1. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 1 Chap 3 Foundations of Scalar Diffraction Theory

2. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 2 Content 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 discuss of the Huygens-Fresnel principle 3.8 Generalization to nonmonochromatic waves 3.9 Diffraction at boundaries 3.10 The angular spectrum of plane waves

3. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 3 3.1 Historical introduction While the theory discussed here is sufficiently general to be applied in other field, such as acoustic-wave and radio- wave propagation, the applications of primary concern will be in the realm of physical optics. To fully understand the properties of optical imaging and data processing system, it is essential that diffraction and the limitation it imposes on system performance be appreciated.

4. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 4 The edge of the aperture is thin enough such that light maybe regarded as unpolarized. In addition the area of the aperture can not be too small. (or be large enough)

5. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 5 Refraction can be defined as the bending of light rays mat takes place when they pass through a region in which there is a gradient of the local velocity of propagation of the wave. The most common example occurs when the light wave encounters a sharp boundary between two regions having different refractive indices.

6. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 6 The Kirchhoff and Rayleigh-Sommerdeld theories share certain major simplifications and approximations. Most important light is treated as a scalar phenomenon, neglecting the fundamentally vectorial nature of the electromagnetic fields.

7. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 7 3.2 From a vector to a scalar theory In the case of diffraction of light by an aperture, the and field modified only at the edges of the aperture where light interacts with the material which the edges are composed of, and the effects extend over only a few wavelengths into the aperture itself.

8. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 8 1.Wave eq. (in vectorial form) (from Maxwell’s eq.) 2. Wave eq. (in scalar forms). 3. Wave eq. (in phasor forms). a.Linear, b.Homogeneous, c.Isotropic, d.Nondispersive, e.Nonmagnetic.

9. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 9 Where the optical disturbance u(p, t) = = U(p) e U(p) = and represents position variable (i.e. r) It follow that (called Helmholtz eq.)

10. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 10 In general Huygens (Huygen’s principle) (Young) Fresnel two assumption Kirchhoff (Fresnel-Kirchhoff formula) Rayletgh-Summerfled Fig. 3.1 Huygens’ envelope construction

11. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 11 3.3 Some mathematical preliminaries 3.3.1 The Helmholtz equation For a monochromatic wave, the scalar field may be written explicitly (3-1) where A(P) and are the amplitude and phase, respectively, of the wave at position P, while v is the optical frequency. If the real disturbance is to represent an optical wave, it must satisfy the scalar wave equation.

12. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 12 The complex function U(P) serves as an adequate description of the disturbance, since the time dependencies known a priori. If (3-1) is substituted in (3-2), it follows that U must obey the time- independent equation. (3-2) (3-3)

13. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 13 3.3.2 Green’s theorem Give U(P), G(p), Let According Gauss’s divergence thm.

14. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 14 Choice of the Green’s func. G(P) Helmholtz eqs. G(P)= (from Huygens-Fresnel principle),

15. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 15 3.3.3 The integral thm. of Helmholtz and Kirchhoff The Kirchhoff formulation of the diffraction problem is based on a certain integral theorem which expresses the solution of the homogenous wave equation at an arbitrary point in terms of the values of the solution and its first derivative on an arbitrary closed surface surrounding that point. The problem is to express the optical disturbance at in terms of its values on the surface S. To solve this problem, we follow Kirchhoff in applying Green’s theorem and in choosing as an auxiliary function a unti-amplitude spherical wave expanding about the point (the so-called free space Green’s function).

16. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 16 Within the volume, the disturbance G, being simply an expanding spherical wave, satisfies the Helmholtz equation as

17. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 17 Fig. 3.2 Surface of integration

18. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 18 Choice of the adequate volume denoted as satisfying the requirement of Green thm. a.The first requirement of Green thm. is that exist in the volume of integration. b. The second requirement…(see Goodman P.42) are continuous within the volume

19. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 19 (3-4) This result is known as the integral theorem of Helmholtz and Kirchhoff, it plays an important role in the development of the scalar theory of diffraction, for it allows the field at any point to be expressed in terms of the “boundary value” of the wave on any closed surface surrounding that point.

20. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 20 3.4 The Kirchhoff formula of diffraction by a planar screen Fig. 3.3 Kirchhoff formula of diffraction by a plane screen

21. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 21 3.4.1 Application of the integral theorem uniformly bounded As R→

22. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 22 3.4.2 The Kirchhoff boundary conditions Kirchhoff accordingly adopted the following assumptions [162]: 1. Across the surface, the field distribution U and its derivative are exactly the same as they would be in the absence of the screen. 2. Over the portion of that lies in the geometrical shadow of the screen, the filed distribution U and its derivative are identically zero. These conditions are commonly knows as the Kirchhoff boundary conditions.

23. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 23 3.4.3 The Fresnel-Kirchhoff diffraction formula Fig. 3.4 Point-source illumination a plane screen.

24. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 24 Note ： (3-5)

25. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 25 (Fowles) (3-6) Let the illumination source be a point source. (3-7)

26. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 26 3.5 The Rayleigh-Sommerfeld formulation diffraction It is a well-known theorem of potential theory that if a two- dimensional potential function and its normal derivative vanish together along any finite curve segment, then that potential function must vanish over the entire plane. Similarly, if a solution of the three-dimension wave equation vanishes on any finite surface element, it must vanish in all space.

27. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 27 3.5.1 Choice of alternative Green’s function The conditions for validity of this equation are: 1.The scalar theory holds. 2. Both U and G satisfy the homogeneous scalar wave equation. Helmoltz equation ） 3. The Sommerfeld radiation condition is satisfied.

28. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 28 ( Kirchhoff spherical wavelet) (3-8) (Sommerfeld choose)

29. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 29 Fig. 3.5 Rayleigh-Sommerfeld formulation of diffraction by a plane screen

30. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 30 (refer to eq. (3-5)) (3-9) Choose Green func.

31. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 31 (3-10) (3-11)

32. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 32 3.5.2 The Rayleigh-Sommerfeld diffraction formula Rayleigh-Sommerfeld diffraction formula (3-12) (3-13) Fresnel-Kirchhoff diffraction formula (3-14) (3-15)

33. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 33 3.6 Comparison of the Kirchhoff and Rayleigh-Sommerfeld theories Diffraction by a planar screen ∴

34. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 34 Kirchhoff (one singular point) Sommerfeld (multiple singular point)

35. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 35 Scalar diffraction formula (general form) (Sommerfeld radiation condition) 2G (Kirchhoff) 2 (Kirchhoff)

36. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 36 A comparison of the above equations leads us to an interesting and surprising conclusion: the Kirchhoff solution is the arithmetic average of the two Rayleigh-Sommerfeld solution. In closing it is worth nothing that, in spite of its internal inconsistencies, there is one sense in which the Kirchhoff theory is more general than the Rayleigh-Sommerfeld theory. The latter requires that the diffracting screens be planar, while the former does not.

37. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 37 3.7 Further discussion of the Huygens- Fresnel principle It expresses the observed field U( ) as a superposition of diverging spherical Huygens-Fresnel wavelet origination from secondary source located at each every point within the aperture. (3-16)

38. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 38 where the impulse response is given explicitly by (3-17)

39. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 39 3.8 Generalization to nonmonochromatic wave (Nonomonochromatic time function) (3-22) (3-23) For nonmonochromatic light where the phasor implies the Fourier transform (FT) of the disturbance with respect to (w, r, t) the temporal frequency.

40. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 40 Monochromatic source Free space (nondispersive medium) Note:

41. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 41 or (3-24) (3-25) Since

42. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 42 Eq. (3-25) becomes or V is the velocity of propagation of the disturbance in a medium of refractive index n (v=c/n), and the relation or has been used. (3-26)

43. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 43 3.9 Diffraction at boundaries A more physical point-of-view, first qualitatively expressed by Thomas Young in 1802, is to 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. The possibility of a new wave origination in the material medium of the rim makes this interpretation a more physical one. The applicability of a boundary diffraction approach in more general diffraction problems was investigated by Maggi [202] and Rubinowicz [249], who showed that the Kirchhoff diffraction formula can indeed be manipulated to yield a form that is equivalent to Young’s ideas. More recently, Miyamoto and Wolf [250] have extended the theory of boundary diffraction.

44. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 44 3.10 The angular spectrum of plane waves Objective: To formulate scalar diffraction theory in a framework that closely resembles the theory of linear, invariant system. As we shall see, (1)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 traveling in different directions always from that plane. (2)The field amplitude at any other point (or across any other parallel plane) can be calculated by adding the contributions of these plane waves, taking due account of the phase shifts they have undergone during propagation.

45. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 45 Young’s experiment

46. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 46 3.10.1 The angular spectrum and its physical interpretation Let the complex field, across z=0 plane, be represented by U(x, y, 0). Our ultimate objective is to calculate the resulting field U(x, y, z). And the inverse Fourier transform of its spectrum (3-27) (3-28) The Fourier transform of U(x, y, 0), i.e., its spectrum Phasor Temporal freq. Spatial freq.

47. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 47 To give physical meaning to the function in the integrand of the above integral, consider the plane wave Where the phasor

48. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 48 Thus, across the plane z=0, We see that The angular spectrum of the disturbance

49. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 49 3.10.2 Propagation of the angular spectrum Consider the angular spectrum of the disturbance (3-29) (3-30) into the Helmholz eq.

50. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 50 gives (3-31) where and for all true direction cosines. An elementary solution of Eq. (3-31) can be written in the form (3-32)

51. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 51 (3-33) where the circ function limits the region of integration to the region within which is satisfied. Note: When, and are no longer interpretable as direction cosines. Eq. (3-32) can be rewritten as Where Finally, the substitution of Eq. (3-32) into Eq. (3-30) yields

52. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 52

53. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 53 3.10.3 Effects of a diffracting aperture on the angular spectrum Define the amplitude transmittance function in the z=0 plane, Then And by the use of the convolution theorem. where

54. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 54 And is again the symbol for convolution. For example, if the incident wave illuminates the diffracting structure normally.

55. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 55 In this case, the transmitted angular spectrum is found directly by Fourier transforming the amplitude transmittance function of the aperture. Note that, if the diffracting structure is an aperture that limits the extent of the field distribution, the is a broadening of the angular spectrum of the disturbance, from smaller the aperture, the broader the angular spectrum behind the aperture.

56. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 56 3.10.4 The propagation phenomenon as a linear spatial filter From Eq. (3-33) in Sec. 3.10.2, we have where and, and we have again explicitly introduced the bandwidth limitation associated with evanescent waves thru the use of a circ. function.

57. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 57 From Eq. (3-32) in Sec. 3.10.2 and Eq. (3-29) Thus the propagation phenomenon may be regarded as a linear, dispersive spatial filter with a finite bandwidth:

58. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 58 (1) Within the circular bandwidth, the modulus of the transfer function is unity but frequency-dependent phase shifts are introduced. (2) The phase dispersion of the system is most significant at high spatial frequency and vanishes as both and approach zero. (3) For any fixed spatial frequency pair the phase dispersion increases as the distance of propagation z increases. In closing we mention the remarkable fact that despite the apparent different of their approaches, the angular spectrum approach and the first Rayleigh-Sommerfeld solution yield identical predictions of diffracted fields.