Abstract

We present an analysis of Fourier-transform arrayed waveguide gratings in the Fresnel diffraction regime. We report a distinct spatial modulation of the interference pattern referred to as the Moiré-Talbot effect. The effect and its influence in a FT AWG device is explained by deriving an original analytical expression for the modulated field, and is also confirmed by numerical simulations using the angular spectrum method to solve the Fresnel diffraction integral. We illustrate the retrieval of spectral information in a waveguide Fourier-transform spectrometer in the presence of the Moiré-Talbot effect. The simulated device comprises two interleaved waveguide arrays each with 180 waveguides and the interference order of 40. It is designed with a Rayleigh spectral resolution of 0.1 nm and 8 nm bandwidth at wavelength λ~1.5 µm. We also demonstrate by numerical simulations that the spectrometer crosstalk is reduced from -20 dB to -40 dB by Gaussian apodization.

© 2007 Optical Society of America

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References

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  1. P. Cheben, Wavelength dispersive planar waveguide devices: Echelle gratings and arrayed waveguide gratings," in Optical Waveguides: From Theory to Applied Technologies, M. L. Calvo and V. Laksminarayanan, eds., (CRC Press, London, 2007) Chap. 5.
  2. P. Cheben, J. H. Schmid, A. Delâge, A. Densmore, S. Janz, B. Lamontagne, J. Lapointe, E. Post, P. Waldron, and D.-X. Xu, "A high-resolution silicon-on-insulator arrayed waveguide grating microspectrometer with submicrometer aperture waveguides," Opt. Express 15, 2299-2306 (2007).
    [CrossRef] [PubMed]
  3. P. Jacquinot, "The luminosity of spectrometers with prisms, gratings, or Fabry Perot etalons," J. Opt. Soc. Am. 44, 761 (1954).
    [CrossRef]
  4. P. B. Fellgett, PhD Thesis, University of Cambridge, (1951).
  5. P. Cheben, I. Powell, S. Janz, and D.-X. Xu, "Wavelength-dispersive device based on a Fourier-transform Michelson-type arrayed waveguide grating," Opt. Lett. 30, 1824-1826 (2005).
    [CrossRef] [PubMed]
  6. J. M. Harlander, F. L. Roesler, Ch. R. Englert, J. G. Cardon, R. R. Conway, Ch. M. Brown, and J. Wimperis, "Robust monolithic ultraviolet interferometer for the SHIMMER instrument on STPSat-1," Appl. Opt. 42, 2829-2834 (2003).
    [CrossRef] [PubMed]
  7. M. Florjańczyk, P. Cheben, S. Janz, A. Scott, B. Solheim, and D.-X. Xu, Planar waveguide spatial heterodyne spectrometer, Proc. Photonics North Conference, 4-7 June, 2007, Ottawa, Canada.
  8. P. Cheben, A. Delâge, L. Erickson, S. Janz, and D.-X. Xu, "Polarization compensation in silicon-on-insulator arrayed waveguide grating devices," in Silicon-based and hybrid optoelectronics III, Proc SPIE 4293, 15-22 (2001).
  9. P. Cheben, D.-X. Xu, S. Janz, A. Delâge, and D. Dalacu, "Birefringence compensation in silicon-on-insulator planar waveguide demultiplexers using a buried oxide layer," Proc SPIE  4997, 181-189 (2003).
  10. M. K. Smit and C. van Dam, "Phasar-based WDM-devices: principles, design, and applications," IEEE J. Sel. Top. Quantum Electron. 2, 236 (1996).
    [CrossRef]
  11. D. Mendlovic, Z. Zalevsky, and N. Konforti, "Computation considerations and fast algorithms for calculating the diffraction integral," J. Mod. Opt. 44, 407 (1997).
    [CrossRef]
  12. H. Hamam and J. L. De Bougrenet de la Tocnaye, "Programmable joint fractional Talbot computer-generated holograms," J. Opt. Soc. Am. A 12, 314 (1995).
    [CrossRef]
  13. H. Hamam and J. L. De Bougrenet de la Tocnaye, "Efficient Fresnel transform algorithm based on fractional Fresnel diffraction," J. Opt. Soc. Am. A 12, 1920 (1995).
    [CrossRef]

2007 (1)

2005 (1)

2003 (2)

J. M. Harlander, F. L. Roesler, Ch. R. Englert, J. G. Cardon, R. R. Conway, Ch. M. Brown, and J. Wimperis, "Robust monolithic ultraviolet interferometer for the SHIMMER instrument on STPSat-1," Appl. Opt. 42, 2829-2834 (2003).
[CrossRef] [PubMed]

P. Cheben, D.-X. Xu, S. Janz, A. Delâge, and D. Dalacu, "Birefringence compensation in silicon-on-insulator planar waveguide demultiplexers using a buried oxide layer," Proc SPIE  4997, 181-189 (2003).

2001 (1)

P. Cheben, A. Delâge, L. Erickson, S. Janz, and D.-X. Xu, "Polarization compensation in silicon-on-insulator arrayed waveguide grating devices," in Silicon-based and hybrid optoelectronics III, Proc SPIE 4293, 15-22 (2001).

1997 (1)

D. Mendlovic, Z. Zalevsky, and N. Konforti, "Computation considerations and fast algorithms for calculating the diffraction integral," J. Mod. Opt. 44, 407 (1997).
[CrossRef]

1996 (1)

M. K. Smit and C. van Dam, "Phasar-based WDM-devices: principles, design, and applications," IEEE J. Sel. Top. Quantum Electron. 2, 236 (1996).
[CrossRef]

1995 (2)

1954 (1)

Brown, Ch. M.

Cardon, J. G.

Cheben, P.

P. Cheben, J. H. Schmid, A. Delâge, A. Densmore, S. Janz, B. Lamontagne, J. Lapointe, E. Post, P. Waldron, and D.-X. Xu, "A high-resolution silicon-on-insulator arrayed waveguide grating microspectrometer with submicrometer aperture waveguides," Opt. Express 15, 2299-2306 (2007).
[CrossRef] [PubMed]

P. Cheben, I. Powell, S. Janz, and D.-X. Xu, "Wavelength-dispersive device based on a Fourier-transform Michelson-type arrayed waveguide grating," Opt. Lett. 30, 1824-1826 (2005).
[CrossRef] [PubMed]

P. Cheben, D.-X. Xu, S. Janz, A. Delâge, and D. Dalacu, "Birefringence compensation in silicon-on-insulator planar waveguide demultiplexers using a buried oxide layer," Proc SPIE  4997, 181-189 (2003).

P. Cheben, A. Delâge, L. Erickson, S. Janz, and D.-X. Xu, "Polarization compensation in silicon-on-insulator arrayed waveguide grating devices," in Silicon-based and hybrid optoelectronics III, Proc SPIE 4293, 15-22 (2001).

Conway, R. R.

Dalacu, D.

P. Cheben, D.-X. Xu, S. Janz, A. Delâge, and D. Dalacu, "Birefringence compensation in silicon-on-insulator planar waveguide demultiplexers using a buried oxide layer," Proc SPIE  4997, 181-189 (2003).

De Bougrenet de la Tocnaye, J. L.

Delâge, A.

P. Cheben, J. H. Schmid, A. Delâge, A. Densmore, S. Janz, B. Lamontagne, J. Lapointe, E. Post, P. Waldron, and D.-X. Xu, "A high-resolution silicon-on-insulator arrayed waveguide grating microspectrometer with submicrometer aperture waveguides," Opt. Express 15, 2299-2306 (2007).
[CrossRef] [PubMed]

P. Cheben, D.-X. Xu, S. Janz, A. Delâge, and D. Dalacu, "Birefringence compensation in silicon-on-insulator planar waveguide demultiplexers using a buried oxide layer," Proc SPIE  4997, 181-189 (2003).

P. Cheben, A. Delâge, L. Erickson, S. Janz, and D.-X. Xu, "Polarization compensation in silicon-on-insulator arrayed waveguide grating devices," in Silicon-based and hybrid optoelectronics III, Proc SPIE 4293, 15-22 (2001).

Densmore, A.

Englert, Ch. R.

Erickson, L.

P. Cheben, A. Delâge, L. Erickson, S. Janz, and D.-X. Xu, "Polarization compensation in silicon-on-insulator arrayed waveguide grating devices," in Silicon-based and hybrid optoelectronics III, Proc SPIE 4293, 15-22 (2001).

Hamam, H.

Harlander, J. M.

Jacquinot, P.

Janz, S.

P. Cheben, J. H. Schmid, A. Delâge, A. Densmore, S. Janz, B. Lamontagne, J. Lapointe, E. Post, P. Waldron, and D.-X. Xu, "A high-resolution silicon-on-insulator arrayed waveguide grating microspectrometer with submicrometer aperture waveguides," Opt. Express 15, 2299-2306 (2007).
[CrossRef] [PubMed]

P. Cheben, I. Powell, S. Janz, and D.-X. Xu, "Wavelength-dispersive device based on a Fourier-transform Michelson-type arrayed waveguide grating," Opt. Lett. 30, 1824-1826 (2005).
[CrossRef] [PubMed]

P. Cheben, D.-X. Xu, S. Janz, A. Delâge, and D. Dalacu, "Birefringence compensation in silicon-on-insulator planar waveguide demultiplexers using a buried oxide layer," Proc SPIE  4997, 181-189 (2003).

P. Cheben, A. Delâge, L. Erickson, S. Janz, and D.-X. Xu, "Polarization compensation in silicon-on-insulator arrayed waveguide grating devices," in Silicon-based and hybrid optoelectronics III, Proc SPIE 4293, 15-22 (2001).

Konforti, N.

D. Mendlovic, Z. Zalevsky, and N. Konforti, "Computation considerations and fast algorithms for calculating the diffraction integral," J. Mod. Opt. 44, 407 (1997).
[CrossRef]

Lamontagne, B.

Lapointe, J.

Mendlovic, D.

D. Mendlovic, Z. Zalevsky, and N. Konforti, "Computation considerations and fast algorithms for calculating the diffraction integral," J. Mod. Opt. 44, 407 (1997).
[CrossRef]

Post, E.

Powell, I.

Roesler, F. L.

Schmid, J. H.

Smit, M. K.

M. K. Smit and C. van Dam, "Phasar-based WDM-devices: principles, design, and applications," IEEE J. Sel. Top. Quantum Electron. 2, 236 (1996).
[CrossRef]

van Dam, C.

M. K. Smit and C. van Dam, "Phasar-based WDM-devices: principles, design, and applications," IEEE J. Sel. Top. Quantum Electron. 2, 236 (1996).
[CrossRef]

Waldron, P.

Wimperis, J.

Xu, D.-X.

P. Cheben, J. H. Schmid, A. Delâge, A. Densmore, S. Janz, B. Lamontagne, J. Lapointe, E. Post, P. Waldron, and D.-X. Xu, "A high-resolution silicon-on-insulator arrayed waveguide grating microspectrometer with submicrometer aperture waveguides," Opt. Express 15, 2299-2306 (2007).
[CrossRef] [PubMed]

P. Cheben, I. Powell, S. Janz, and D.-X. Xu, "Wavelength-dispersive device based on a Fourier-transform Michelson-type arrayed waveguide grating," Opt. Lett. 30, 1824-1826 (2005).
[CrossRef] [PubMed]

P. Cheben, D.-X. Xu, S. Janz, A. Delâge, and D. Dalacu, "Birefringence compensation in silicon-on-insulator planar waveguide demultiplexers using a buried oxide layer," Proc SPIE  4997, 181-189 (2003).

P. Cheben, A. Delâge, L. Erickson, S. Janz, and D.-X. Xu, "Polarization compensation in silicon-on-insulator arrayed waveguide grating devices," in Silicon-based and hybrid optoelectronics III, Proc SPIE 4293, 15-22 (2001).

Zalevsky, Z.

D. Mendlovic, Z. Zalevsky, and N. Konforti, "Computation considerations and fast algorithms for calculating the diffraction integral," J. Mod. Opt. 44, 407 (1997).
[CrossRef]

Appl. Opt. (1)

IEEE J. Sel. Top. Quantum Electron. (1)

M. K. Smit and C. van Dam, "Phasar-based WDM-devices: principles, design, and applications," IEEE J. Sel. Top. Quantum Electron. 2, 236 (1996).
[CrossRef]

J. Mod. Opt. (1)

D. Mendlovic, Z. Zalevsky, and N. Konforti, "Computation considerations and fast algorithms for calculating the diffraction integral," J. Mod. Opt. 44, 407 (1997).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Opt. Express (1)

Opt. Lett. (1)

SPIE (1)

P. Cheben, A. Delâge, L. Erickson, S. Janz, and D.-X. Xu, "Polarization compensation in silicon-on-insulator arrayed waveguide grating devices," in Silicon-based and hybrid optoelectronics III, Proc SPIE 4293, 15-22 (2001).

SPIE (1)

P. Cheben, D.-X. Xu, S. Janz, A. Delâge, and D. Dalacu, "Birefringence compensation in silicon-on-insulator planar waveguide demultiplexers using a buried oxide layer," Proc SPIE  4997, 181-189 (2003).

Other (3)

M. Florjańczyk, P. Cheben, S. Janz, A. Scott, B. Solheim, and D.-X. Xu, Planar waveguide spatial heterodyne spectrometer, Proc. Photonics North Conference, 4-7 June, 2007, Ottawa, Canada.

P. Cheben, Wavelength dispersive planar waveguide devices: Echelle gratings and arrayed waveguide gratings," in Optical Waveguides: From Theory to Applied Technologies, M. L. Calvo and V. Laksminarayanan, eds., (CRC Press, London, 2007) Chap. 5.

P. B. Fellgett, PhD Thesis, University of Cambridge, (1951).

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Figures (9)

Fig. 1.
Fig. 1.

General schematics of a FT AWG microspectrometer with two interleaved AWGs. Wavefronts W 1 and W 2 originating from AWG1 and AWG2 propagate with a wavelength dependent tilt angle β(λ) and -β(λ), respectively.

Fig. 2.
Fig. 2.

Interferogram (top) at a Talbot plane and its Fourier spectrum (center and bottom, in linear and logarithmic scales, respectively). Wavelength λ=1502 nm. The peaks (a–g) correspond to the spatial frequencies of 2α (a), fd ±2α j (b–d) and 2 fd ±2αj (e–g). Right panel shows the calculated spectra for the input light with spectral widths (FWHM) of Δλ=0 nm (monochromatic), 0.05 nm and 0.1 nm.

Fig. 3.
Fig. 3.

The waveguide pitch influence on spectral retrieval. The waveguide pitch 3 µm (top), 4 µm (center), and 5 µm (bottom). The terms (a)–(g) of Eq. (8) are identified to help visualize the effect of varying pitch on different spatial frequencies.

Fig. 4.
Fig. 4.

Light interference in the combiner (silicon) and the free propagation (air) regions. a) Talbot effect at Littrow wavelength λ L =1500 nm; b) Moiré-Talbot effect at λ=1508 nm. The silicon-air interface (spectrometer chip edge) is located at z=125 µm, as it is indicated by the grey line in (a) and (b). Inset in figure (a) shows the intensity distribution at the fractional Talbot plane zT /4.

Fig. 5.
Fig. 5.

Interferogram and calculated spectrum for an FT AWG device. Unapodized input field. The interferogram is at the Talbot plane z=zT (top), calculated from Eq. (8). Calculated spectrum (using the FFT algorithm) for wavelengths: 1502 nm, 1502.3 nm, 1504 nm, 1505 nm, 1506 nm and 1508 nm in linear (center) and logarithmic (bottom) scales.

Fig. 6.
Fig. 6.

Interferogram and calculated spectrum for FT AWG device. Gaussian apodized input field. The interferogram is at Talbot plane z=zT (top), calculated from Eq. (8). Calculated spectrum (using the FFT algorithm) for wavelengths: 1502 nm, 1502.3 nm, 1504 nm, 1505 nm, 1506 nm and 1508 nm in linear (center) and logarithmic (bottom) scales.

Fig. 7.
Fig. 7.

Interferogram (top) at the fractional Talbot plane zT /4 calculated from Eq. (14) and its Fourier spectrum (center and bottom, in linear and logarithmic scales, respectively). Unapodized input field. Wavelength λ=1502 nm. The peaks (a–f) correspond to the spatial frequencies of 2α (a), fd ±2α (b,c) and 2fd ±2α j (d–f).

Fig. 8.
Fig. 8.

Interferogram and calculated spectrum for FT AWG device. Gaussian apodized input field. The interferogram is at Talbot plane z=zT /4 (top), calculated from Eq. (14). Calculated spectrum (using the FFT algorithm) for wavelengths: 1502 nm, 1502.3 nm, 1504 nm, 1505 nm, 1506 nm and 1508 nm in linear (center) and logarithmic (bottom) scales.

Fig. 9.
Fig. 9.

Influence of interferogram sampling on spectrum retrieval. Gaussian unapodized input field. The interferogram (top) at Talbot plane z=zT . Calculated spectra for the wavelength of 1502 nm and different sampling frequencies (bottom); sampling periods 2 µm (a), 4 µm (b) and 5 µm (c).

Equations (20)

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Δ ϕ = 2 π m p n g ( λ λ L ) n 0 λ L
sin β = ( λ λ L ) n g m p d n 0 n eff ,
T p ( x ) = t ( x ) exp ( i 2 π α p x ) ,
t ( x ) = q = + a q exp ( i 2 π qx d )
W p ( x , z ) = exp ( i k n eff z ) i λ z T p ( x i ) exp ( i π n eff λ z ( x x i ) 2 ) d x i
= Φ ( z ) exp ( i 2 π α p x ) q = + a q exp ( i 2 π qx d ) exp ( i π λ z q 2 d 2 n eff ) exp ( i 2 π λ z α p q d n eff ) ,
Φ ( z ) = exp ( i k n eff z ) exp ( i π λ z α p 2 n eff ) .
W p ( x , z l ) = Φ ( z l ) t ( x + 2 d 2 l α p ) exp ( i 2 π α p x )
= Φ ( z l ) T p ( x + 2 d 2 l α p ) exp ( i 2 π 2 ( d α p ) 2 l ) .
I ( x , z T l ) = W 1 ( x , z T l ) + W 2 ( x , z T l ) 2 = t ( x + 2 d 2 l α ) + t ( x 2 d 2 l α ) exp ( i 2 π 2 α x ) 2
= t ( x + 2 d 2 l α ) 2 + t ( x 2 d 2 l α ) 2 + 2 t ( x + 2 d 2 l α ) t ( x 2 d 2 l α ) cos ( 2 π 2 α x ) .
W p ( x , z T η ζ ) = Φ ( z T η ζ ) a = 0 ζ 2 1 B ( a , η , ζ ) T p ( x d 2 + 2 η a ζ d + 2 η ζ d 2 α p ) ,
B ( a , η , ζ ) = 2 ζ b = 0 ζ 2 1 exp ( i 2 π 2 ba ζ ) exp ( i π ( 2 b 2 η ζ + b ) )
W p ( x , z T 2 ) = Φ ( z T 2 ) T p ( x d 2 + d 2 α p ) ,
W p ( x , z T 4 ) = Φ ( z T 4 ) a = 0 1 B ( a , 1 , 4 ) T p ( x d 2 + a 2 d + 1 2 d 2 α p )
= Φ ( z T 4 ) 2 ( exp ( i π 4 ) T p ( x d 2 + d 2 α p 2 ) + exp ( i π 4 ) T p ( x + d 2 α p 2 ) ) .
I ( x , z T 4 ) = 1 2 exp ( i π 4 ) [ T 1 ( x d 2 + d 2 α 2 ) + T 2 ( x d 2 d 2 α 2 ) ]
+ exp ( i π 4 ) [ T 1 ( x + d 2 α 2 ) + T 2 ( x d 2 α 2 ) ] 2 .
I ( x , z T 4 ) = ( 1 + cos ( 2 π 2 α x ) ) ×
exp ( i π 4 ) t ( x d 2 ) [ cos ( d π α ) + tan ( 2 π α x ) sin ( d π α ) ] + exp ( i π 4 ) t ( x ) 2 .

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