Part 1: Impact of sidewall Roughness on Integrated Bragg Gratings

Transcription

1 Part 1: Impact of sidewall Roughness on Integrated Bragg Gratings Alexandre D. Simard Silicon Nanophotonic Course 2012

2 AFM measurement shows that the top-down process leaves vertical stripes, which justify the use of the effective index approximation. The sidewall fluctuations follows a normal distribution. y z h x Si SiO 2 w 0 15 nm 7.5 nm [1] K. P. Yap et al., SOI waveguide fabrication process development using star coupler scattering loss measurements, in Proceedings of SPIE, nm AFM : Atomic force microscopy (the resolution is ~ 0.1-1nm) 2

3 Outline 1) Sidewall roughness model 2) Emulator Experiments Simulations 3) Numerical analysis of apodized uniform gratings 4) Weak grating analysis Unapodized grating Gaussian-apodized grating 3

4 Outline 1) Sidewall roughness model 2) Emulator Experiments Simulations 3) Numerical analysis of apodized uniform gratings 4) Weak grating analysis Unapodized grating Gaussian-apodized grating 4

5 The sidewall roughness follows a normal distribution and its spectral density function (G x ) is obtained with a Fourier transform of its autocorrelation function (R x ). z x 1 L/2 R x ( z) = lim L x( z) x( z z) dz L + L/2 R x z = Lc ( z) σ 2 exp 2σ L G x ( f z ) = 1 + 4π 2 c Lc f z G x /σ 2 (nm) L c = 10 nm L c = 50 nm L c = 100 nm f z (µm) -1 Typical values : σ ~ 2 nm - 4 nm // L c ~ 30 nm, 300 nm, 3000 nm? 5

6 Outline 1) Sidewall roughness model 2) Emulator Experiments Simulations 3) Numerical analysis of apodized uniform gratings 4) Weak grating analysis Unapodized grating Gaussian-apodized grating 6

7 To analyze the impact of sidewall roughness on IBGs, the roughness perturbation must be transposed to a detuning perturbation Waveguide sidewall perturbation (σ and L c ) δ ( z) 2 π n( z, λ) π 1 θ ( z) = λ Λ 2 z σ δ π 2 CFEM n ref Λ σ 7

8 To analyze the impact of sidewall roughness on IBGs, the roughness perturbation must be transposed to a detuning perturbation Waveguide sidewall perturbation (σ and L c ) δ ( z) 2 π n( z, λ) π 1 θ ( z) = λ Λ 2 z Effective index perturbation (σ neff and L c ) σ δ π 2 CFEM n ref Λ σ Detuning perturbation (σ δ and L c ) Emulator 8

9 To analyze the impact of sidewall roughness on IBGs, the roughness perturbation must be transposed to a detuning perturbation Waveguide sidewall perturbation (σ and L c ) δ ( z) 2 π n( z, λ) π 1 θ ( z) = λ Λ 2 z σ neff ~ 2C FEM σ Effective index perturbation (σ neff and L c ) σ δ π 2 CFEM n ref Λ σ n=c FEM w Detuning perturbation (σ δ and L c ) Emulator 9

10 To analyze the impact of sidewall roughness on IBGs, the roughness perturbation must be transposed to a detuning perturbation Waveguide sidewall perturbation (σ and L c ) n eff (λ) = n 0 + n 1 λ δ ( z) 2 π n( z, λ) π 1 θ ( z) = λ Λ 2 z σ neff ~ 2C FEM σ Effective index perturbation (σ neff and L c ) σ δ π 2 CFEM n ref Λ σ n=c FEM w Detuning perturbation (σ δ and L c ) Emulator 10

11 To analyze the impact of sidewall roughness on IBGs, the roughness perturbation must be transposed to a detuning perturbation Waveguide sidewall perturbation (σ and L c ) n eff (λ) = n 0 + n 1 λ δ ( z) 2 π n( z, λ) π 1 θ ( z) = λ Λ 2 z σ neff ~ 2C FEM σ Effective index perturbation (σ neff and L c ) σ δ π 2 CFEM n ref Λ σ n=c FEM w Detuning perturbation (σ δ and L c ) Emulator 11

12 The sidewall roughness high frequencies are not relevant to analyze the IBG spectral response, so the Gaussian noise can be filtered a second time by a low-pass filter having f th as a threshold frequency randn(length(z),1)/sqrt(mean(diff(z))) Generating a white Gaussian noise (WGN) Identify the ideal grating structure (δ and κ) G x /σ 2 (nm) L c = 10 nm L c = 50 nm L c = 100 nm Simulate the noisy grating structure using the couple mode theory f z (µm) -1 12

13 The sidewall roughness high frequencies are not relevant to analyze the IBG spectral response, so the Gaussian noise can be filtered a second time by a low-pass filter having f th as a threshold frequency randn(length(z),1)/sqrt(mean(diff(z))) Generating a white Gaussian noise (WGN) Identify the ideal grating structure (δ and κ) Filtering with a Lorentzian function having an amplitude of 2L c σ 2 δ G x /σ 2 (nm) L c = 10 nm L c = 50 nm L c = 100 nm Simulate the noisy grating structure using the couple mode theory f z (µm) -1 13

14 The sidewall roughness high frequencies are not relevant to analyze the IBG spectral response, so the Gaussian noise can be filtered a second time by a low-pass filter having f th as a threshold frequency randn(length(z),1)/sqrt(mean(diff(z))) Generating a white Gaussian noise (WGN) Identify the ideal grating structure (δ and κ) Filtering with a Lorentzian function having an amplitude of 2L c σ 2 δ Low pass filtering with a threshold frequency of f c G x /σ 2 (nm) L c = 10 nm L c = 50 nm L c = 100 nm Simulate the noisy grating structure using the couple mode theory f z (µm) -1 14

15 The sidewall roughness high frequencies are not relevant to analyze the IBG spectral response, so the Gaussian noise can be filtered a second time by a low-pass filter having f th as a threshold frequency randn(length(z),1)/sqrt(mean(diff(z))) Generating a white Gaussian noise (WGN) Identify the ideal grating structure (δ and κ) Generating δ and κ vectors with N > 1/2f c Filtering with a Lorentzian function having an amplitude of 2L c σ 2 δ Low pass filtering with a threshold frequency of f c G x /σ 2 (nm) L c = 10 nm L c = 50 nm L c = 100 nm Simulate the noisy grating structure using the couple mode theory f z (µm) -1 15

16 The Gaussian noise can be low-pass filtered because high frequencies have out-of-band impact on the IBG spectral response 2π n = n( λ) + ncos z + φ sin 2π fm z Λ ( ) 2π n = n( λ) + n Jm ( φ ) cos z + m2π fm z m= Λ fm ΛλB λ = 1 2n Λ + f Λ g M f M 2ng 2ng λ = 2 λ λ B 1 λb B λ A. D. Simard, N. Ayotte, Y. Painchaud, S. Bedard, et S. LaRochelle, "Impact of Sidewall Roughness on Integrated Bragg Gratings," Journal of Lightwave Technology, vol. 29, n. 24, p , Dec

17 The Gaussian noise can be low-pass filtered because high frequencies have out-of-band impact on the IBG spectral response 2π n = n( λ) + ncos z + φ sin 2π fm z Λ ( ) 2π n = n( λ) + n Jm ( φ ) cos z + m2π fm z m= Λ fm ΛλB λ = 1 2n Λ + f Λ g M f M 2ng 2ng λ = 2 λ λ B 1 λb B λ A. D. Simard, N. Ayotte, Y. Painchaud, S. Bedard, et S. LaRochelle, "Impact of Sidewall Roughness on Integrated Bragg Gratings," Journal of Lightwave Technology, vol. 29, n. 24, p , Dec

18 The Gaussian noise can be low-pass filtered because high frequencies have out-of-band impact on the IBG spectral response 2π n = n( λ) + ncos z + φ sin 2π fm z Λ ( ) 2π n = n( λ) + n Jm ( φ ) cos z + m2π fm z m= Λ fm ΛλB λ = 1 2n Λ + f Λ g M f M 2ng 2ng λ = 2 λ λ B 1 λb B λ A. D. Simard, N. Ayotte, Y. Painchaud, S. Bedard, et S. LaRochelle, "Impact of Sidewall Roughness on Integrated Bragg Gratings," Journal of Lightwave Technology, vol. 29, n. 24, p , Dec

19 The Gaussian noise can be low-pass filtered because high frequencies have out-of-band impact on the IBG spectral response 2π n = n( λ) + ncos z + φ sin 2π fm z Λ ( ) 2π n = n( λ) + n Jm ( φ ) cos z + m2π fm z m= Λ fm ΛλB λ = 1 2n Λ + f Λ g M f M 2ng 2ng λ = 2 λ λ B 1 λb B λ A. D. Simard, N. Ayotte, Y. Painchaud, S. Bedard, et S. LaRochelle, "Impact of Sidewall Roughness on Integrated Bragg Gratings," Journal of Lightwave Technology, vol. 29, n. 24, p , Dec

20 The Gaussian noise can be low-pass filtered because high frequencies have out-of-band impact on the IBG spectral response f c = 30 x 10 3 m -1 f c = 60 x 10 3 m -1 20

21 Outline 1) Sidewall roughness model 2) Emulator Experiments Simulations 3) Numerical analysis of apodized uniform gratings 4) Weak grating analysis Unapodized grating Gaussian-apodized grating 21

22 A set of simulation can provide the fraction of gratings that meet specific requirement with the presence roughness. If the yield is not sufficient, a design modification can be done without requiring fabrications and measurements, hence saving time and effort. σ = 4 nm, L c = 300 nm 22

23 A comparison between uniform and Gaussian apodized uniform grating shows that the improvement of the side-lobe suppression ratio is considerably diminished by the roughness Unapodized grating 23

24 A comparison between uniform and Gaussian apodized uniform grating shows that the improvement of the side-lobe suppression ratio is considerably diminished by the roughness Unapodized grating Gaussian-apodized grating 24

25 Outline 1) Sidewall roughness model 2) Emulator Experiments Simulations 3) Numerical analysis of apodized uniform gratings 4) Weak grating analysis Unapodized grating Gaussian-apodized grating 25

26 An analytic derivation of the average grating gives fruitful quantitative information of the grating and the noise parameters impact on the IBG spectral response. L/ 2 r = dzκ ( z) e L/ 2 L/2 iδ ( z) r = dzκ ( z) e e L/ 2 2 i dz ' δ ( z ') 0 z 2 i dz ' δ ( z ') L/ 2 z L/ 2 ζ iδ ( z) 2 i dz ' δ ( z ') iδ ( ζ ) 2 i dζ ' δ ( ζ ') 0 0 E [ rr *] = E dzκ ( z) e e dζκ ( ζ ) e e L/ 2 L/ 2 L/2 L/2 iδ ( ζ ) iδ ( z) E rr * dz dζ E e e κ( ζ ) e κ( z) e [ ] { 0 0 = L/2 L/2 L/2 L/2 0 z ζ 2 i dζ ' δ ( ζ ') 2 i dz ' δ ( z ') [ ] { ( ) 0 E rr * dz dζ 1 E δ ( z) δ ( ζ ) κ ( ζ ) e κ( z) e 0 + L/2 L/2 ζ L : Grating length z 2 i dζ ' δ ( ζ ') 2 i dz ' δ ( z ') z [ ] E rr R R * + n 26

27 An analytic derivation of the average grating gives fruitful quantitative information of the grating and the noise parameters impact on the IBG spectral response. L/ 2 r = dzκ ( z) e L/ 2 L/2 iδ ( z) r = dzκ ( z) e e L/ 2 z 2 i dz ' δ ( z ') 0 z 2 i dz ' δ ( z ') L/ 2 z L/ 2 ζ iδ ( z) 2 i dz ' δ ( z ') iδ ( ζ ) 2 i dζ ' δ ( ζ ') 0 0 E [ rr *] = E dzκ ( z) e e dζκ ( ζ ) e e L/ 2 L/ 2 L/2 L/2 iδ ( ζ ) iδ ( z) E rr * dz dζ E e e κ( ζ ) e κ( z) e [ ] { 0 0 = L/2 L/2 L/2 L/2 0 ζ 2 i dζ ' δ ( ζ ') 2 i dz ' δ ( z ') [ ] { ( ) 0 E rr * dz dζ 1 E δ ( z) δ ( ζ ) κ ( ζ ) e κ( z) e 0 + L/2 L/2 ζ L : Grating length z 2 i dζ ' δ ( ζ ') 2 i dz ' δ ( z ') z [ ] E rr R R * + n 27

28 An analytic derivation of the average grating gives fruitful quantitative information of the grating and the noise parameters impact on the IBG spectral response. L/ 2 r = dzκ ( z) e L/ 2 L/2 iδ ( z) r = dzκ ( z) e e L/ 2 z 2 i dz ' δ ( z ') 0 z 2 i dz ' δ ( z ') L/ 2 z L/ 2 ζ iδ ( z) 2 i dz ' δ ( z ') iδ ( ζ ) 2 i dζ ' δ ( ζ ') 0 0 E [ rr *] = E dzκ ( z) e e dζκ ( ζ ) e e L/ 2 L/ 2 L/2 L/2 iδ ( ζ ) iδ ( z) E rr * dz dζ E e e κ( ζ ) e κ( z) e [ ] { 0 0 = L/2 L/2 L/2 L/2 0 ζ 2 i dζ ' δ ( ζ ') 2 i dz ' δ ( z ') [ ] { ( ) 0 E rr * dz dζ 1 E δ ( z) δ ( ζ ) κ ( ζ ) e κ( z) e 0 + L/2 L/2 ζ L : Grating length z 2 i dζ ' δ ( ζ ') 2 i dz ' δ ( z ') z [ ] E rr R R * + n 28

29 An analytic derivation of the average grating gives fruitful quantitative information of the grating and the noise parameters impact on the IBG spectral response. L/ 2 r = dzκ ( z) e L/ 2 L/2 iδ ( z) r = dzκ ( z) e e L/ 2 z 2 i dz ' δ ( z ') 0 z 2 i dz ' δ ( z ') L/ 2 z L/ 2 ζ iδ ( z) 2 i dz ' δ ( z ') iδ ( ζ ) 2 i dζ ' δ ( ζ ') 0 0 E [ rr *] = E dzκ ( z) e e dζκ ( ζ ) e e L/ 2 L/ 2 L/2 L/2 iδ ( ζ ) iδ ( z) E rr * dz dζ E e e κ( ζ ) e κ( z) e [ ] { 0 0 = L/2 L/2 L/2 L/2 0 ζ 2 i dζ ' δ ( ζ ') 2 i dz ' δ ( z ') [ ] { ( ) 0 E rr * dz dζ 1 E δ ( z) δ ( ζ ) κ ( ζ ) e κ( z) e 0 + L/2 L/2 ζ L : Grating length z 2 i dζ ' δ ( ζ ') 2 i dz ' δ ( z ') z [ ] E rr R R * + n 29

30 The noise contribution is linearly proportional to L and L c, but has a quadratic dependency in function of κl, σ and C FEM. [ ] E rr R R * + n π CFEMσ κ L 1 2 Rn 2Lc cos L 2 n ref δ ( Lδ ) sin 2 = ( δ ) Λ 2 2Lδ κl = 1, L c = 40 nm, σ = 2 nm L = 1 mm, L c = 40 nm, σ = 2 nm Rn (x10-3 ) Wavelength (nm) L = 1 mm L = 2 mm L = 3 mm Rn (x10-3 ) 0-50 κl = 1 κl = Wavelength (nm) κl = 1/2 30

31 The noise contribution is linearly proportional to L and L c, but has a quadratic dependency in function of κl, σ and C FEM. [ ] E rr R R * + n π CFEMσ κ L 1 2 Rn 2Lc cos L 2 n ref δ ( Lδ ) sin 2 = ( δ ) Λ 2 2Lδ κl = 1, L c = 40 nm, σ = 2 nm L = 1 mm, L c = 40 nm, σ = 2 nm Rn (x10-3 ) Wavelength (nm) L = 1 mm L = 2 mm L = 3 mm Rn (x10-3 ) 0-50 κl = 1 κl = Wavelength (nm) κl = 1/2 31

32 To prove the validity of this derivation, the mean of a thousand simulated uniform gratings is compared to the analytic expression of E[rr*] L L c σ 32

33 However, to analyze the spectral distortion, it is the variance of the spectral response that should be analysed. [ ] E rr * rr * = E dze κe dz ' e κe L/2 L/2 [ * *] [ *] 2 σ = L/2 iδ L/2 ( z) 2iδ i ( z ' d z δ ) 2 iδ d z ' L/2 iδ L/2 ( ζ ) 2i δ i ( ' d ζ δ ζ ) 2 iδ dζ ' dζ e κe dζ ' e κe L/2 L/2 ( ) 2 R E rr rr E rr σ FEM max ( ) L L ( L) 2 R c C n ref σ Λ κ 33

34 However, to analyze the spectral distortion, it is the variance of the spectral response that should be analysed. [ ] E rr * rr * = E dze κe dz ' e κe L/2 L/2 [ * *] [ *] 2 σ = L/2 iδ L/2 ( z) 2iδ i ( z ' d z δ ) 2 iδ d z ' L/2 iδ L/2 ( ζ ) 2i δ i ( ' d ζ δ ζ ) 2 iδ dζ ' dζ e κe dζ ' e κe L/2 L/2 ( ) 2 R E rr rr E rr σ FEM max ( ) L L ( L) 2 R c C n ref σ Λ κ 34

35 However, to analyze the spectral distortion, it is the variance of the spectral response that should be analysed. [ ] E rr * rr * = E dze κe dz ' e κe L/2 L/2 [ * *] [ *] 2 σ = L/2 iδ L/2 ( z) 2iδ i ( z ' d z δ ) 2 iδ d z ' L/2 iδ L/2 ( ζ ) 2i δ i ( ' d ζ δ ζ ) 2 iδ dζ ' dζ e κe dζ ' e κe L/2 L/2 ( ) 2 R E rr rr E rr σ FEM max ( ) L L ( L) 2 R c C n ref σ Λ κ 35

36 However, to analyze the spectral distortion, it is the variance of the spectral response that should be analysed. [ ] E rr * rr * = E dze κe dz ' e κe L/2 L/2 [ * *] [ *] 2 σ = L/2 iδ L/2 ( z) 2iδ i ( z ' d z δ ) 2 iδ d z ' L/2 iδ L/2 ( ζ ) 2i δ i ( ' d ζ δ ζ ) 2 iδ dζ ' dζ e κe dζ ' e κe L/2 L/2 ( ) 2 R E rr rr E rr σ FEM max ( ) L L ( L) 2 R c C n ref σ Λ κ 36

37 However, to analyze the spectral distortion, it is the variance of the spectral response that should be analysed. [ ] E rr * rr * = E dze κe dz ' e κe L/2 L/2 [ * *] [ *] 2 σ = L/2 iδ L/2 ( z) 2iδ i ( z ' d z δ ) 2 iδ d z ' L/2 iδ L/2 ( ζ ) 2i δ i ( ' d ζ δ ζ ) 2 iδ dζ ' dζ e κe dζ ' e κe L/2 L/2 ( ) 2 R E rr rr E rr σ FEM max ( ) L L ( L) 2 R c C n ref σ Λ κ 37

38 What s next? FEM max ( σ ) ( κ L) 2 R C n ref σ Λ L L c 38

39 What s next? σ FEM max ( ) ( L) 2 R C n ref σ Λ κ L c L 39

40 What s next? Design Fabrication max FEM ( σ ) ( κ L) 2 R C n ref σ Λ L c L 40

41 What s next? Design Fabrication SIDE VIEW max FEM ( σ ) ( κ L) 2 R C n ref σ Λ L c L CFEM [1] S. Selvaraja et. al., Subnanometer Linewidth Uniformity in Silicon Nanophotonic Waveguide Devices Using CMOS Fabrication Technology, IEEE J. Sel. Top. Quantum Electron., vol. 16, n. 1, p ,

42 Conclusion We examined the importance of sidewall roughness on IBGs spectral response. We proposed a technique to emulate IBG spectral responses in the presence of an imperfect waveguide. We presented an analytic study of sidewall roughness. This analysis shows why the noise has a relatively small impact on the reflection strength at the Bragg wavelength while larger distortions appear on each side of the main reflection peak. We have shown that the noise contribution to the grating distortion increases as the square of κl, linearly with σ and C FEM, and as a square root of L and L c. 42

43 Part 2: Characterization of Integrated Bragg Gratings Alexandre D. Simard Silicon Nanophotonic Course 2012

44 Due to the sidewall roughness, the design of long integrated Bragg gratings (IBG) on Silicon-on-Insulator (SOI) must be done carefully. The use of hybrid multimode/singlemode waveguide considerably decrease the phase noise caused by sidewall roughness 44

45 Outline 1) Spectral Measurement 45

46 Outline 1) Spectral Measurement 2) Grating Reconstruction 46

47 Outline 1) Spectral Measurement 3) Silicon layer thickness characterization 2) Grating Reconstruction 47

48 Outline 1) Spectral Measurement 3) Silicon layer thickness characterization 2) Grating Reconstruction 4) Apodisation Characterization 48

49 Outline 1) Spectral Measurement 3) Silicon layer thickness characterization 2) Grating Reconstruction 4) Apodisation Characterization 49

50 To analyze IBGs reflection spectrum, a time filtering must be done to suppress the interference with spurious reflections A. D. Simard, Y. Painchaud, and S. LaRochelle, "Characterization of Integrated Bragg Grating Profiles," Bragg Gratings, Photosensitivity and Poling, Colorado Springs,

51 To analyze IBGs reflection spectrum, a time filtering must be done to suppress the interference with spurious reflections Cleaved fibre reflections A. D. Simard, Y. Painchaud, and S. LaRochelle, "Characterization of Integrated Bragg Grating Profiles," Bragg Gratings, Photosensitivity and Poling, Colorado Springs,

52 Outline 1) Spectral Measurement 3) Silicon layer thickness characterization 2) Grating Reconstruction 4) Apodisation Characterization 52

53 An integral layer peeling algorithm was applied to the complex reflection spectrum of the measured IBG 53

54 An integral layer peeling algorithm was applied to the complex reflection spectrum of the measured IBG ILP A. Rosenthal, and M. Horowitz, IEEE J. Quantum Electron., vol. 39, no. 8, pp , Aug

55 Only the low spatial frequencies have an impact on the grating spectral response. The high frequency components are only noise 2π n = n( λ) + ncos z + φ sin 2π fm z Λ ( ) 2π n = n( λ) + n Jm ( φ ) cos z + m2π fm z m= Λ fm ΛλB λ = 1 2n Λ + f Λ g M f M 2ng 2ng λ = 2 λ λ B 1 λb B λ A. D. Simard, N. Ayotte, Y. Painchaud, S. Bedard, et S. LaRochelle, "Impact of Sidewall Roughness on Integrated Bragg Gratings," Journal of Lightwave Technology, vol. 29, n. 24, p , Dec

56 Since high spatial frequencies have an out-of-band impact on the IBG spectral response, the spatial profile must be low-pass filtered f c f c = 30 x 10 3 m -1 2n g λ λ 2 B Measurement noise level -35 db f c = 60 x 10 3 m db 56

57 Since high spatial frequencies have an out-of-band impact on the IBG spectral response, the spatial profile must be low-pass filtered 57

58 Since high spatial frequencies have an out-of-band impact on the IBG spectral response, the spatial profile must be low-pass filtered f c = 3 x 10 3 m -1 58

59 Outline 1) Spectral Measurement 3) Silicon layer thickness characterization 2) Grating Reconstruction 4) Apodisation Characterization 59

60 The use of hybrid singlemode/multimode IBG strongly reduces the phase noise caused by sidewall roughness which allows characterizing the low frequency components of the thickness of the silicon layer λb n = = CWTV h 2Λ h = ( λ ( ) ) B z λb 2ΛC WTV 60

61 The use of hybrid singlemode/multimode IBG strongly reduces the phase noise caused by sidewall roughness which allows characterizing the low frequency components of the thickness of the silicon layer λb n = = CWTV h 2Λ Obtained using a Finite element mode solver h = ( λ ( ) ) B z λb 2ΛC WTV 61

62 The use of hybrid singlemode/multimode IBG strongly reduces the phase noise caused by sidewall roughness which allows characterizing the low frequency components of the thickness of the silicon layer λb n = = CWTV h 2Λ Obtained using a Finite element mode solver h = ( λ ( ) ) B z λb 2ΛC WTV 62

63 The use of hybrid singlemode/multimode IBG strongly reduces the phase noise caused by sidewall roughness which allows characterizing the low frequency components of the thickness of the silicon layer λb n = = CWTV h 2Λ Obtained using a Finite element mode solver h = ( λ ( ) ) B z λb 2ΛC WTV σ = 0.11 nm 63

64 Outline 1) Spectral Measurement 3) Silicon layer thickness characterization 2) Grating Reconstruction 4) Apodisation Characterization 64

65 We tested two approaches that controlled the apodization profile while letting unchanged the effective index. 1) Superposition apodization 2π n = neff ( λ) + neff f ( z)cos z + θ( z) Λ neff, LG 2π neff, RG 2π n = neff ( λ) + cos z + θlg ( z) + F( z) + cos z + θrg ( z) F( z) 2 Λ LG 2 Λ RG 2π n = neff ( λ) + neff cos ( F ( z) ) cos z + θ ( z) Λ ( f z ) F ( z) = arccos ( ) Out of phase In-phase A. D. Simard, N. Belhadj, Y. Painchaud, and S. LaRochelle, «Apodized Silicon-on-Insulator Bragg Gratings," IEEE Photon. Technol. Lett.,

66 We tested two approaches that controlled the apodization profile while letting unchanged the effective index. 1) Superposition apodization 2π n = neff ( λ) + neff f ( z)cos z + θ( z) Λ neff, LG 2π neff, RG 2π n = neff ( λ) + cos z + θlg ( z) + F( z) + cos z + θrg ( z) F( z) 2 Λ LG 2 Λ RG 2π n = neff ( λ) + neff cos ( F ( z) ) cos z + θ ( z) Λ ( f z ) F ( z) = arccos ( ) A. D. Simard, N. Belhadj, Y. Painchaud, and S. LaRochelle, «Apodized Silicon-on-Insulator Bragg Gratings," IEEE Photon. Technol. Lett.,

67 We tested two approaches that controlled the apodization profile while letting unchanged the effective index. 2) Phase apodization 2π n = neff ( λ) + neff f ( z)cos z + θ( z) Λ 2π n = neff + neff cos z + θ ( z) + φ( z)sin ωm z Λ ( ) 2π n = neff + neff J m ( φ ( z) ) cos z + mω M z + θ ( z) m= Λ A. D. Simard, N. Belhadj, Y. Painchaud, and S. LaRochelle, «Apodized Silicon-on-Insulator Bragg Gratings," IEEE Photon. Technol. Lett.,

68 We tested two approaches that controlled the apodization profile while letting unchanged the effective index. 2) Phase apodization 2π n = neff ( λ) + neff f ( z)cos z + θ( z) Λ 2π n = neff + neff cos z + θ ( z) + φ( z)sin ωm z Λ ( ) 2π n = neff + neff J m ( φ ( z) ) cos z + mω M z + θ ( z) m= Λ A. D. Simard, N. Belhadj, Y. Painchaud, and S. LaRochelle, «Apodized Silicon-on-Insulator Bragg Gratings," IEEE Photon. Technol. Lett.,

69 We tested two approaches that controlled the apodization profile while letting unchanged the effective index. 2) Phase apodization 2π n = neff ( λ) + neff f ( z)cos z + θ( z) Λ 2π n = neff + neff cos z + θ ( z) + φ( z)sin ωm z Λ ( ) 2π n = neff + neff J m ( φ ( z) ) cos z + mω M z + θ ( z) m= Λ A. D. Simard, N. Belhadj, Y. Painchaud, and S. LaRochelle, «Apodized Silicon-on-Insulator Bragg Gratings," IEEE Photon. Technol. Lett.,

70 We successfully characterized the precision of those two apodisation techniques 1) Superposition apodization 2) Phase apodization 70

71 Conclusion We extracted the IBG spectral responses by time windowing of the device impulse response thereby eliminating unwanted broadband reflections. We showed that the filtering of high spatial frequencies of the reconstructed profiles was appropriate for noise suppression. Using the phase information of the grating, we examined the silicon layer thickness variation. Using the amplitude information of the grating, we examined the precision of different apodization technique 71