Chapter 3 Foundations of Scalar Diffraction Theory

Prof. Hsuan-Ting Chang

September 10, 2019

Prof. Hsuan-Ting Chang Chapter 3 Foundations of Scalar Diffraction Theory Chapter 3 Foundations of Scalar Diffraction Theory

3.1 Historical Introduction 3.2 From a vector to a scalar theory 3.3 Some mathematical preliminaries 3.4 The Kirchhoff formulation of Diffraction by a planar screen 3.5 The Rayleigh-Sommerfeld formulation of diffraction 3.6 Comparison of the Kirchhoff and Rayleigh-Sommerfeld theories 3.7 Further discussion of the Huygens-Fresnel principle 3.8 Generalization of nonmonochromatic waves 3.9 Diffraction at boundaries 3.10 The angular spectrum of plane waves

Prof. Hsuan-Ting Chang Chapter 3 Foundations of Scalar Diffraction Theory 3.1 Historical Introduction

1. Refraction: the bending of light rays that takes place when they pass through a region in which there is a gradient of the local velocity of propagation of the wave. 2. Reflection: light rays are bent at a metallic or dielectric interface. The angle of reflection is always equal to the angle of incidence. 3. Diffraction: any deviation of light rays from rectilinear paths which cannot be interpreted as reflection or refraction. 4. Diffraction is caused by the confinement of the lateral extent of a wave, and is most appreciable when that confinement is to sizes comparable with a wavelength of the radiation being used.

Prof. Hsuan-Ting Chang Chapter 3 Foundations of Scalar Diffraction Theory Figure 3.3

The transition from light to shadow was gradual rather than abrupt.

Prof. Hsuan-Ting Chang Chapter 3 Foundations of Scalar Diffraction Theory Figure 3.4

If each point on the wavefront of a disturbance were considered to be a new source of a “secondary” spherical disturbance, then the wavefront at a later instant could be found by constructing the “envelope” of the secondary wavelets.

Prof. Hsuan-Ting Chang Chapter 3 Foundations of Scalar Diffraction Theory 3.2 From a Vector to a Scalar Theory – 1

1. Maxwell’s Equations:

∂H~ ▽ × E~ = −µ ∂t ∂E~ ▽ × H~ = ǫ (3.2) ∂t ▽ · ǫE~ = 0 ▽ · µH~ = 0

2. E~ : electric field; H~ : magnetic filed; µ and ǫ are the permeability and permittivity, respectively, of the medium in which the wave is propagating. 3. E~ and H~ are functions of both the position P and time t.

Prof. Hsuan-Ting Chang Chapter 3 Foundations of Scalar Diffraction Theory 3.2 From a Vector to a Scalar Theory – 2

4. The symbols × and · represent a vector cross product and a vector dot product, respectively, while ∂ ∂ ∂ ▽ = ~i + ~j + ~k, ∂x ∂y ∂z where ~i,~j and k~ are unit vectors in the x, y and z direction, respectively. 5. Properties of the medium in which the wave is propagating: isotropic – its properties are independent of the direction of ploarization of the wave, homogeneous – the permittivity is constant throughout the region of propagation, nondispersive – the permittivity is independent of wavelength over the wavelength region occupied by the propagation wave, nonmagnetic – the magnetic permeability is always equal to µ0, the vacuum permeability. Prof. Hsuan-Ting Chang Chapter 3 Foundations of Scalar Diffraction Theory 3.2 From a Vector to a Scalar Theory – 3

6. Apply the ▽× operation to the left and right sides of the first equation for ~ε, we make use of the vector identity

▽ × (▽ × ~ε)= ▽(▽ · ~ε) − ▽2~ε. (3.3)

7. If the propagation medium is linear, isotropic, homogeneous (constant ǫ), and nondispersive, substitution of the two Maxwell’s equations for ~ε in Eq. (3.3) yields

n2 ∂2~ε ▽2~ε − =0, (3.4) c2 ∂t2 where n is the refractive index of the medium.

Prof. Hsuan-Ting Chang Chapter 3 Foundations of Scalar Diffraction Theory 3.2 From a Vector to a Scalar Theory – 4

8. The magnetic field satisfies an identical equation,

n2 ∂2H~ ▽2H~ − =0. c2 ∂t2 9. Since the vector wave equation is obeyed by both ~ε and H~ , an identical scalar wave equation is obeyed by all components of those vectors.

10. Thus, for exampel, εX obeys the equation n2 ∂2ε ▽2ε − X =0, X c2 ∂t2

and similarly for εY ,εZ , HX , HY , and HZ .

Prof. Hsuan-Ting Chang Chapter 3 Foundations of Scalar Diffraction Theory 3.2 From a Vector to a Scalar Theory – 5

11. Therefore, it is possible to summarize the behavior of all components of E~ and H~ through a single scalar wave equation, n2 ∂2u(P, t) ▽2u(P, t) − =0, (3.7) c2 ∂t2 where u(P, t) represents any of the scalar field components. 12. If the aperture has an area that is large compared with a wavelength, the coupling effects of the boundary conditions on the E~ and H~ fields will be small. 13. The scalar theory is accurate provided that the diffracting structures are large compared with the wavelength of light.

Prof. Hsuan-Ting Chang Chapter 3 Foundations of Scalar Diffraction Theory 3.3 Some Mathematical Preliminaries - 1

1. 3.3.1 Helmholtz Equation 2. For a monochromatic wave, the scalar field is

u(P, t)= A(P) cos[2πνt + φ(P)], (3.9)

where A(P) and φ(P) are the amplitude and phase, respectively, of the wave at position P, while ν is the optical frequency. 3. A more compact form

u(P, t)= Re{U(P) exp(−j2πνt)}, (3.10)

where U(P) is a complex function of position (phasor),

U(P)= A(P) exp[−jφ(P)].

Prof. Hsuan-Ting Chang Chapter 3 Foundations of Scalar Diffraction Theory 3.3 Some Mathematical Preliminaries - 2

4. If Eq. (3.10) is substituted in Eq. (3.12)=Eq. (3.7), it follows that U must obey the time-independent equation (Helmholtz equation):

(▽2 + k2)U =0. (3.13)

Here k is termed the wave number and is given by ν 2π k =2πn = , c λ and λ is the wavelength in the dielectric medium.

Prof. Hsuan-Ting Chang Chapter 3 Foundations of Scalar Diffraction Theory 3.3.2 Green’s Theorem

1. GOAL: Calculation of the complex disturbance U at an observation point in space. 2. Let U(P) and G(P) be any two complex-valued functions of position, and let S be a closed surface surrounding a volume V . If U, G, and their first and second partial derivatives are single-valued and continuous within and on S, then we have ∂G ∂U (U ▽2 G − G ▽2 U)dv = (U − G )ds Z Z ZV Z ZS ∂n ∂n where ∂/∂n signifies a partial derivative in the outward normal direction at each point on S. 3. This theorem is the prime foundation of scalar diffraction theory.

Prof. Hsuan-Ting Chang Chapter 3 Foundations of Scalar Diffraction Theory 3.3.3 The Integral Theorem of Helmholtz and Kirchhoff - 1

1. Let P0 be the point of observation, S be an arbitrary closed surface surrounding P0.

2. To express the optical disturbance at P0 in terms of its values on the surface S, we follow Kirchhoff in applying Green’s theorem and in choosing as an auxiliary function G, a unit-amplitude spherical wave expanding about P0 (free space Green’s function).

3. The value of G at an arbitrary point P1 is

exp(jkr01) G(P1)= , r01

where r01 is the length of the vector ~r01 pointing from P0 to P1.

Prof. Hsuan-Ting Chang Chapter 3 Foundations of Scalar Diffraction Theory Fig 3.5

Prof. Hsuan-Ting Chang Chapter 3 Foundations of Scalar Diffraction Theory 3.3.3 The Integral Theorem of Helmholtz and Kirchhoff - 2

4. To be legitimately used in Green’s theorem, the function G must be continuous within the enclosed volume V . Therefore to exclude the discontinuity at P0, a small spherical surface Sǫ, of radius ǫ, is inserted about the point P0. 5. The volume of integration V ′ being that volume lying between S and Sǫ, and the surface of integration being the composite ′ surface S = S + Sǫ.

Prof. Hsuan-Ting Chang Chapter 3 Foundations of Scalar Diffraction Theory 3.3.3 The Integral Theorem of Helmholtz and Kirchhoff - 3

6. The disturbance G satisfies the Helmholtz equation (▽2 + k2)G =0 (3.18) Then substituting Eqns. (3.13) and (3.18) in the left-hand side of Green’s theorem,

(U▽2G−G▽2U)dv = − (UGk2−GUk2)dv ≡ 0 Z Z ZV ′ Z Z ZV ′ The theorem reduces to ∂G ∂U (U − G )ds =0 Z ZS′ ∂n ∂n or ∂G ∂U ∂G ∂U − (U − G )ds = (U − G )ds. Z ZSǫ ∂n ∂n Z ZS ∂n ∂n

Prof. Hsuan-Ting Chang Chapter 3 Foundations of Scalar Diffraction Theory 3.3.3 The Integral Theorem of Helmholtz and Kirchhoff - 4

′ 7. For general point P1 on S , (n~: outward normal)

exp(jkr01) G(P1)= r01 and

∂G(P1) ∂~r01 ∂G(P1) 1 exp(jkr01) = = cos(n~, ~r01)(jk − ) . ∂n ∂n~ ∂r01 ~r01 r01 (3.20) ∂~r01 where ∂n~ = cos(n~, ~r01). 8. For the particular case of P1 on Sǫ, cos(n~, ~r01)= −1,

ejkǫ ∂G(P ) ejkǫ 1 G(P )= and 1 = ( − jk). 1 ǫ ∂n ǫ ǫ

Prof. Hsuan-Ting Chang Chapter 3 Foundations of Scalar Diffraction Theory 3.3.3 The Integral Theorem of Helmholtz and Kirchhoff - 5

9. Letting ǫ → 0, the continuity of U at P0 allow us to write ∂G ∂U lim (U −G )ds ǫ→ 0 Z ZSǫ ∂n ∂n

2 exp(jkǫ) 1 ∂U(P0) exp(jkǫ) = lim 4πǫ U(P0) ( − jk) − ǫ→0  ǫ ǫ ∂n ǫ 

=4πU(P0). 10. Finally,

1 ∂U exp(jkr01) ∂ exp(jkr01) U(P0)= − U ds. 4π Z ZS  ∂n  r01  ∂n  r01  (3.21) The integral theorem of Helmholtz and Kirchhoff; it plays an important role in the development of the scalar theory of diffraction. Prof. Hsuan-Ting Chang Chapter 3 Foundations of Scalar Diffraction Theory 3.3.3 The Integral Theorem of Helmholtz and Kirchhoff - 6

11. The field at any point P0 can be expressed in terms of the “boundary values” of the wave on any closed surface surrounding that point.

Prof. Hsuan-Ting Chang Chapter 3 Foundations of Scalar Diffraction Theory Fig 3.6

Prof. Hsuan-Ting Chang Chapter 3 Foundations of Scalar Diffraction Theory 3.4 The Kirchhoff Formulation of Diffraction by a Planar Screen - 1

1. 3.4.1 Application of the Integral Theorem (for S2 plane) 2. As shown in Figure 3.6, following the Kirchhoff, the closed surface S is chosen to consist of two parts. Let a plane surface, S1, lying directly behind the diffraction screen, be joined and closed by a large spherical cap, S2, of radius R and centered at the observation point P0.

3. Since S = S1 + S2, then 1 ∂U ∂G U(P0)= (G − U )ds 4π Z ZS1+S2 ∂n ∂n where exp(jkr ) G = 01 . r01

Prof. Hsuan-Ting Chang Chapter 3 Foundations of Scalar Diffraction Theory 3.4.1 Application of the Integral Theorem (for S2 plane) - 2

4. Since both U and G will fall off as 1/R, the integrand will ultimately vanish, yielding a contribution of zero from the surface integral over S2. 5. However, the area of integration increases as R2, so this argument is incomplete. Evidently, a more careful investigation is required before the contribution from S2 can be disposed of.

6. On S2, exp(jkR) G = R and, from Eq. (3.20) and cos(n~, R) = 1, we obtain ∂G 1 exp(jkR) =(jk − ) ≈ jkG (for large R). ∂n R R

Prof. Hsuan-Ting Chang Chapter 3 Foundations of Scalar Diffraction Theory 3.4.1 Application of the Integral Theorem (for S2 plane) - 3

7. The integration reduces to ∂U ∂U [G − U(jkG)]ds = G( − jkU)R2dω, Z ZS2 ∂n ZΩ ∂n

where Ω is the solid angle subtended by S2 at P0.

8. Now |RG| = | exp(jkR)| = 1 is uniformly bounded on S2 and 1 Gds = RG × R ds. The entire integral over S2 will vanish as R becomes arbitrary large, provided that ∂U ∂U lim [G −U(jkG)]ds = lim [ −U(jk)]Gds =0, R→∞ R→∞ Z ZS2 ∂n Z ZS2 ∂n and this required that (see next page)

Prof. Hsuan-Ting Chang Chapter 3 Foundations of Scalar Diffraction Theory 3.4.1 Application of the Integral Theorem (for S2 plane) - 4

1 ∂U lim ( − jkU)ds =0 R→∞ R Z ZS2 ∂n ∂U lim R( − jkU)=0 (3.22) R→∞ ∂n 9. This requirement is known as the Sommerfeld radiation condition (for S2 plane) and is satisfied if U vanishes at least as fast as a diverging spherical wave. The integral over S2 will yield a contribution of precisely zero.

Prof. Hsuan-Ting Chang Chapter 3 Foundations of Scalar Diffraction Theory 3.4.2 The Kirchhoff Boundary Conditions (for S1 plane) - 1

1. After considering Sommerfeld radiation condition, the disturbance at P0 over the infinite plane S1 is expressed by, 1 ∂U ∂G U(P0)= (G − U )ds. (3.23) 4π Z ZS1 ∂n ∂n 2. The major contribution to the integral (3.23) arises from the point of S1 located within the aperture Σ, where we expect the integrand to be largest. 3. Kirchhoff boundary conditions: (1). Across the surface Σ, the field distribution U and its derivative ∂U/∂n are exactly the same as they would be in the absence of the screen. (2). Over the portion of S1 that lies in the geometric shadow of the screen, the field distribution U and its derivative ∂U/∂n are identically zero.

Prof. Hsuan-Ting Chang Chapter 3 Foundations of Scalar Diffraction Theory 3.4.2 The Kirchhoff Boundary Conditions (for S1 plane) - 2

4. Thus (3.23) is reduced to 1 ∂U ∂G U(P0)= (G − U )ds. (3.24) 4π Z ZΣ ∂n ∂n 5. However, it is important to realize that neither can be exactly true. The screen will inevitably perturb the fields on Σ to some degree. In addition, fields will inevitably extend behind the screen for a distance of several wavelengths. 6. However, if the dimensions of the aperture are large compared with a wavelength, the fringing effects can be safely neglected, and the two boundary conditions can be used.

Prof. Hsuan-Ting Chang Chapter 3 Foundations of Scalar Diffraction Theory 3.4.3 The Fresnel-Kirchhoff Diffraction Formula (for S1 plane) - 1

1. A further simplification for U(P0) is obtained by noting k ≫ 1/r01, (3.20) becomes

∂G(P1) 1 exp(jkr01) = cos(n~, ~r01)(jk − ) ∂n ~r01 r01

exp(jkr01) ≈ jk cos(n~, ~r01) . (3.25) r01 2. Then

1 exp(jkr01) ∂U U(P0)= − jkU cos(n~, ~r01) ds. 4π Z ZΣ r01  ∂n  (3.26)

Prof. Hsuan-Ting Chang Chapter 3 Foundations of Scalar Diffraction Theory Fig 3.7

Prof. Hsuan-Ting Chang Chapter 3 Foundations of Scalar Diffraction Theory 3.4.3 The Fresnel-Kirchhoff Diffraction Formula (for S1 plane) - 2 3. (Figure 3.7) Now suppose that the aperture is illuminated by a single spherical wave,

A exp(jkr21) U(P1)= r21

arising from a point source P2, a distance r21 from P1. 4. If r21 is many optical wavelengths, then

U(P0)= A exp[jk(r + r )] cos(n~, ~r ) − cos(n~, ~r ) 21 01 [ 01 21 ]ds. jλ Z ZΣ r21r01 2 (3.27) This result, which holds only for an illumination consisting of a single point source, is known as the Fresnel-Kirchhoff diffraction formula. Prof. Hsuan-Ting Chang Chapter 3 Foundations of Scalar Diffraction Theory 3.4.3 The Fresnel-Kirchhoff Diffraction Formula (for S1 plane) - 3

5. The reciprocity theorem of Helmholtz: A point source at P0 will produce at P2 the same effect that a point source of equal intensity placed at P2 will produce at P0. 6. Finally, an interesting interpretation of the diffraction formula (3.27) is ′ exp(jkr01) U(P0)= U (P1) ds, (3.28) Z ZΣ r01 where

′ U (P1)= 1 A exp(jkr ) cos(n~, ~r ) − cos(n~, ~r ) [ 21 ][ 01 21 ]. (3.29) jλ r21 2

Prof. Hsuan-Ting Chang Chapter 3 Foundations of Scalar Diffraction Theory 3.4.3 The Fresnel-Kirchhoff Diffraction Formula (for S1 plane) - 4

7. The light field at P0 arises from an infinity of fictitious “secondary” point sources located within the aperture itself. ′ 8. The secondary sources U (P1) have amplitudes and phases, that are related to (1). The illumination wavefront (scaling factor 1/λ) (2). The angles of illumination and observation (obliquity cos(n~,~r01)−cos(n~,~r21) factor −1 ≤ 2 ≤ 1) (3). Phase lead (90o) of the secondary source.

Prof. Hsuan-Ting Chang Chapter 3 Foundations of Scalar Diffraction Theory 3.5 The Rayleigh-Sommerfeld Formulation of Diffraction - 1

1. There are certain internal inconsistencies in the theory which motivated a search for a more satisfactory mathematical development. 2. The difficulties of the Kirchhoff theory stem from the fact that boundary conditions must be imposed on both the field strength and its normal derivative. 3. Potential theory: if a 2-D potential function and its normal derivative vanish together along any finite curve segment, then that potential function must vanish over the entire plane. 4. Similarily, if a solution of the 3-D wave equation vanishes on any finite surface element, it must vanish in all space.

Prof. Hsuan-Ting Chang Chapter 3 Foundations of Scalar Diffraction Theory 3.5 The Rayleigh-Sommerfeld Formulation of Diffraction - 2

5. The two Kirchhoff boundary conditions together imply that the field is zero everywhere behind the aperture, which contradicts the known physical situation. 6. The Fresnel-Kirchhoff diffraction formula can be shown to fail to reproduce the assumed boundary conditions as the observation point approaches the screen or aperture. 7. The inconsistencies of the Kirchhoff theory were removed by Sommerfeld, who eliminated the necessity of imposing boundary values on both the disturbance and its normal derivative simultaneously.

Prof. Hsuan-Ting Chang Chapter 3 Foundations of Scalar Diffraction Theory 3.5.1 Choice of Alternative Green’s Functions - 1

1. Consider again Eq. (3.23) 1 ∂U ∂G U(P0)= ( G − U )ds. 4π Z ZS1 ∂n ∂n The conditions for validity are: 1. The scalar theory holds. 2. Both U and G satisfy the homogeneous scalar wave equation. 3. The Sommerfeld radiation condition, Eq. (3.22), is satisfied.

Prof. Hsuan-Ting Chang Chapter 3 Foundations of Scalar Diffraction Theory 3.5.1 Choice of Alternative Green’s Functions - 2

2. Suppose that the Green’s function G of the Kirchhoff theory were modified either G or ∂G/∂n vanishes over the entire surface S1. The inconsistencies of the Kirchhoff theory would be eliminated. 3. Suppose G is generated not only by a point source located at P0, but also simultaneously by a second point source at a position P˜0, which is a mirror image of P0 on the opposite side of the screen. Let the source at P˜0 be of the same wavelength as the source at P0, and suppose that the two sources are oscillating with a 180o phase difference.

Prof. Hsuan-Ting Chang Chapter 3 Foundations of Scalar Diffraction Theory Fig 3.8

Prof. Hsuan-Ting Chang Chapter 3 Foundations of Scalar Diffraction Theory 3.5.1 Choice of Alternative Green’s Functions - 3

4. (Figure 3.8)

exp(jkr01) exp(jkr˜01) G−(P1)= − . r01 r˜01

This function vanishes (G−(P1) = 0) on the plane aperture Σ,

−1 ∂G− UI (P0)= U ds. 4π Z ZΣ ∂n We refer to this solution as the first Rayleigh-Sommerfeld solution.

5. Letr ˜01 be the distance from P˜0 to P1.

∂G−(P1) 1 exp(jkr01) = cos(n~, ~r01)(jk − ) ∂n r01 r01

1 exp(jkr˜01) − cos(n~,~r˜01)(jk − ) . r˜01 r˜01

Prof. Hsuan-Ting Chang Chapter 3 Foundations of Scalar Diffraction Theory 3.5.1 Choice of Alternative Green’s Functions - 4

6. For P1 on S1,

r01 =r ˜01, cos(n~, ~r01)= − cos(n~,~r˜01),

and r01 ≫ λ,

∂G−(P1) exp(jkr01) =2jk cos(n~, ~r01) , (3.35) ∂n r01 7.

∂G−(P1) ∂G(P1) −1 ∂G =2 and UI (P0)= U ds. ∂n ∂n 2π Z ZΣ ∂n

Prof. Hsuan-Ting Chang Chapter 3 Foundations of Scalar Diffraction Theory 3.5.1 Choice of Alternative Green’s Functions - 5

8. An alternative and equally valid Green’s function is found by allowing the two point sources to oscillate in phase,

exp(jkr01) exp(jkr˜01) G+(P1)= + . (3.37) r01 r˜01 9. The normal derivative of this function vanishes across the screen and aperture, leading to the second Rayleigh-Sommerfeld solution, 1 ∂U UII (P0)= G+ds. (3.38) 4π Z ZΣ ∂n

Prof. Hsuan-Ting Chang Chapter 3 Foundations of Scalar Diffraction Theory 3.5.1 Choice of Alternative Green’s Functions - 6

10. On Σ and under the condition that r01 ≫ λ, G+ is twice the Kirchhoff Green’s function G,

G+ =2G,

and 1 ∂U UII (P0)= Gds. 2π Z ZΣ ∂n

Prof. Hsuan-Ting Chang Chapter 3 Foundations of Scalar Diffraction Theory 3.5.2 The Rayleigh-Sommerfeld Diffraction Formula - 1

1. Substitute G− for G in Eq. (3.23). Using (3.35),

1 exp(jkr01) UI (P0)= U(P1) cos(n~, ~r01)ds, (3.40) jλ Z ZS1 r01

(where r01 ≫ λ). 2. The Kirchhoff boundary conditions may now be applied to U alone, yielding

1 exp(jkr01) UI (P0)= U(P1) cos(n~, ~r01)ds. (3.41) jλ Z ZΣ r01 Since no boundary conditions need be applied to ∂U/∂n, the inconsistencies of the Kirchhoff theory have been removed. 3. If Eq. (3.37) is used,

1 ∂U(P1) exp(jkr01) UII (P0)= ds. (3.42) 2π Z ZΣ ∂n r01

Prof. Hsuan-Ting Chang Chapter 3 Foundations of Scalar Diffraction Theory 3.5.2 The Rayleigh-Sommerfeld Diffraction Formula - 2

4. If the illumination of the aperture in all cases is a spherical wave diverging from a point source at position P2:

exp(jkr21) U(P1)= A . r21

5. Using G−, we obtain

A exp[jk(r21 + r01)] UI (P0)= cos(n~, ~r01)ds. (3.43) jλ Z ZΣ r21r01 This result is known as the Rayleigh-Sommerfeld diffraction formula.

6. Using G+, and assuming that r21 ≫ λ,

A exp[jk(r21 + r01)] UII (P0)= − cos(n~, ~r21)ds, (3.44) jλ Z ZΣ r21r01 o where the angle between n~ and r~21 is greater than 90 . Prof. Hsuan-Ting Chang Chapter 3 Foundations of Scalar Diffraction Theory 3.6 Comparison of The Kirchhoff and Rayleigh-Sommerfeld Theorems - 1 1. Let GK : the Green’s function for Kirchhoff theory, G− and G+: the Green’s functions for the two Rayleigh-Sommerfeld formulations

2. On the surface Σ, G+ =2GK and ∂G−/∂n =2∂GK /∂n. 3. The Kirchhoff solution is the arithmetic average of the two Rayleigh-Smmerfeld solutions. 1 U(P )= [U (P )+ U (P )]. 0 2 I 0 II 0 4. The results from the Rayleigh-Sommerfeld theory differ from the Fresnel-Kirchhoff diffraction formula only through what is known as the obliquity factor ψ, which is the angular dependence introduced by the cosine terms.

Prof. Hsuan-Ting Chang Chapter 3 Foundations of Scalar Diffraction Theory 3.6 Comparison of The Kirchhoff and Rayleigh-Sommerfeld Theorems - 2 5. For all cases we can write

A exp[jk(r21 + r01)] U(P0)= ψds, jλ Z ZΣ r21r01 where 1 2 [cos(n~, ~r01) − cos(n~, ~r21)], Kirchhoff theory (3.27) ψ =  cos(n~, ~r01), First Rayleigh-Sommerfeld solution (3.43)  − cos(n~, ~r21), Second Rayleigh-Sommerfeld solution (3.44) 6. For the special case of an infinite distant point source producing normally incident plane wave illumination, 1 2 [1 + cos θ], Kirchhoff theory ψ =  cos θ, First Rayleigh-Sommerfeld solution  1, Second Rayleigh-Sommerfeld solution

where θ is the angle between the vectors n~ and ~r01. Prof. Hsuan-Ting Chang Chapter 3 Foundations of Scalar Diffraction Theory 3.6 Comparison of The Kirchhoff and Rayleigh-Sommerfeld Theorems - 3

7. Comments: ◮ Kirchhoff solution and the two Rayleigh-Sommerfeld solutions to be essentially the same provided the aperture diameter is much greater than a wavelength. ◮ When only small angles θ are involved, all three solutions are identical. ◮ The Kirchhoff theory is more general than the Rayleigh-Sommerfeld theory since the latter requires that the diffracting screens be planar, while the former does not. ◮ We generally use the first Rayleigh-Sommerfeld solution because of its simplicity.

Prof. Hsuan-Ting Chang Chapter 3 Foundations of Scalar Diffraction Theory 3.7 Further Discussion of The Huygens-Fresnel Principle - 1

1. The Huygens-Fresnel principle, as predicted by the first Rayleigh-Sommerfeld solution (Eq. 3.41):

1 exp(jkr01) U(P0)= U(P1) cos θds. (3.51) jλ Z ZΣ r01

2. The observed field U(P0) is a superposition of diverging spherical waves exp(jkr01/r01) originating from secondary sources located at each and every point P1 within the aperture Σ. 3. The secondary source P1 has the following properties: ◮ Its complex amplitude is proportional to the amplitude of U(P0) at the corresponding point. ◮ Its amplitude is inversely proportional to λ. ◮ It has a phase lead 90o than the incident wave. ◮ Each secondary source has a directivity pattern cos θ.

Prof. Hsuan-Ting Chang Chapter 3 Foundations of Scalar Diffraction Theory 3.7 Further Discussion of The Huygens-Fresnel Principle - 2

4. The Huygens-Fresnel principle is nothing more than a superposition integral.

U(P0)= h(P0, P1)U(P1)ds, Z ZΣ

where the impulse response h(P0, P1) is

1 exp(jkr01) h(P0, P1)= cos θ. jλ r01 We will find it is also space-invariant in Chapter 4.

Prof. Hsuan-Ting Chang Chapter 3 Foundations of Scalar Diffraction Theory 3.8 Generalization to Nonmonochromatic Waves - 1

1. The more general case of a nonmonochromatic disturbance will now be considered briefly.

2. Consider the scalar disturbance u(P0, t) observed behind an aperture when a disturbance u(P1, t) is incident on that aperture.

Prof. Hsuan-Ting Chang Chapter 3 Foundations of Scalar Diffraction Theory 3.8 Generalization to Nonmonochromatic Waves - 2

3. The time functions may be expressed in terms of their inverse

FTs: ∞ u(P1, t)= U(P1, ν) exp(j2πνt)dν Z−∞ ∞ u(P0, t)= U(P0, ν) exp(j2πνt)dν Z−∞ 4. By the exchange of variables ν′ = −ν, yielding ∞ ′ ′ ′ u(P1, t) = U(P1, −ν ) exp(−j2πν t)dν Z−∞ ∞ ′ ′ ′ u(P0, t) = U(P0, −ν ) exp(−j2πν t)dν (3.55) Z−∞

Prof. Hsuan-Ting Chang Chapter 3 Foundations of Scalar Diffraction Theory 3.8 Generalization to Nonmonochromatic Waves - 3

5. The nonmonochromatic time functions u(P1, t) and u(P0, t) may be regarded as the linear combination of monochromatic time functions of the type represented in Eq. (3.10). 6. The monochromatic elementary functions are of various frequencies ν′, the complex amplitudes of the disturbance at ′ ′ ′ frequency ν being simply U(P1, −ν ) and U(P0, −ν ).

Prof. Hsuan-Ting Chang Chapter 3 Foundations of Scalar Diffraction Theory 3.8 Generalization to Nonmonochromatic Waves - 4

7. Eq. (3.51) is used to write

′ U(P0, −ν )

′ ′ ν ′ exp(j2πν r01/v) = −j U(P1, −ν ) cos(n~, ~r01)ds, v Z ZΣ r01 (3.56) where v = c/n = ν′λ. 8. Subtitution of (3.56) into Eq. (3.55),

u(P0, t)=

∞ cos(n~, ~r01) ′ ′ ′ r01 ′ −j2πν U(P1, −ν ) exp[−j2πν (t− )]dν ds. Z ZΣ 2πvr01 Z−∞ v

Prof. Hsuan-Ting Chang Chapter 3 Foundations of Scalar Diffraction Theory 3.8 Generalization to Nonmonochromatic Waves - 5

9. Finally, the identity d u(P , t) dt 1 ∞ d ′ ′ ′ = U(P1, −ν ) exp(−j2πν t)dν dt Z−∞ ∞ ′ ′ ′ ′ = −j2πν U(P1, −ν ) exp(−j2πν t)dν Z−∞ can be used to write

u(P0, t)

cos(n~, ~r01) d r01 = u(P1, t − )ds. (3.57) Z ZΣ 2πvr01 dt v

Prof. Hsuan-Ting Chang Chapter 3 Foundations of Scalar Diffraction Theory 3.8 Generalization to Nonmonochromatic Waves - 6

10. The wave disturbance at point P0 is linearly proportional to the time derivative of the disturbance at each P1 on the aperture. 11. An understanding of diffraction of monochromatic waves can be used directly to synthesize the results for much more general nonmonochromatic waves.

Prof. Hsuan-Ting Chang Chapter 3 Foundations of Scalar Diffraction Theory 3.9 Diffraction at Boundaries –1

1. Regard the observed field as consisting of a superposition of the incident wave transmitted through the aperture unperturbed, and a diffracted wave originating at the rim of the aperture. 2. Young’s ideas: The field in the geometrical shadow of the screen has the form of a cylindrical wave originating on the rim of the screen. 3. In the directly illuminated region behind the plane of the screen the field is a superposition of this cylindrical wave with the directly transmitted wave. 4. The Kirchhoff diffraction formula can indeed be manipulated to yield a form that is equivalent to Young’s ideas.

Prof. Hsuan-Ting Chang Chapter 3 Foundations of Scalar Diffraction Theory 3.9 Diffraction at Boundaries –2

5. The field behind a diffracting obstacle is found by the principles of geometrical optics, modified by the inclusion of “diffracted rays” that originate at certain points on the obstacle itself.

Prof. Hsuan-Ting Chang Chapter 3 Foundations of Scalar Diffraction Theory 3.10 The Angular Spectrum of Plane Waves

1. To formulate scalar diffraction theory in a framework that closely resembles the theory of linear, invariant systems. 2. If the complex field distribution of a monochromatic disturbance is Fourier-analyzed across any plane, the various spatial Fourier components can be identified as plane waves travelling in different directions away from that plane.

Prof. Hsuan-Ting Chang Chapter 3 Foundations of Scalar Diffraction Theory 3.10.1 The Angular Spectrum and Its Physical Interpretation - 1 1. Let the complex field across that z = 0 plane be represented by U(x, y, 0). Our objective is to calculate the resulting field U(x, y, z) that appears across a second, parallel plane a distance z to the right of the first plane. 2. Across the z = 0 plane, the function U has a 2-D Fourier transform given by ∞ A(fX , fY , 0) = U(x, y, 0) exp[−j2π(fX x + fY y)]dxdy. Z Z−∞ 3. Frequency decomposition: we write U as an inverse Fourier transform of its spectrum, ∞ U(x, y, 0) = A(fX , fY , 0) exp[j2π(fX x + fY y)]dfX dfY . Z Z−∞ (3.59)

Prof. Hsuan-Ting Chang Chapter 3 Foundations of Scalar Diffraction Theory 3.10.1 The Angular Spectrum and Its Physical Interpretation - 2

4. Consider a simple plane wave propagating with wave vector k~, where k~ has magnitude 2π/λ and has direction cosines (α,β,γ) (Fig. 3.9)

p(x, y, z; t) = exp[j(k~ · ~r − 2πνt)]

where ~r = xxˆ + yyˆ + zzˆ is a position vector, while ~ 2π k = λ (αxˆ + βyˆ + γzˆ). 5. Dropping the time dependence,

2π 2π P(x, y, z) = exp(jk~ · ~r)= ej λ (αx+βy)ej λ γz .

Note that γ = 1 − α2 − β2. p

Prof. Hsuan-Ting Chang Chapter 3 Foundations of Scalar Diffraction Theory Fig 3.9

Prof. Hsuan-Ting Chang Chapter 3 Foundations of Scalar Diffraction Theory 3.10.1 The Angular Spectrum and Its Physical Interpretation - 3

6. Across the plane z = 0, exp[j2π(fX x + fY y)] may be regarded as representing a plane wave propagating with direction cosines

2 2 α = λfX , β = λfY , γ = 1 − (λfX ) − (λfY ) . (3.62) p 7. In the Fourier decomposition of U, the complex amplitude of the plane-wave component with spatial frequencies (fX , fY ) is simply A(fX , fY ;0)dfX dfY , evaluated at (fX = α/λ, fY = β/λ). 8. For this reason, the function α β ∞ α β A( , ;0) = U(x, y, 0) exp[−j2π( x + y)]dxdy λ λ Z Z−∞ λ λ (3.63) is called the angular spectrum of the disturbance U(x, y, 0).

Prof. Hsuan-Ting Chang Chapter 3 Foundations of Scalar Diffraction Theory 3.10.2 Propagation of the Angular Spectrum - 1

1. Consider the angular spectrum of U across a plane parallel to the (x, y) plane but at a distance z from it. α β ∞ α β A( , ; z)= U(x, y, z) exp[−j2π( x + y)]dxdy. λ λ Z Z−∞ λ λ (3.64) 2. Note that U can be written as ∞ α β α β α β U(x, y, z)= A( , ; z) exp[j2π( x + y)]d d . Z Z−∞ λ λ λ λ λ λ (3.65) 3. Recall the Helmholtz equation ▽2U + k2U =0 at all source-free points. Then A must satisfy the differential equation d 2 α β 2π α β A( , ; z)+( )2[1 − α2 − β2]A( , ; z)=0. dz 2 λ λ λ λ λ

Prof. Hsuan-Ting Chang Chapter 3 Foundations of Scalar Diffraction Theory 3.10.2 Propagation of the Angular Spectrum - 2

4. An elementary solution: α β α β 2π A( , ; z)= A( , ;0)exp(j 1 − α2 − β2z). (3.66) λ λ λ λ λ p 5. If the direction cosines (α, β) satisfy

α2 + β2 < 1, (3.67)

the effect of propagation over distance z is simply a change of the relative phases of the various components of the angular spectrum (relative phase delays are introduced).

Prof. Hsuan-Ting Chang Chapter 3 Foundations of Scalar Diffraction Theory 3.10.2 Propagation of the Angular Spectrum - 3

6. If α2 + β2 > 1, α and β are no longer interpretable as direction cosines. The equation is rewritten α β α β A( , ; z)= A( , ;0)exp(−µz), (3.68) λ λ λ λ

2π 2 2 where µ = λ α + β − 1. 7. Since µ is a positivep real number, these wave components are rapidly attenuated by the propagation phenomenon. Such components are called evanescent waves.

Prof. Hsuan-Ting Chang Chapter 3 Foundations of Scalar Diffraction Theory 3.10.2 Propagation of the Angular Spectrum - 4

8. Finally, ∞ α β 2π U(x, y, z)= A( , ;0)exp(j 1 − α2 − β2z) λ λ λ Z Z−∞ p α β α β × exp[j2π( x + y)]d d , (3.69) λ λ λ λ where the evanescent wave phenomenon will effectively limit the region with which Eq. (3.67) is satisfied. 9. No angular spectrum components beyond the evanescent wave cutoff contribute to U(x, y, z).

Prof. Hsuan-Ting Chang Chapter 3 Foundations of Scalar Diffraction Theory 3.10.3 Effects of a Diffracting Aperture on the Angular Spectrum - 1 1. Define the amplitude transmittance function of the aperture as the ratio of the transmitted field amplitude Ut (x, y; 0) to the incident field amplitude Ui (x, y;0) at each (x, y) in the z =0 plane, Ut (x, y;0) tA(x, y)= . (3.70) Ui (x, y;0) 2. Use the convolution theorem to relate the angular spectrum Ai (α/λ,β/λ) of the incident field and the angular spectrum At (α/λ,β/λ) of the transmitted field, α β α β α β A ( , ) = [A ( , ) ⊗ T ( , )], (3.71) t λ λ i λ λ λ λ where α β ∞ α β T ( , )= tA(x, y) exp[−j2π( x + y)]dxdy. λ λ Z Z−∞ λ λ

Prof. Hsuan-Ting Chang Chapter 3 Foundations of Scalar Diffraction(3 Theory.72) 3.10.3 Effects of a Diffracting Aperture on the Angular Spectrum - 2

3. For the case of a unit amplitude plane wave illuminating the diffracting structure normally, α β α β A ( , )= δ( , ) i λ λ λ λ and α β α β α β α β A ( , )= δ( , ) ⊗ T ( , )= T ( , ). t λ λ λ λ λ λ λ λ 4. The transmitted angular spectrum is found directly by Fourier transforming the amplitude transmittance function of the aperture. The smaller the aperture, the broader the angular spectrum behind the aperture.

Prof. Hsuan-Ting Chang Chapter 3 Foundations of Scalar Diffraction Theory 3.10.4 The Propagation Phenomenon as a Linear Spatial Filter - 1

1. The propagation phenomenon acts as a linear space-invariant system and is characterized by a relatively simple transfer function. 2. It is now more convenient to leave the spectra as functions of spatial frequencies (fX , fY ) (Eq. 3.62). 3. Let the spatial spectrum of U(x, y, z) again be represented by A(fX , fY ; z).

∞ U(x, y, z)= A(fX , fY ; z) exp[j2π(fX x + fY y)]dfX dfY . Z Z−∞

Prof. Hsuan-Ting Chang Chapter 3 Foundations of Scalar Diffraction Theory 3.10.4 The Propagation Phenomenon as a Linear Spatial Filter - 2

4. From Eq. (3.69) ∞ 2π U(x, y, z)= A(f , f ;0)exp(j 1 − (λf )2 − (λf )2z) X Y λ X Y Z Z−∞ p

× exp[j2π(fX x + fY y)]dfX dfY . 5. A comparison of the above two equations shows that z A(f , f ; z)= A(f , f ;0) × exp(j2π 1 − (λf )2 − (λf )2). X Y X Y λ X Y p (3.73)

Prof. Hsuan-Ting Chang Chapter 3 Foundations of Scalar Diffraction Theory 3.10.4 The Propagation Phenomenon as a Linear Spatial Filter - 3

6. Finally, the transfer function of the wave propagation phenomenon is seen to be z H(f , f ) = exp[j2π 1 − (λf )2 − (λf )2] (3.74) X Y λ X Y p

Prof. Hsuan-Ting Chang Chapter 3 Foundations of Scalar Diffraction Theory 3.10.4 The Propagation Phenomenon as a Linear Spatial Filter - 3

7. The propagation phenomenon may be regarded as a linear, dispersive spatial filter with a finite bandwidth. The transmittance of the filter is zero outside a circular region of radius λ−1 in the frequency plane. Within the circular bandwidth, the modulus of the transfer function is unity but frequency-dependent phase shifts are introduced. 8. The phase dispersion is most significant at high spatial frequencies and vanishes as both fX and fY approach zero. In addition, it increases as z increases. 9. The angular spectrum approach and the first Rayleigh-Sommerfeld solution yield identical predictions of diffracted field.

Prof. Hsuan-Ting Chang Chapter 3 Foundations of Scalar Diffraction Theory