1. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 1 Chap 4 Fresnel and Fraunhofer Diffraction

2. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 2 Content 4.1 Background 4.2 The Fresnel approximation 4.3 The Fraunhofer approximation 4.4 Examples of Fraunhofer diffraction patterns

3. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 3

4. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 4 4.1 Background These approximations, which are commonly made in many fields that deal with wave propagation, will be referred to as Fresnel and Fraunhofer approximations. In accordance with our view of the wave propagation phenomenon as a “system”, we shall attempt to find approximations that are valid for a wide class of “input” field distributions.

5. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 5 4.1.1 The intensity of a wave field Poynting’s thm. When calculation a diffraction pattern, we will general regard the intensity of the pattern as the quantity we are seeking.

6. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 6 4.1.2 The Huygens-Fresnel principle in rectangular coordinates Before we introducing a series of approximations to the Huygens- Fresnel principle, it will be helping to first state the principle in more explicit from for the case of rectangular coordinates. As shown in Fig. 4.1, the diffracting aperture is assumed to lie in the plane, and is illuminated in the positive z direction. According to Eq. (3-41), the Huygens-Fresnel principle can be stated as (1)

7. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 7 Fig. 4.1 Diffraction geometry

8. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 8 and therefore the Huygens-Fresnel principle can be rewritten (2) (3)

9. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 9 There have been only two approximations in reaching this expression. 1.One is the approximation inherent in the scalar theory

10. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 10 4.2 Fresnel Diffraction Recall, the mathematical formulation of the Huygens-Fresnel, the first Rayleigh- Sommerfeld sol. The Fresnel diffraction means the Fresnel approximation to diffraction between two parallel planes. We can obtain the approximated result. (1)

11. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 11 e (Why?) (wave propagation) wave propagation z Aperture Plane Observation Plane Corresponding to The quadratic-phase exponential with positive phase i.e,,for z>0

12. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 12 Note:The distance from the observation point to an aperture point Using the binominal expansion, we obtain the approximation to =b

13. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 13 as the term is sufficiently small. The first Rayleigh Sommerfeld sol for diffraction between two parallel planes is then approximated by

14. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 14 ( ), the r 01 in denominator of the integrand is supposed to be well approximated by the first term only in the binomial expansion, i.e, In addition, the aperture points and the observation points are confined to the (, ) plane and the (x,y) plane,respectively. ) Thus, we see

15. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 15 Furthermore, Eq(1) can be rewritten as (2a) where the convolution kernel is (2b) Obviously, we may regard the phenomenon of wave propagation as the behavior of a linear system.

16. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 16 Another form of Eq.(1) is found if the term is factored outside the integral signs, it yields (3) which we recognize (aside from the multiplicative factors) to be the Fourier transform of the complex field just to the right of the aperture and a quadratic phase exponential.

17. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 17 We refer to both forms of the result Eqs. (1) and (3), as the Fresnel diffraction integral. When this approximation is valid, the observer is said to be in the region of Fresnel diffraction or equivalently in the near field of the aperture. Note: In Eq(1),the quadratic phase exponential in the integrand

18. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 18 do not always have positive phase for z>0.Its sign depends on the direction of wave propagation. (e.g, diverging of converging spherical waves) In the next subsection,we deal with the problem of positive or negative phase for the quadratic phase exponent.

19. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 19 4.2.1 Positive vs. Negative Phases Since we treat wave propagation as the behavior of a linear system as described in chap.3 of Goodman), it is important to descries the direction of wave propagation. As a example of description of wave propagation direction, if we move in space in such a way as to intercept portions of a wavefield (of wavefronts ) that were emitted earlier in time.

20. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 20 z z z t t t In the above two illustrations, we assume the wave speed v=zc/tc where zc and tc are both fixed real numbers.

21. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 21 In the case of spherical waves, Diverging spherical waveConverging spherical wave

22. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 22 Consider the wave func.,where and r >0 and If,then (  Positive phase) implies a diverging spherical wave. Or if

23. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 23 implies a converging spherical wave. (  Negative phase) Note ： For spherical wave,we say they are diverging or converging ones instead or saying that they are emitted “earlier in time ” or “later in time”.

24. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 24 The term standing for the time dependence of a traveling wave implies that we have chosen our phasors to rotate in the clockwise direction. “ Earlier in time ” Positive phase Specifically, for a time interval t c >0, we see the following relations,

25. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 25 Therefore, we have the following seasonings ： “Earlier in time ” Positive phase (e.g., diverging spherical waves) “Later in time” Negative phase (e.g., converging spherical waves) Note ： “Earlier in time ” means the general statement that if we move in space in such a way as to intercept wavefronts (or portions of a wave-field ) that were emitted earlier in time.

26. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 26 Propagation direction Spatial distribution of wavefronts To describe the direction of wave propagation for plane waves, we cannot use the term diverging or “converging”.Instead.we employ the general statement,for the following situations.

27. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 27 The phasor of a plane wave,, (where multiplied by the time dependence gives, where We may say that,if we move in the positive y direction, the argument of the exponential increases in a positive sense, and thus we are moving to a portion of the wave that was emitted earlier in time. >0)

28. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 28 Propagation direction In a similar fashion, we may deal with the situation for

29. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 29 Note ： Show that the Huygens-Fresnel principle can be expressed by Recall the wave field at observation point P 0 (1)

30. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 30 For the first Rayleigh – Sommerfeld solution,the Green func. Note we put the subscript “ - ”, i.e, G- to signify this kind of Green func. Substituting Eq(2) into Eq.(1) gives (2)

31. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 31 (3) (4) or where the Green func. proposed by Kirchhoff

32. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 32 The term in the integrand of Eq.(4)

33. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 33 as or Finally, substituting Eq.(5) into Eq.(4) yields (5)

34. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 34 4.2.2 Accuracy of Fresnel Approximation Recall Fresnel diffraction integral observation point (fixed) Aperture point (varying withΣ) Parabolic wavelet …(4.14) We compare it with the exact formula Spherical wavelet where (or )

35. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 35 since the binomial expansion where The max.approx.error (i.e.,( ) max ) and the corresponding error of the exponential is maximized at the phase (or approximately 1 radian)

36. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 36 A sufficient condition for accuracy would be <<1 For example (ξ,η)

37. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 37 or This sufficient condition implies that the distance z must be relatively much larger than

38. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 38 Since the binomial expansion (high order term) where we can see that the sufficient condition leads to a sufficient small value of b However, this condition is not necessary. In the following, we will give the next comment that accuracy can be expected for much smaller values of z (i.e., the observation point (x, y) can be located at a relatively much shorter distance to an arbitrary aperture point on the ( ξ, η ) plane)

39. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 39 We basically malcr use of the argument that for the convolution integral of Eq.(4-14), if the major contribution to the integral comes from points ( ξ, η ) for which ξ ≒ x and η ≒ y, then the values of the HOTs of the expansion become sufficiently small.(That is, as ( ξ, η ) is close to (x, y) gives a relatively small value Consequently, can be well approximated by. )

40. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 40 In addition it is found that the convolution integral of Eq.(4-14), or where and,

41. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 41 can be governed by the convolution integral of the function with a second function (i.e., U(ξ,η)) that is smooth and slowly varying for the rang –2 < X < 2 and –2 < Y < 2. Obviously, outside this range, the convolution integral does not yield a significant addition. ( Note For one dimensional case is governed by we can see that is well approximated by

42. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 42 Finally, it appears that the majority of the contribution to the convolution integral for the range -∞ < X < ∞ and -∞ < Y < ∞ or the aperture area Σ comes from that for a square in the (ξ,η) plane with width and centered on the point ξ= x,η= y (i.e., the range –2 < <2 and –2< <2 or < and < ) As a result within the square area, the expansion as well approximated, since is small enough.

43. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 43 From another point of view, since the Fresnel diffraction integral Corresponding square area yields a good approximation to the exact formula where

44. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 44 we may say that for the Fresnel approximation (for the aperture area Σ or the corresponding square area) to give accurate results, it is not necessary that the HOTs of the expansion be small, only that they do not change the value of the Fresnel diffraction integral significantly. Note ： From Goodman’s treatment (P.69 70), we see that can well approximate or Where the width of the diffracting aperture is larger than the length of the region –2 < X < 2

45. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 45 For the scaled quadratic-phase exponential of Eqs.(4-14) and Eq.(4-16), the corresponding conclusion is that the majority of the contribution to the convolution integral comes from a square in the ( ξ, η ) plane, with width and centered on the point ( ξ = x, η = y) In effect, 1.When this square lie entirely within the open portion of the aperture, the field observed at distance z is, to a good approximation, what it would be if the aperture were not present. (This is corresponding to the “light” region)

46. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 46 2.When the square lies entirely behind the obstruction of the aperture, then the observation point lies in a region that is, to a good approximation, dark due to the shadow of the aperture. 3.When the square bridges the open and obstructed parts of the aperture, then the observed field is in the transition (or gray) region between light and dark. For the case of a one-dimensional rectangular slit, boundaries among the regions mentioned above can be shown to be parabolas, as illustrated in the following figure.

47. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 47

48. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 48 Thus, the upper (or lower) boundary between the transition (or gray) region and the light region can be expressed by (or ) The light region W – x ≧, x ≧ 0 W + x ≧, x ＜ 0

49. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 49 4.2.3 The Fresnel approximation and the Angular Spectrum In this subsection, we will see that the Fourier transform of the Fresnel diffraction impression response identical to the transfer func. of the wave propagation phenomenon in the angular spectrum method of analysis, under the condition of small angles. From Eqs.(4-15)and (4-16), We have Where the convolution kernel (or impulse response) is

50. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 50 The FT of the Fresnel diffraction impulse response becomes The integral term can be rewritten a where and

51. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 51 (because the exponents where as a result, =1 P q

52. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 52 so On the other hand, the transfer function of the wave propagation phenomenon in the angular spectrum method of analysis is expressed by under the condition of small angles (as noted below the term) can be approximated by (because )

53. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 53 (Note: because For Fresnel approximation, the sufficient condition ma be The obliquity factor then approaches 1 That is, is small angle

54. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 54 Which is the transfer function of the wave propagation phenomenon in the angular spectrum method of analysis under the condition of small angles. Therefore, we have shown that the FT of the Fresnel diffraction impulse response

55. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 55 4.2.4 Fresnel Diffraction between Confocal Spherical surfaces.

56. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 56 as are all very close to zero, (i.e, the paraxial condition) Recall the Rayleigh Sommerfeld sol, (for the paraxial condition

57. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 57 as a result, for the paraxial region, This Fresnel diffraction eq. expresses the field observed on the right hand spherical cap as the FT of the filed U(x,y) on the left-hand spherical cap. Comparison of the result with Eq(4-17),the Fresnel diffraction integral (including Fourier-transform-like operation) (including the paraxial representation of spherical phase)

58. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 58 quadratic phase parabolic phase Note: Recall

59. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 59 The two quadratic phase factors in Eq(4-17)are in fact simply paraxial representations of spherical phase surfaces, (since the Rayleigh Sommerfeld sol. can be applied only to the planar screens), and it is therefore reasonable that moving to the spheres has eliminated them. For the diffraction between two spherical caps, it is not really valid to use the Rayleigh-Sommerfeld result as the basis for the calculation (only for the diffraction between two parallel planes). However, the Kirchhoff analysis remains valid, and its predictions are the same as those of the Rayleigh-Sommerfeld approach provided paraxial conditions hold.

60. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 60 4.3 The Fraunhofer approximation From Eq(4-17), We see If the exponent We have (4-17)

61. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 61 The observed filed strength U(x,y) can be found directly from a FT of the aperture function itself (because ) That is, Eq.(4-17)with the Fraunhofer approximation becomes (Aside from the multiplicative phase factors, this expression is simply the FT of the aperture distribution) where (4-26) (4-25)

62. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 62 Note ： Recall the different forms of Fresnel diffraction integral where the Fresnel diffraction impulse response (4-16) and that of Eq(4-17)

63. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 63 Comparison of Eqs(4-15)and (4-16) with Eqs.(4-25)and (4-26) tell us that there is no transfer function for the Fraunhofer (or far-field) diffraction since Eqs(4-25) and (4-26) do not include impulse response. Nonetheless, since Fraunhofer diffraction is only a special case of Fresnel diffraction, the transfer function Eq(4-21) remains valid throughout both the Fresnel and the Fraunhofer regimes. That is, it is always possible to calculate diffracted field in the Fraunhofer region by retaining the full accuracy of the Fresnel approximation. Treating the wave propagation phenomenon as a linear system

64. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 64 4.4 Examples of Fraunhofer diffraction patterns 4.4.1 Rectangular Aperture If the aperture is illuminated by a unit-amplitude, normally incident, monochromatic plane wave, then the field distribution across the aperture is equal to the transmittance function.Thus using Eq.(4-25), the Fraunhofer diffraction pattern is seen to be

65. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 65 4.4.2 Circular Aperture Suggests that the Fourier transform of Eq.(4-25) be rewritten as a Fourier-Bessel transform. Thus if is the radius coordinate in the observation plane, we have

66. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 66 4.4.3 Thin Sinusoidal Amplitude Grating In practice, diffracting objects can be far more complex. In accord with our earlier definition (3-68),the amplitude transmittance of a screen is defined as the ratio of the complex field amplitude immediately behind the screen to the complex amplitude incident on the screen. Until now,our examples have involved only transmittance functions of the form

67. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 67 Spatial patterns of phase shift can be introduced by means of transparent plates of varying thickness, thus extending the realizable values of t A to all points within or on the unit circle in the complex plane. As an example of this more general type of diffracting screen, consider a thin sinusoidal amplitude grating defined by the amplitude transmittance function (4-33) where for simplicity we have assumed that the grating structure is bounded by a square aperture of width 2w. The parameter m represents the peak-to-peak change of amplitude transmittance across the screen, and f 0 is the spatial frequency of the grating.

68. Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 68 4.4.4 Thin sinusoidal phase grating Binary phase grating