One-edge Diffraction. Chapter 1. Direction of incidence. Reflected ray 4. Screen y

Transcription

1 Chapter One-edge Diffration The time domain impulse response for edge diffration is derived from Eqn.. aording to Fig... ϕ < ϕ < π is the angle between y and r. ϕ < ϕ < π is the angle between y and the diretion of inidene. Refleted ray 4 r Sreen y Diretion of inidene x Figure.: The wave sattered by a perfetly onduting half-plane. } u jkr osϕϕ + j = e u kr/π os[ϕϕ/] e jπτ / jkr osϕ+ϕ + j dτ+e If r, ϕ and ϕ are determined, then the first term of RHS of Eqn.. is onsidered first. kr/π os[ϕ+ϕ/] e jπτ / dτ. jkr osϕϕ + j H k = e kr/π os[ϕϕ/] e jπτ / dτ.

2 CHAPTER. ONE-EDGE DIFFRACTION If kr/π os[ϕ ϕ /] >, aording to the Fresnel Integral, [ jkr osϕϕ + j H k = e j = e jkr osϕϕ jkr osϕϕ + j e kr/π os[ϕϕ /] kr/π os[ϕϕ /] e jπτ / dτ ].3 e jπτ / dτ.4 Let x = jπτ. So, and, π x = + jτ.5 dτ = + j dx.6 π Let = + jkr os[ϕ ϕ /]. So H k an be simplified as H k = e jkr osϕϕ e jkr osϕϕ π e x / dx.7 Beause erf x = π x et dt, Eqn..7 an be further rewritten as, H k = e jkr osϕϕ ejkr osϕϕ erf.8 From the defintion of, = jkr [osϕ ϕ + ], so, H k = e e jkr e e jkr erf.9 Let k = w, τ a = r[osϕϕ+] and p = jw, where is the speed of light. So, Eqn..9 is hanged to H p = e p r τa πτ a πτ a e τap erf τ a p e p r. Beause of the Laplae transform pairs δ t τ e pτ and t+a domain impulse response for the first item of RHS of Eqn.. is τ a πτ a t u t πa e ap erf a p t u t r t + τ a Re p, the time. where τ a = τ a,i = r[osϕϕ+]. If kr/π os[ϕ ϕ /] =, aording to the Fresnel Integral, H k = ejkr. Further, h t an be obtained, h t = δ t r.3

3 CHAPTER. ONE-EDGE DIFFRACTION If kr/π os[ϕ ϕ /] <, aording to the Fresnel Integral, jkr osϕϕ + j H k = e kr/π os[ϕϕ /] Based on Eqn..5, Eqn..6 and = + jkr os[ϕ ϕ /], Eqn..4 an be simplified as e jπτ / dτ.4 H k = e e jkr erf.5 Further, h t an be obtained, h t = πτ a t u t r t + τ a.6 where τ a = τ a,i = r[osϕϕ+]. If gt is the input signal, then the output signal is s t t = g t h t.7 Let ht = t t+τ a u t. Beause t goes to infinity when t approah, it is infeasible to alulate ht g t diretly. Aording to power series expansion, when t τ a <, h t = t τ a + t u t.8 τ a = t n u t.9 τ a t n= = τ a n= τ a τ a n t n u t. Aording to the property of onvolution, h t g t = d t dt = d dt hτdτ g t τ a n= n τ a n + t n+ u t g t.. So, in this way s t t is obtained. In the other way, beause k = w, H w w = H w from Eqn.., sw t an be ahieved from the following fomula, s w t = IFT {H w w FT {g t}}.3 where, FT { } means Fourier transform and IFT { } means inverse fourier transform. s t t and s w t should be exatly the same from theoretial point of view. If g t is the seond order Gaussian pulse Fig.., t t g t= 4π exp π.4 α α and φ = 7π/4, φ = π/4 as well as r = m, then the distorted terms of s t t and s w t are shown in Fig..3. Similarly, the time domain impulse response for the seond term of RHS of Eqn.. an be obtained as follows.

4 CHAPTER. ONE-EDGE DIFFRACTION The nd Order Gaussian Pulse Time ns Figure.: The seond order Gaussian pulse. 5 x 3 4 S t t t Time ns Figure.3: The distorted terms of s t t and s w t.

5 CHAPTER. ONE-EDGE DIFFRACTION If kr/π os[ϕ + ϕ /] >, then τ a πτ a t u t r t + τ a.5 If kr/π os[ϕ + ϕ /] =, then h t = δ t r.6 If kr/π os[ϕ + ϕ /] <, then h t = πτ a t u t r t + τ a.7 where τ a = τ a,r = r[osϕ+ϕ+]. If g t is the seond order Gaussian pulse Fig.. and φ = 7π/4, φ = π/4 as well as r = m, then the distorted terms of s t t and s w t for this ase are shown in Fig x 3 4 S t t t Time ns Figure.4: The distorted terms of s t t and s w t. In sum, based on the results shown above and the knowledge of geometrial optis, the onlusion is easily obtained as follows: In the Zone < ϕ + ϕ < π of Fig.., the inidene ray, the refleted ray and two overlapped diffrated rays are observed, i.e. τ a,i +δ t τ a,r t u t + t u t δ t r t + τ a,i πτ t + τ a,r a,r πτ a,i.8

6 CHAPTER. ONE-EDGE DIFFRACTION In the Zone ϕ ϕ < π and π < ϕ + ϕ of Fig.., the inidene ray and two overlapped diffrated rays are observed, i.e. τ a,i t u t t u t r.9 πτ t + τ a,i a,i πτ t + τ a,r a,r In the Zone 3 π < ϕ ϕ < π ϕ of Fig.., only two overlapped diffrated rays are observed, i.e. h t = t u t + t u t r πτ t + τ a,i a,i πτ t + τ a,r a,r.3 In the Zone 4 ϕ + ϕ = π of Fig.., the inidene ray, the attenuated refleted ray and one diffrated ray are observed, i.e. τ a,i + δ t τ a,r t u t r.3 t + τ a,i πτ a,i In the Zone 5 ϕ ϕ = π of Fig.., the attenuated inidene ray and one diffrated ray are observed, i.e. h t = r t δ τ a,i + t u t r πτ t + τ a,r a,r.3 In these situations, τ a,i = r[osϕϕ+] and τ a,r = r[osϕ+ϕ+]. In order to verify the above onlusion, g t whih is the seond order gaussian pulse Fig.. is used as the input the signal and the output signal s w t is observed. In the Zone of Fig.., if φ = π/3, φ = π/4 as well as r = m, then r = 3.333ns, r τ a,i =.863ns and r τ a,r = 3.9ns. s w t is shown in Fig..5. The zoomed in version of s w t at.863ns is shown in Fig..6. The zoomed in version of s w t at 3.9ns is shown in Fig..7. The zoomed in version of s w t at 3.333ns is shown in Fig..8. In the Zone of Fig.., if φ = π, φ = π/4 as well as r = m, then r = 3.333ns and r τ a,i =.357ns. s w t is shown in Fig..9. The zoomed in version of s w t at.357ns is shown in Fig... The zoomed in version of s w t at 3.333ns is shown in Fig... In the Zone 3 of Fig.., if φ = 7π/4, φ = π/4 as well as r = m, then r = 3.333ns. s w t is shown in Fig... The zoomed in version of s w t at 3.333ns is shown in Fig..3. In the Zone 4 of Fig.., if φ = 3π/4, φ = π/4 as well as r = m, then r = 3.333ns, r τ a,i = ns and r τ a,r = 3.333ns. s w t is shown in Fig..4. The zoomed in version of s w t at ns is shown in Fig..5. The zoomed in version of s w t at 3.333ns is shown in Fig..6. In the Zone 5 of Fig.., if φ = 5π/4, φ = π/4 as well as r = m, then r = 3.333ns and r τ a,i = 3.333ns. s w t is shown in Fig..7. The zoomed in version of s w t at 3.333ns is shown in Fig..8.

7 CHAPTER. ONE-EDGE DIFFRACTION t aused by the first term t aused by the seond term Time ns Figure.5: s w t in Zone.

8 CHAPTER. ONE-EDGE DIFFRACTION t aused by the first term t aused by the seond term Time ns Figure.6: The zoomed in version of s w t at.863ns in Zone.

9 CHAPTER. ONE-EDGE DIFFRACTION t aused by the first term t aused by the seond term Time ns Figure.7: The zoomed in version of s w t at 3.9ns in Zone.

10 CHAPTER. ONE-EDGE DIFFRACTION.3. t aused by the first term t aused by the seond term Time ns Figure.8: The zoomed in version of s w t at 3.333ns in Zone.

11 CHAPTER. ONE-EDGE DIFFRACTION t aused by the first term t aused by the seond term Time ns Figure.9: s w t in Zone.

12 CHAPTER. ONE-EDGE DIFFRACTION t aused by the first term t aused by the seond term Time ns Figure.: The zoomed in version of s w t at.357ns in Zone.

13 CHAPTER. ONE-EDGE DIFFRACTION..8.6 t aused by the first term t aused by the seond term Time ns Figure.: The zoomed in version of s w t at 3.333ns in Zone.

14 CHAPTER. ONE-EDGE DIFFRACTION 5 x 3 4 t aused by the first term t aused by the seond term Time ns Figure.: s w t in Zone 3.

15 CHAPTER. ONE-EDGE DIFFRACTION 5 x 3 4 t aused by the first term t aused by the seond term Time ns Figure.3: The zoomed in version of s w t at 3.333ns in Zone 3.

16 CHAPTER. ONE-EDGE DIFFRACTION..8 t aused by the first term t aused by the seond term Time ns Figure.4: s w t in Zone 4.

17 CHAPTER. ONE-EDGE DIFFRACTION..8 t aused by the first term t aused by the seond term Time ns Figure.5: The zoomed in version of s w t at ns in Zone 4.

18 CHAPTER. ONE-EDGE DIFFRACTION.5.4 t aused by the first term t aused by the seond term Time ns Figure.6: The zoomed in version of s w t at 3.333ns in Zone 4.

19 CHAPTER. ONE-EDGE DIFFRACTION.5.4 t aused by the first term t aused by the seond term Time ns Figure.7: s w t in Zone 5.

20 CHAPTER. ONE-EDGE DIFFRACTION.5.4 t aused by the first term t aused by the seond term Time ns Figure.8: The zoomed in version of s w t at 3.333ns in Zone 5.

21 Chapter Two-edge Diffration The time domain impulse response for two-edge diffration with a hole between two edges is onsidered here aording to Fig... ϕ is the angle between the right side of sreen and r. ϕ is the angle between the right side of sreen and the diretion of inidene. The diameter of the hole is d. If A is hosen as the referene point, then the time delay when the inidene ray arrives at B is τ d = d os ϕ. Refleted rays 8 7 Diretion of inidene Sreen 3 4 B 5 d r A 6 Sreen 9 Figure.: The wave sattered by two perfetly onduting half-planes with a hole between two edges. The phenomenons of diffration and refletion in A is studied first. In this ase, the inidene ray is onsidered.

22 CHAPTER. TWO-EDGE DIFFRACTION In the Zone of Fig.., the inidene ray, the refleted ray and two overlapped diffrated rays are observed, i.e. τ +δ t τ t u t + t + τ πτ πτ t + τ t u t In the Zone, Zone 3, Zone 5 and Zone 8 of Fig.., the inidene ray and two overlapped diffrated rays are observed, i.e. τ t u t t u t r. t + τ t + τ πτ πτ δ t r. In the Zone 4 and Zone 9 of Fig.., only two overlapped diffrated rays are observed beause the inidene ray is bloked by the left side of the sreen, i.e. h t = t u t t u t r.3 t + τ t + τ πτ πτ In the Zone 6 of Fig.., only two overlapped diffrated rays are observed, i.e. h t = t u t + t + τ πτ πτ t + τ t u t r.4 In the Zone 7 of Fig.., the inidene ray, the attenuated refleted ray are observed and one diffrated ray are observed, i.e. τ + δ t τ t u t r.5 t + τ πτ In the Zone of Fig.., the attenuated inidene ray and one diffrated ray are observed, i.e. h t = r t δ τ + t u t t + τ πτ r.6 In these situations, τ = r[osϕϕ+] and τ = r[osϕ+ϕ+]. The phenomenons of diffration and refletion in B is more omplex ompared with A. In this ase, r B whih is the distane between observation point and B is. ϕ,b = π ϕ. r sin ϕ sin artan r sin ϕ d+r os ϕ If < ϕ < π, then ϕ B = { π artan r sin ϕ d+r os ϕ artan r sin ϕ d+r os ϕ r sin ϕ d+r os ϕ r sin ϕ d+r os ϕ <.7 If π < ϕ < π, then ϕ B = π artan r sin ϕ d+r os ϕ π artan r sin ϕ d+r os ϕ r sin ϕ d+r os ϕ r sin ϕ d+r os ϕ <.8

23 CHAPTER. TWO-EDGE DIFFRACTION τ = rb[osϕbϕ,b+] and τ = rb[osϕb+ϕ,b+]. Meanwhile, the inidene ray is not onsidered here. In the Zone, Zone, Zone 5, Zone 6, Zone 7 and Zone of Fig.., two overlapped diffrated rays are observed, i.e. h t = t u t t u t rb t + τ t + τ + τ d πτ πτ.9 In Zone 3 of Fig.., the refleted ray and two overlapped diffrated rays are observed, i.e. B τ + τ d t u t + t + τ πτ πτ t + τ t u t B δ t + τ d. In Zone 4 of Fig.., two overlapped diffrated rays are observed, i.e. h t = t u t + t + τ πτ πτ t + τ t u t rb + τ d. In Zone 8 of Fig.., the attenuated refleted ray and one diffrated ray are observed, i.e. h t = t δ rb τ + τ d πτ t + τ t u t B + τ d. In Zone 9 of Fig.., the attenuated inidene ray and one diffrated ray are observed, i.e. h t = t δ rb τ a,i + τ d + πτ t + τ t u t B + τ d.3 In sum, the time domain impulse response for two-edge diffration with a hole between two edges an be obtained aording to the priniple of superposition of the field as follows. In the Zone of Fig.., πτ πτ t u t t + τ t u t t + τ τ + δ t τ + πτ πτ t u t r t + τ t u t rb t + τ + τ d.4 In the Zone of Fig.., τ πτ πτ t u t t + τ t u t t + τ πτ πτ t u t r t + τ t u t rb t + τ + τ d.5

24 CHAPTER. TWO-EDGE DIFFRACTION In the Zone 3 of Fig.., τ B δ t τ + τ d πτ πτ t u t t + τ t u t t + τ + πτ πτ t u t r t + τ t u t rb t + τ + τ d.6 In the Zone 4 of Fig.., h t = πτ πτ t u t t + τ t u t t + τ + πτ πτ t u t r t + τ t u t rb t + τ + τ d.7 In the Zone 5 of Fig.., τ πτ πτ t u t t + τ t u t t + τ πτ πτ t u t r t + τ t u t rb t + τ + τ d.8 In the Zone 6 of Fig.., h t = πτ πτ t u t t + τ t u t t + τ + πτ πτ t u t r t + τ t u t rb t + τ + τ d.9 In the Zone 7 of Fig.., τ + δ t τ πτ t u t t + τ πτ πτ t + τ t u t r t u t rb t + τ + τ d. In the Zone 8 of Fig.., τ + δ t πτ t u t t + τ rb τ + τ d πτ πτ t u t r t + τ t + τ t u t B. + τ d

25 CHAPTER. TWO-EDGE DIFFRACTION In the Zone 9 of Fig.., h t = + δ t πτ t u t t + τ rb τ a,i + τ d + πτ πτ t u t r t + τ B + τ d t + τ t u t. In the Zone of Fig.., πτ h t = r t δ τ + t u t t + τ πτ πτ t + τ t u t r t u t rb t + τ + τ d.3