Abstract

A novel method for measuring local stress distributions and birefringence of films on substrates and planar optical waveguides, with submicrometric resolution, is presented. The technique relies on a reflective tomographic configuration, applied in conjunction with a polarimetric setup, which processes the stress-induced change of the state of polarization of a laser probe beam reflected at the waveguide–substrate (film–substrate) interface. By this means, theoretically foreseen stress behavior can be experimentally verified and spurious or induced local stress variations in integrated optics components can also be brought into evidence. The feasibility of the proposed method has been verified by reconstructing the two-dimensional axial stress distribution in the 4×2μm2 core region of a doped silica-on-silicon optical waveguide.

© 2008 Optical Society of America

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  1. S. M. Ojha, C. Cureton, T. Bricheno, S. Day, D. Moule, A. J. Bell, and J. Taylor, “Simple method of fabricating polarization-insensitive and very low crosstalk AWG grating devices,” Electron. Lett. 34, 78-79 (1998).
    [CrossRef]
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    [CrossRef]
  3. S. Suzuki, S. Sumida, Y. Inoue, M. Ishii, and Y. Ohmori, “Polarization-insensitive arrayed-waveguide gratings using dopant-rich silica-based glass with thermal expansion adjusted to Si substrate,” Electron. Lett. 33, 1173-1174 (1997).
    [CrossRef]
  4. E. Wildermuth, Ch. Nadler, M. Lanker, W. Hunziker, and H. Melchior, “Penalty-free polarization compensation of SiO2/Si arrayed waveguide grating wavelength multiplexers using stress release grooves,” Electron. Lett. 34, 1161-1163 (1998).
    [CrossRef]
  5. H. Takahashi, Y. Hibino, Y. Ohmori, and M. Kawachi, “Polarization-insensitive arrayed-waveguide wavelength multiplexer with birefringence compensating film,” IEEE Photonics Technol. Lett. 5, 707-709 (1993).
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  19. S.-I. Ro, H.-C. Kim, and Y.-W. Park, “Residual stress measuring system for optical fiber,” European patent 1,411,334 (April 21, 2004), http://v3.espacenet.com.
  20. Y. Park, T.-J. Ahn, Y. H. Kim, W.-T. Han, U.-C. Peak, and D. Y. Kim, “Measurement method for profiling the residual stress and the strain-optic coefficient of an optical fiber,” Appl. Opt. 41, 21-26 (2002).
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  22. J. R. Shewchuk, “An introduction to the conjugate gradient method without the agonizing pain,” http://www.cs.cmu.edu/~quake-papers/painless-conjugate-gradient.pdf. (August 1994).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  34. M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 2002).
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2006 (1)

M. Huang, “The influence of light propagation direction on the stress-induced polarization dependence of silicon waveguides,” IEEE Photonics Technol. Lett. 18, 1314-1316 (2006) and references therein.
[CrossRef]

2004 (2)

M. Huang, “Analytical solutions for thermal stresses in buried channel waveguides,” IEEE J. Quantum Electron. 40, 1562-1568 (2004).
[CrossRef]

M. Ferrario, S. M. Pietralunga, M. Torregiani, and M. Martinelli, “Modification of local stress-induced birefringence in low-PMD spun fibers evaluated by high-resolution optical tomography,” IEEE Photonics Technol. Lett. 16, 2634-2636 (2004).
[CrossRef]

2003 (3)

2002 (1)

2001 (1)

M. Fukuzawa and M. Yamada, “Photoelastic characterization of Si wafers by scanning infrared polariscope,” J. Cryst. Growth 229, 22-25 (2001).
[CrossRef]

2000 (3)

1998 (3)

X. Cheng and D. Boas, “Diffuse optical reflection tomography with continuous-wave illumination,” Opt. Express 3, 118-123 (1998).
[CrossRef] [PubMed]

S. M. Ojha, C. Cureton, T. Bricheno, S. Day, D. Moule, A. J. Bell, and J. Taylor, “Simple method of fabricating polarization-insensitive and very low crosstalk AWG grating devices,” Electron. Lett. 34, 78-79 (1998).
[CrossRef]

E. Wildermuth, Ch. Nadler, M. Lanker, W. Hunziker, and H. Melchior, “Penalty-free polarization compensation of SiO2/Si arrayed waveguide grating wavelength multiplexers using stress release grooves,” Electron. Lett. 34, 1161-1163 (1998).
[CrossRef]

1997 (1)

S. Suzuki, S. Sumida, Y. Inoue, M. Ishii, and Y. Ohmori, “Polarization-insensitive arrayed-waveguide gratings using dopant-rich silica-based glass with thermal expansion adjusted to Si substrate,” Electron. Lett. 33, 1173-1174 (1997).
[CrossRef]

1996 (1)

X. Chang, “Relationship between ray distribution and reconstructed velocity image in reflection tomography,” J. Appl. Geophys. 35, 145-150 (1996).
[CrossRef]

1994 (1)

H. H. Yaffe, C. H. Henry, R. F. Kazarinov, and M. A. Milbrodt, “Polarization-independent silica-on-silicon Mach-Zehnder interferometers,” J. Lightwave Technol. 12, 64-67 (1994).
[CrossRef]

1993 (1)

H. Takahashi, Y. Hibino, Y. Ohmori, and M. Kawachi, “Polarization-insensitive arrayed-waveguide wavelength multiplexer with birefringence compensating film,” IEEE Photonics Technol. Lett. 5, 707-709 (1993).
[CrossRef]

1992 (1)

A. E. Puro and K.-J. E. Kell, “Complete determination of stress in fiber preforms of arbitrary cross section,” J. Lightwave Technol. 10, 1010-1014 (1992).
[CrossRef]

1986 (1)

Abe, T.

Ahn, T.-J.

Ankiewicz, A.

Aslund, M.

Bell, A. J.

S. M. Ojha, C. Cureton, T. Bricheno, S. Day, D. Moule, A. J. Bell, and J. Taylor, “Simple method of fabricating polarization-insensitive and very low crosstalk AWG grating devices,” Electron. Lett. 34, 78-79 (1998).
[CrossRef]

Bell, M. R.

P. D. Walker and M. R. Bell, “Subsurface permittivity estimation from ground-penetrating radar measurements,” in Proceedings of IEEE International Radar Conference (IEEE, 2000), pp. 341-346.

Boas, D.

Bording, R. P.

D. Churchill, S. Padina, and R. P. Bording, “Seismic tomography as a high performance application,” in Proceedings of the 20th International Symposium on High-Performance Computing in an Advanced Collaborative Enviromental (HPCS'06) (IEEE, 2006), pp. 32-39.
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 2002).

Bricheno, T.

S. M. Ojha, C. Cureton, T. Bricheno, S. Day, D. Moule, A. J. Bell, and J. Taylor, “Simple method of fabricating polarization-insensitive and very low crosstalk AWG grating devices,” Electron. Lett. 34, 78-79 (1998).
[CrossRef]

Canning, J.

Cardimona, S.

S. Cardimona, “Subsurface investigation using ground penetrating radar,” http://www.dot.ca.gov/hq/esc/geotech/gg/geophysics2002/059cardimona_%20radar_overview.pdf. (Retrieved March 26, 2008).

Chang, X.

X. Chang, “Relationship between ray distribution and reconstructed velocity image in reflection tomography,” J. Appl. Geophys. 35, 145-150 (1996).
[CrossRef]

Cheng, X.

Chu, T.

M. Yamada and T. Chu, “Microscopic observation of strain induced in heteropitaxial layers with reflection type of infrared polariscope,” J. Cryst. Growth 210, 102-106 (2000).
[CrossRef]

Churchill, D.

D. Churchill, S. Padina, and R. P. Bording, “Seismic tomography as a high performance application,” in Proceedings of the 20th International Symposium on High-Performance Computing in an Advanced Collaborative Enviromental (HPCS'06) (IEEE, 2006), pp. 32-39.
[CrossRef]

Crocco, L.

L. Crocco and F. Soldovieri, “GPR prospecting in a layered medium via microwave tomography,” Ann. Geophys. (Germany) 43, 559-572 (2003).

Cureton, C.

S. M. Ojha, C. Cureton, T. Bricheno, S. Day, D. Moule, A. J. Bell, and J. Taylor, “Simple method of fabricating polarization-insensitive and very low crosstalk AWG grating devices,” Electron. Lett. 34, 78-79 (1998).
[CrossRef]

Dainese, M.

Day, S.

S. M. Ojha, C. Cureton, T. Bricheno, S. Day, D. Moule, A. J. Bell, and J. Taylor, “Simple method of fabricating polarization-insensitive and very low crosstalk AWG grating devices,” Electron. Lett. 34, 78-79 (1998).
[CrossRef]

Du-Nour, O.

O. Du-Nour and Y. Ish-Shalom, “Method and apparatus for measuring stress in semiconductor wafer,” U.S. patent 0,098,704 (May 29, 2003), http://v3.espacenet.com.

Fernando, H.

Ferrario, M.

M. Ferrario, S. M. Pietralunga, M. Torregiani, and M. Martinelli, “Modification of local stress-induced birefringence in low-PMD spun fibers evaluated by high-resolution optical tomography,” IEEE Photonics Technol. Lett. 16, 2634-2636 (2004).
[CrossRef]

Fukuzawa, M.

M. Fukuzawa and M. Yamada, “Photoelastic characterization of Si wafers by scanning infrared polariscope,” J. Cryst. Growth 229, 22-25 (2001).
[CrossRef]

M. Yamada, K. Ito, and M. Fukuzawa, “Photoelastic characterization of undoped semi-insulating GaAs wafers with a high-spatial-resolution infrared polariscope,” in Proceedings of the IEEE SIMC-9 (IEEE, 1996), pp. 177-180.

Gdoutos, E. E.

P. S. Theocaris and E. E. Gdoutos, “Description of polarized light,” in Matrix Theory of Photoelasticity (Springer-Verlag, 1979), pp. 20-44.

Han, W.-T.

Henry, C. H.

H. H. Yaffe, C. H. Henry, R. F. Kazarinov, and M. A. Milbrodt, “Polarization-independent silica-on-silicon Mach-Zehnder interferometers,” J. Lightwave Technol. 12, 64-67 (1994).
[CrossRef]

Hibino, Y.

H. Takahashi, Y. Hibino, Y. Ohmori, and M. Kawachi, “Polarization-insensitive arrayed-waveguide wavelength multiplexer with birefringence compensating film,” IEEE Photonics Technol. Lett. 5, 707-709 (1993).
[CrossRef]

Huang, M.

M. Huang, “The influence of light propagation direction on the stress-induced polarization dependence of silicon waveguides,” IEEE Photonics Technol. Lett. 18, 1314-1316 (2006) and references therein.
[CrossRef]

M. Huang, “Analytical solutions for thermal stresses in buried channel waveguides,” IEEE J. Quantum Electron. 40, 1562-1568 (2004).
[CrossRef]

Hunziker, W.

E. Wildermuth, Ch. Nadler, M. Lanker, W. Hunziker, and H. Melchior, “Penalty-free polarization compensation of SiO2/Si arrayed waveguide grating wavelength multiplexers using stress release grooves,” Electron. Lett. 34, 1161-1163 (1998).
[CrossRef]

Inoue, Y.

S. Suzuki, S. Sumida, Y. Inoue, M. Ishii, and Y. Ohmori, “Polarization-insensitive arrayed-waveguide gratings using dopant-rich silica-based glass with thermal expansion adjusted to Si substrate,” Electron. Lett. 33, 1173-1174 (1997).
[CrossRef]

Ishii, M.

S. Suzuki, S. Sumida, Y. Inoue, M. Ishii, and Y. Ohmori, “Polarization-insensitive arrayed-waveguide gratings using dopant-rich silica-based glass with thermal expansion adjusted to Si substrate,” Electron. Lett. 33, 1173-1174 (1997).
[CrossRef]

Ish-Shalom, Y.

O. Du-Nour and Y. Ish-Shalom, “Method and apparatus for measuring stress in semiconductor wafer,” U.S. patent 0,098,704 (May 29, 2003), http://v3.espacenet.com.

Ito, K.

M. Yamada, K. Ito, and M. Fukuzawa, “Photoelastic characterization of undoped semi-insulating GaAs wafers with a high-spatial-resolution infrared polariscope,” in Proceedings of the IEEE SIMC-9 (IEEE, 1996), pp. 177-180.

Jeong, J. R.

Y. S. Kim, J. R. Jeong, and S. C. Shin, “Apparatus for measuring stress in a thin film and method of manufacturing a probe used therefor,” U.S. patent 6,476,906 (November 5, 2002), http://v3.espacenet.com.

Kak, A. C.

A. C. Kak and M. Slaney, “Aliasing artifacts and noise in CT images,” in Principles of Computerized Tomographic Imaging (IEEE, 1988), pp. 177-201.

A. C. Kak and M. Slaney, “Algorithms for reconstruction with nondiffracting sources,” in Principles of Computerized Tomographic Imaging (IEEE, 1988), pp. 49-112.

A. C. Kak and M. Slaney, “Reflection tomography,” in Principles of Computerized Tomographic Imaging (IEEE, 1988), pp. 297-322.

Kawachi, M.

H. Takahashi, Y. Hibino, Y. Ohmori, and M. Kawachi, “Polarization-insensitive arrayed-waveguide wavelength multiplexer with birefringence compensating film,” IEEE Photonics Technol. Lett. 5, 707-709 (1993).
[CrossRef]

Kazarinov, R. F.

H. H. Yaffe, C. H. Henry, R. F. Kazarinov, and M. A. Milbrodt, “Polarization-independent silica-on-silicon Mach-Zehnder interferometers,” J. Lightwave Technol. 12, 64-67 (1994).
[CrossRef]

Kell, K.-J. E.

A. E. Puro and K.-J. E. Kell, “Complete determination of stress in fiber preforms of arbitrary cross section,” J. Lightwave Technol. 10, 1010-1014 (1992).
[CrossRef]

Kilian, A.

Kim, D. Y.

Kim, H.-C.

S.-I. Ro, H.-C. Kim, and Y.-W. Park, “Residual stress measuring system for optical fiber,” European patent 1,411,334 (April 21, 2004), http://v3.espacenet.com.

Kim, J.-H.

J.-H. Kim, S.-H. Oh, Y.-W. Park, U.-C. Peak, and D. Y. Kim, “Apparatus and method for measuring residual stress and photoelastic effect of optical fiber,” U.S. patent 01,26,944 (September 12, 2002), http://v3.espacenet.com.

Kim, Y. H.

Kim, Y. S.

Y. S. Kim, J. R. Jeong, and S. C. Shin, “Apparatus for measuring stress in a thin film and method of manufacturing a probe used therefor,” U.S. patent 6,476,906 (November 5, 2002), http://v3.espacenet.com.

Kirchhof, J.

Koga, H.

Kuhlow, B.

Kurtz, D. S.

D. S. Kurtz, “Apparatus for rapid in-situ X-ray stress measurement during thermal cycling of semiconductor wafers,” U.S. patent 5,848,122 (December 8, 1998), http://v3.espacenet.com.

Lanker, M.

E. Wildermuth, Ch. Nadler, M. Lanker, W. Hunziker, and H. Melchior, “Penalty-free polarization compensation of SiO2/Si arrayed waveguide grating wavelength multiplexers using stress release grooves,” Electron. Lett. 34, 1161-1163 (1998).
[CrossRef]

Li, C.

Martinelli, M.

M. Ferrario, S. M. Pietralunga, M. Torregiani, and M. Martinelli, “Modification of local stress-induced birefringence in low-PMD spun fibers evaluated by high-resolution optical tomography,” IEEE Photonics Technol. Lett. 16, 2634-2636 (2004).
[CrossRef]

Melchior, H.

E. Wildermuth, Ch. Nadler, M. Lanker, W. Hunziker, and H. Melchior, “Penalty-free polarization compensation of SiO2/Si arrayed waveguide grating wavelength multiplexers using stress release grooves,” Electron. Lett. 34, 1161-1163 (1998).
[CrossRef]

Milbrodt, M. A.

H. H. Yaffe, C. H. Henry, R. F. Kazarinov, and M. A. Milbrodt, “Polarization-independent silica-on-silicon Mach-Zehnder interferometers,” J. Lightwave Technol. 12, 64-67 (1994).
[CrossRef]

Mitsunaga, Y.

Moule, D.

S. M. Ojha, C. Cureton, T. Bricheno, S. Day, D. Moule, A. J. Bell, and J. Taylor, “Simple method of fabricating polarization-insensitive and very low crosstalk AWG grating devices,” Electron. Lett. 34, 78-79 (1998).
[CrossRef]

Nadler, Ch.

E. Wildermuth, Ch. Nadler, M. Lanker, W. Hunziker, and H. Melchior, “Penalty-free polarization compensation of SiO2/Si arrayed waveguide grating wavelength multiplexers using stress release grooves,” Electron. Lett. 34, 1161-1163 (1998).
[CrossRef]

Oh, S.-H.

J.-H. Kim, S.-H. Oh, Y.-W. Park, U.-C. Peak, and D. Y. Kim, “Apparatus and method for measuring residual stress and photoelastic effect of optical fiber,” U.S. patent 01,26,944 (September 12, 2002), http://v3.espacenet.com.

Ohmori, Y.

S. Suzuki, S. Sumida, Y. Inoue, M. Ishii, and Y. Ohmori, “Polarization-insensitive arrayed-waveguide gratings using dopant-rich silica-based glass with thermal expansion adjusted to Si substrate,” Electron. Lett. 33, 1173-1174 (1997).
[CrossRef]

H. Takahashi, Y. Hibino, Y. Ohmori, and M. Kawachi, “Polarization-insensitive arrayed-waveguide wavelength multiplexer with birefringence compensating film,” IEEE Photonics Technol. Lett. 5, 707-709 (1993).
[CrossRef]

Ojha, S. M.

S. M. Ojha, C. Cureton, T. Bricheno, S. Day, D. Moule, A. J. Bell, and J. Taylor, “Simple method of fabricating polarization-insensitive and very low crosstalk AWG grating devices,” Electron. Lett. 34, 78-79 (1998).
[CrossRef]

Padina, S.

D. Churchill, S. Padina, and R. P. Bording, “Seismic tomography as a high performance application,” in Proceedings of the 20th International Symposium on High-Performance Computing in an Advanced Collaborative Enviromental (HPCS'06) (IEEE, 2006), pp. 32-39.
[CrossRef]

Park, Y.

Park, Y.-W.

S.-I. Ro, H.-C. Kim, and Y.-W. Park, “Residual stress measuring system for optical fiber,” European patent 1,411,334 (April 21, 2004), http://v3.espacenet.com.

J.-H. Kim, S.-H. Oh, Y.-W. Park, U.-C. Peak, and D. Y. Kim, “Apparatus and method for measuring residual stress and photoelastic effect of optical fiber,” U.S. patent 01,26,944 (September 12, 2002), http://v3.espacenet.com.

Peak, U.-C.

Pietralunga, S. M.

M. Ferrario, S. M. Pietralunga, M. Torregiani, and M. Martinelli, “Modification of local stress-induced birefringence in low-PMD spun fibers evaluated by high-resolution optical tomography,” IEEE Photonics Technol. Lett. 16, 2634-2636 (2004).
[CrossRef]

Przyrembel, G.

Puro, A. E.

A. E. Puro and K.-J. E. Kell, “Complete determination of stress in fiber preforms of arbitrary cross section,” J. Lightwave Technol. 10, 1010-1014 (1992).
[CrossRef]

Ro, S.-I.

S.-I. Ro, H.-C. Kim, and Y.-W. Park, “Residual stress measuring system for optical fiber,” European patent 1,411,334 (April 21, 2004), http://v3.espacenet.com.

Sahu, J. K.

Shewchuk, J. R.

J. R. Shewchuk, “An introduction to the conjugate gradient method without the agonizing pain,” http://www.cs.cmu.edu/~quake-papers/painless-conjugate-gradient.pdf. (August 1994).

Shin, S. C.

Y. S. Kim, J. R. Jeong, and S. C. Shin, “Apparatus for measuring stress in a thin film and method of manufacturing a probe used therefor,” U.S. patent 6,476,906 (November 5, 2002), http://v3.espacenet.com.

Slaney, M.

A. C. Kak and M. Slaney, “Reflection tomography,” in Principles of Computerized Tomographic Imaging (IEEE, 1988), pp. 297-322.

A. C. Kak and M. Slaney, “Algorithms for reconstruction with nondiffracting sources,” in Principles of Computerized Tomographic Imaging (IEEE, 1988), pp. 49-112.

A. C. Kak and M. Slaney, “Aliasing artifacts and noise in CT images,” in Principles of Computerized Tomographic Imaging (IEEE, 1988), pp. 177-201.

Soldovieri, F.

L. Crocco and F. Soldovieri, “GPR prospecting in a layered medium via microwave tomography,” Ann. Geophys. (Germany) 43, 559-572 (2003).

Sumida, S.

S. Suzuki, S. Sumida, Y. Inoue, M. Ishii, and Y. Ohmori, “Polarization-insensitive arrayed-waveguide gratings using dopant-rich silica-based glass with thermal expansion adjusted to Si substrate,” Electron. Lett. 33, 1173-1174 (1997).
[CrossRef]

Suzuki, S.

S. Suzuki, S. Sumida, Y. Inoue, M. Ishii, and Y. Ohmori, “Polarization-insensitive arrayed-waveguide gratings using dopant-rich silica-based glass with thermal expansion adjusted to Si substrate,” Electron. Lett. 33, 1173-1174 (1997).
[CrossRef]

Takahashi, H.

H. Takahashi, Y. Hibino, Y. Ohmori, and M. Kawachi, “Polarization-insensitive arrayed-waveguide wavelength multiplexer with birefringence compensating film,” IEEE Photonics Technol. Lett. 5, 707-709 (1993).
[CrossRef]

Taylor, J.

S. M. Ojha, C. Cureton, T. Bricheno, S. Day, D. Moule, A. J. Bell, and J. Taylor, “Simple method of fabricating polarization-insensitive and very low crosstalk AWG grating devices,” Electron. Lett. 34, 78-79 (1998).
[CrossRef]

Theocaris, P. S.

P. S. Theocaris and E. E. Gdoutos, “Description of polarized light,” in Matrix Theory of Photoelasticity (Springer-Verlag, 1979), pp. 20-44.

Torregiani, M.

M. Ferrario, S. M. Pietralunga, M. Torregiani, and M. Martinelli, “Modification of local stress-induced birefringence in low-PMD spun fibers evaluated by high-resolution optical tomography,” IEEE Photonics Technol. Lett. 16, 2634-2636 (2004).
[CrossRef]

Walker, P. D.

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Figures (16)

Fig. 1
Fig. 1

Schematic detail of the guiding layer, composed of an upper and lower cladding and the core region.

Fig. 2
Fig. 2

Ray refraction due to the strong refractive index difference between the silicon substrate and the silica layers.

Fig. 3
Fig. 3

Polarimetric reflection tomographic setup with counter-rotating arms and detail of the optical waveguide in the beaker.

Fig. 4
Fig. 4

(a) Prospective view of the Poincaré sphere for visualizing the input beam’s SOP evolution while passing through the forward arm of the tomographich setup of Fig. 3. (b) Top view of the SOP evolution on the Poincaré sphere.

Fig. 5
Fig. 5

Poincaré sphere visualization of the light’s SOP evolution while passing through the overall reflection tomographic setup, depending on the angle β: (a) prospective view and (b) top view of the Poincaré sphere.

Fig. 6
Fig. 6

Effect of reflection at the cladding–substrate interface: Intensities of the vertical ( I V ) and horizontal ( I H ) components of the reflected light are no longer equal. (a) The ratio γ varies depending on the projection angle θ. (b, c) Poincaré sphere visualization of the light’s SOP evolution due to reflection for two different incident angles θ.

Fig. 7
Fig. 7

Ray path inside the real waveguide (a) and in the mirrored one (b) as seen by the tomographic algorithm.

Fig. 8
Fig. 8

(a) Prospective view of the Poincaré sphere for visualizing the input beam’s SOP evolution while passing through the reflection tomographic setup of Fig. 3 when the incident angle θ exceeds Brewster’s angle θ B . (b) Top view of the SOP evolution on the Poincaré sphere.

Fig. 9
Fig. 9

Measured phase retardation profile at θ = 45 ° .

Fig. 10
Fig. 10

Synogram.

Fig. 11
Fig. 11

(a) Reconstructed two-dimensional axial stress distribution in the optical Si O 2 Si waveguide core region and (b) top view of the stress distribution.

Fig. 12
Fig. 12

(a) Evaluation of the stress homogeneity along the propagation direction z. (b) Phase retardation projections taken at different z positions for θ = 50 ° . (c) Resulting average phase retardation profile and corresponding standard deviation σ. (d) Resulting two-dimensional average stress profile section and standard deviation σ.

Fig. 13
Fig. 13

Simulations performed to evaluate the noise level in reconstructed stress distributions: (a) stress level in the ideal waveguide, (b) stress distribution reconstructed with 46 phase projections with no noise added, (c) stress profile reconstructed with 46 phase projections with a white Gaussian-distributed noise added.

Fig. 14
Fig. 14

Representation of an elliptically polarized light by point P on the Poincaré sphere.

Fig. 15
Fig. 15

SOP path ( D E ¯ ) induced by Fresnel reflection on the Poincaré sphere when θ < θ B : prospective view (a) and top view (b).

Fig. 16
Fig. 16

SOP path ( D E ¯ ) induced by Fresnel reflection on the Poincaré sphere when θ > θ B : prospective view (a) and top view (b).

Equations (22)

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Δ n eff n h 2 n 0 n eff A w [ ( C 2 C 1 ) ( σ x x σ z z ) n 0 2 n h 2 A t ( C 2 C 1 ) ( σ y y σ z z ) ] .
σ u c = E u c * ( α s α u c ) Δ T ,
σ l c = E l c * ( α s α l c ) Δ T .
σ x x = ( 1 E core + k y y 1 E u c ) [ α u c α core k y x ( α s α u c ) ] + ( ν core E core + k y x ν u c E u c ) [ α u c α core + k y y ( α s α u c ) ] ( 1 E core + k y y 1 E u c ) ( 1 E core + k x x 1 E u c ) ( ν core E core + k y x ν u c E u c ) ( ν core E core + k x y ν u c E u c ) Δ T ,
σ y y = ( 1 E core + k x x 1 E u c ) [ α u c α core k y y ( α s α u c ) ] + ( ν core E core + k x y ν u c E u c ) [ α u c α core + k y x ( α s α u c ) ] ( 1 E core + k y y 1 E u c ) ( 1 E core + k x x 1 E u c ) ( ν core E core + k y x ν u c E u c ) ( ν core E core + k x y ν u c E u c ) Δ T .
σ x x core σ y y core 4 ( 1 + ν u c ) ( α s α u c ) Δ T 1 + ν core E core + ( 1 + ν u c ) ( 3 4 ν u c ) E u c .
σ z z = E core ( α s α core ) Δ T + ν core ( σ x x + σ y y ) .
δ θ ( x ) = C + σ z z ( x , y ) d y ,
I ( X ) = I 0 sin 2 [ ( 90 ° 4 β δ θ ) 2 ]
δ θ = 90 ° 4 β ¯ .
s 0 = A x 2 + A y 2 , s 1 = A x 2 A y 2 ,
s 2 = 2 A x A y cos δ , s 3 = 2 A x A y sin δ .
s 1 = s 0 cos 2 ω cos 2 ψ , s 2 = s 0 cos 2 ω sin 2 ψ ,
s 3 = s 0 sin 2 ω .
s 1 s 0 = A x 2 A y 2 A x 2 + A y 2 = γ 1 γ + 1
D O ̂ Q + = E F ̂ L .
s 3 D s 2 D = t g 2 ω = t g δ 2 .
s 3 D s 2 D = s 3 E s 2 E .
t g δ 2 = E G ¯ F G ¯ = s 3 E s 2 E .
t g δ 2 = t g δ 2 .
s 2 B = s 2 E , s 3 B = s 3 E ,
t g δ B = t g δ 2 .

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