = U l(x,y)p(x,y)exp[ j k 2f (x2 +y 2 )]. (5 26)

37 5.3.1 The impulse response of a positive lens Finally, using the Fresnel diffraction equation (4-14) to account for propagation over distance z 2, 1 h(u,v;ξ,η) = U l jλz (x,y) 2 k exp{j [(u x) 2 +(v y) 2 ]}dxdy, 2z 2 (5 27) where constant phase factors have been dropped.

38 5.3.1 The impulse response of a positive lens Combining Eqs. (5-25) (5-27) and neglecting a pure phase factor, h(u,v;ξ,η) = 1 k exp[j (u 2 +v 2 k )]exp[j (ξ 2 +η 2 )] λ 2 z 1 z 2 2z 2 2z 1 P(x,y)exp[j k 2 ( z 1 z 2 f )(x2 +y 2 )] exp{ jk[( ξ z 1 + u )x +( η + v z 2 z 1 z 2)y]}dxdy. This is the relation between the object U o and the image U i.

39 5.3.2 Eliminating Quadratic Phase Factor: The Lens Law Two quadratic phase terms are independent of the lens coordinates (x, y): k exp[j (u 2 +v 2 k )] and exp[j (ξ 2 +η 2 )], 2z 2 2z 1 while one term depends on the lens coordinates: exp[j k 2 ( z 1 z 2 f )(x2 +y 2 )] 2. We first choose the distance z 2 to the image plane so that the term in the last term above will vanish. That is, This is the classical lens law z 1 z 2 f = 0

40 5.3.2 Eliminating Quadratic Phase Factor: The Lens Law Consider the quadratic phase factor that depends only on the coordinates (u,v). This term can be ignored under either of two conditions: 3.1 It is the intensity distribution in the image plane that is of interest, in which case the phase distribution associated with the image is of no consequence. = A very usual case. 3.2 The image field distribution is of interest, but the image is measured on a spherical surface, centered at the point where the optical axis pierces the thin lens, and of radius z 2.

41 5.3.2 Eliminating Quadratic Phase Factor: The Lens Law Finally, consider the quadratic phase factor in the object coordinates (ξ,η). It has the potential to affect the result of that integration significantly. There are three different conditions under which this term can be neglected: 4.1 The object exists on the surface of a sphere of radius z 1 centered on the point where the optical axis pierces the thin lens. 4.2 The object is illuminated by a spherical wave that is converging towards the point where the optical axis pierces the lens (Fig. 5.9). 4.3 The phase of the quadratic phase factor changes by an amount that is only a small fraction of a radian within the region of the object that contributes significantly to the field at the particular image point (u, v) (Fig. 5.10).

42 Figure 5.9

43 Figure 5.10

44 5.3.2 Eliminating Quadratic Phase Factor: The Lens Law The end result of these arguments is a simplified expression for the impulse response of the imaging system. 1 h(u,v;ξ,η) = P(x,y) λ 2 z 1 z 2 exp{ jk[( ξ z 1 + u z 2 )x +( η z 1 + v z 2 )y]}dxdy.

45 5.3.2 Eliminating Quadratic Phase Factor: The Lens Law Defining the magnification of the system by M = z 2 z 1, the minus sign being included to remove the effects of image inversion, we find a final simplified form for the impulse response, 1 h(u,v;ξ,η) = P(x,y) λ 2 z 1 z 2 exp{ j 2π λz 2 [(u Mξ)x +(v Mη)y]}dxdy. (5 33)

46 5.3.2 Eliminating Quadratic Phase Factor: The Lens Law Thus, if the lens law is satisfied, the impulse response is seen to be given (up to an extra scaling factor 1/λz 1 ) by the Fraunhofer diffraction pattern of the lens aperture, centered on image coordinates (u = Mξ,v = Mη).

47 5.3.3 The Relation Between Object and Image If the imaging system is perfect, then the image is simply an inverted and scaled replication of the object. Thus according to geometric optics, the image and object would be related by 2. Under this case, we have U i (u,v) = 1 M U o( u M, v M ). h(u,v;ξ,η) 1 M δ(ξ u M,η v M ). 3. To include the effects of diffraction, we return to the expression (5-33) for the impulse response of the imaging system. The impulse response is that of a linear space-variant system, so the object and image are related by a superposition integral but not by a convolution integral.

48 5.3.3 The Relation Between Object and Image To reduce the object-image relation to a convolution equation, we must normalize the object coordinates to remove inversion and magnification. Let ˆξ = Mξ and ˆη = Mη. Equation (5-33) can be reduced to h(u,v; ˆξ,ˆη) 1 = P(x,y) λ 2 z 1 z 2 exp{ j 2π λz 2 [(u ˆξ)x +(v ˆη)y]}dxdy, (5 36) which only depends on the differences of coordinates (u ˆξ,v ˆη).

49 5.3.3 The Relation Between Object and Image Let ˆx = x λz 2,ŷ = y λz 2,ĥ = 1 h. Then the object-image M relationship becomes U i (u,v) = ĥ(u ˆξ,v 1 ˆη)[ M U o( ˆξ M, ˆη M )]dˆξdˆη, or where U i (u,v) = ĥ(u,v) U g(u,v) U g (u,v) = 1 M U o( u M, v M ) is the geometric optics prediction of the image, and ĥ(u,v) = P(λz 2ˆx,λz 2 ŷ)exp[ j2π(uˆx +vŷ)]dˆxdŷ is the point-spread function introduced by diffraction.

50 5.3.3 The Relation Between Object and Image Two main conclusions obtained: (1) The ideal image produced by a diffraction-limited optical system is a scaled and inverted version of the object. (2) The effect of diffraction is to convolve that ideal image with the Fraunhofer diffraction pattern of the lens pupil. 7. The smoothing operation associated with the convolution can strongly attenuate the fine details of the object, with a corresponding loss of image fidelity resulting.

51 5.4 Analysis of Complex Coherent Optical Systems The number of integrations grows as the number of free-space regions grows, and the complexity of the calculations increases as the number of lenses included grows. The introduction of a certain operator notation that is useful in analyzing complex systems.

52 5.4.1 An Operator Notation 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 geometry of the optical system being analyzed. 3. Parameters are included within square brackets [ ] following the operator. The operators act on the quantities contained in curly brackets { }.

53 5.4.1 An Operator Notation Basic operators are given as follows: Multiplication by a quadratic-phase exponential. The operator Q is defined as Q[c]{U(x)} = e j k 2 cx2 U(x), where k = 2π/λ and c is an inverse length. The inverse of Q[c] is Q[ c]. Scaling by a constant. Symbol: V, V[b]{U(x)} = b 1/2 U(bx), where b is dimensionless. The inverse of V[b] is V[1/b].

54 5.4.1 An Operator Notation - 3 Fourier transformation. Symbol: F F{U(x)} = Free-space propagation. Symbol: R R[d]{U(x 1 )} = 1 jλd U(x)e j2πfx dx. U(x 1 )e j k 2d (x2 x1)2 dx 1, where d is the distance of propagation and x 2 is the coordinate that applies after propagation. The inverse of R[d] is R[ d].

55 5.4.1 An Operator Notation Some simple and useful properties are listed below: 5.1 V[t 2 ]V[t 1 ] = V[t 2 t 1 ] 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 transforms are in the forward direction. 5.4 Q[c 2 ]Q[c 1 ] = Q[c 2 +c 1 ] 5.5 R[d] = F 1 Q[ λ 2 d]f A statement that free-space propagation over distance d can be analyzed either by a Fresnel diffraction equation or by a sequence of Fourier transformation, multiplication by the transfer function of free space, and inverse Fourier transformation. 5.6 Q[c]V[t] = V[t]Q[ c t ] 2

56 5.4.1 An Operator Notation Two slightly more sophisticated relations are R[d] = Q[ 1 d ]V[ 1 λd ]FQ[1 ], (5.51) d which is a statement that the Fresnel diffraction operation is equivalent to premultiplication by a quadratic-phase exponential, a properly scaled Fourier transform, and postmultiplication by a quadratic-phase exponential, and V[ 1 λf ]F = R[f]Q[ 1 f ]R[f], which is a statement that the fields across the front and back focal planes of a positive lens are related by a properly scaled Fourier transform, with no quadratic-phase exponential multiplier.

57 5.4.1 An Operator Notation Table 5.1 summarizes many useful relations between operators.

58 5.4.2 Application of the operator approach to some optical systems Figure 5.11 shows the first example: The goal is to determine the relationship between the complex field across a plane S 1 just to the left of lens L 1, and the complex field across a plane S 2 just to the right of the lens L The first operation on the wave takes place as it passes through L 1. Q[ 1 f ] 3. The second operation is propagation through space over distance f. R[f] 4. The third operation is passage through the lens L 2. Q[ 1 f ]

59 Figure 5.11

60 5.4.2 Application of the operator approach to some optical systems The entire sequence of operations can be represented by a system operation S, By means of Eq. (5.51), S = Q[ 1 f ]R[f]Q[ 1 f ] S = Q[ 1 f ]Q[1 f ]V[ 1 λf ]FQ[1 f ]Q[ 1 f ] = V[ 1 λf ]F, where the relations Q[ 1 f ]Q[1] = f Q[1 f ]Q[ 1 ] = 1 have been f used to simplify the equation.

61 5.4.2 Application of the operator approach to some optical systems This system of two lenses separated by their common focal length f performs a scaled optical Fourier transform, without quadratic-phase exponentials in the result, similar to the focal-plane-to-focal-plane relationship derived earlier. 7. The result explicitly in terms of the input and output fields, U f (u) = 1 λf U 0 (x)e j k f xu dx, where U 0 is the field just to the left of L 1 and U f is the field just to the right of L 2.

62 5.4.2 Application of the operator approach to some optical systems The second example that contains only a single lens is shown in Fig Here the object or the input to the system, located distance d to the left of the lens, is illuminated by a diverging spherical wave, emanating from a point that is distance z 1 > d to the left of the lens. 10. The output of interest here will be in the plane where the point source is imaged, at distance z 2 to the right of the lens, where z 1,z 2, and the focal length f of the lens satisfy the lens law, z1 1 +z2 1 f 1 = 0.

63 Figure 5.12

64 5.4.2 Application of the operator approach to some optical systems The system of operators describing this system is S = R[z 2 ]Q[ 1 f ]R[d]Q[ 1 z 1 d ] Q[ 1 z 1 d ] represents the fact that the input is illuminated by a diverging 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 z Apply the lens law immediately, replacing Q[ 1 ] by f Q[ z1 1 z2 1 ].

65 5.4.2 Application of the operator approach to some optical systems There are several different ways to simplify this sequence of operators. 14. Use the relationship in the 4th row and 3rd column of Table 5.1. R[z 2 ]Q[ 1 1 ] = Q[ z 1 +z 2 ]V[ z 1 ]R[ z z 1 z 2 z2 2 1 ]. z The two remaining adjacent R operators can now be combined using the relation given in 4th row and 4th column of Table 5.1. S = Q[ z 1 +z 2 ]V[ z 1 1 ]R[d z z2 2 1 ]Q[ z 2 z 1 d ].

66 5.4.2 Application of the operator approach to some optical systems Next Eq. (5.51) is applied to write 1 1 R[d z 1 ] = Q[ ]V[ d z 1 λ(d z 1 ) ]FQ[ 1 ]. d z Substitution of this result yields an operation system S = Q[ z 1 +z 2 ]V[ z ]Q[ ]V[ z2 2 z 2 d z 1 λ(d z 1 ) ]F. 18. The last steps are to apply the relation (5.50) to invert the order of V and Q operators in the middle of the chain, following which the two adjacent V operators and the two adjacent Q operators can be combined. The final result becomes [ ] [ ] (z1 +z 2 )d z 1 z 2 z 1 S = Q z2 2(d z V F. 1) λz 2 (z 1 d)

67 5.4.2 Application of the operator approach to some optical systems A more conventional statement of the relationship between in the input field U 1 (ξ) and the output field U 2 (u) is U 2 (u) = exp[j k 2 (z 1 +z 2 )d z 1 z 2 z 2 2 (d z 1) u 2 ] λz 2 (z 1 d) z 1 2πz 1 U 1 (ξ)exp[ j λz 2 (z 1 d) uξ]dξ. (5.57) The field U 2 (u) is again seen to be a Fourier transform of the input amplitude distribution.

68 5.4.2 Application of the operator approach to some optical systems The results reveal some important general facts not explicitly evident in our earlier analyses: The Fourier transform plane need 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 transform operation is always the quadratic-phase factor that would result at the transform plane from a point source of light located on the optical axis in the plane of the input transparency.

69 5.4.2 Application of the operator approach to some optical systems A few general comments about the operator method of analysis: Advantage it allows a methodical approach to complex calculations that might otherwise be difficult to treat by the conventional methods. Drawbacks (1) Being one step more abstract than the diffraction integrals it replaces, the operator method is one step further from the physics of the experiment under analysis. (2) To save time with the operator approach, it is necessary that one be rather familiar with the operator relations of Table 5.1. Good intuition about which operation relations to use on a given problem comes only after experience with the method.