Abstract

The total internal reflection (TIR) grating is an integrated optical diffraction grating designed to achieve high efficiency for the retrodiffracted order by use of total internal reflection twice within a groove of the grating rather than by use of metalized grooves. Numerical calculations are presented for both TE and TM polarizations of incident light. When the TIR grating was used in the -mth-order Littrow mount with m>13, the diffraction efficiency was found to decrease linearly with 1/m. The polarization dependence of the retrodiffraction efficiency exceeds 3 dB for TIR gratings formed in silica glass (n=1.5) but is very small for gratings with InP-based technology (n=3.2).

© 2000 Optical Society of America

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  1. M. K. Smit, “New focusing and dispersive planar component based on an optical phased array,” Electron. Lett. 24, 385–386 (1988).
    [CrossRef]
  2. K. A. McGreer, “Arrayed waveguide gratings for wavelength routing,” IEEE Commun. Mag. 36, 62–68 (1998).
    [CrossRef]
  3. J. B. D. Soole, A. Scherer, H. P. LeBlanc, N. C. Andreadakis, R. Bhat, M. A. Koza, “Monolithic InP/InGaAsP/InP grating spectrometer for the 1.48–1.56 μm wavelength range,” Appl. Phys. Lett. 58, 1949–1951 (1991).
    [CrossRef]
  4. C. Cremer, N. Emeis, M. Schier, G. Heise, G. Ebbinghaus, L. Stoll, “Grating spectrograph integrated with photodiode array in InGaAsP/InGaAs/InP,” IEEE Photon. Technol. Lett. 4, 108–110 (1992).
    [CrossRef]
  5. Z. J. Sun, K. A. McGreer, J. N. Broughton, “Integrated concave grating WDM demultiplexer with 0.144 nm channel spacing,” Electron. Lett. 33, 1140–1142 (1997).
    [CrossRef]
  6. Z. J. Sun, K. A. McGreer, J. N. Broughton, “Demultiplexer with 120 channels and 0.29-nm channel spacing,” IEEE Photonics Technol. Lett. 10, 90–93 (1998).
    [CrossRef]
  7. J.-J. He, B. Lamontagne, A. Delage, L. Erikson, M. Davies, E. S. Koteles, “Monolithic integrated wavelength demultiplexer based on a waveguide Rowland circle grating in InGaAsP/InP,” J. Lightwave Technol. 16, 631–638 (1998).
    [CrossRef]
  8. M. S. D. Smith, K. A. McGreer, “Diffraction gratings utilizing total internal reflection facets in Littrow configuration,” IEEE Photonics Technol. Lett. 11, 84–86 (1999).
    [CrossRef]
  9. S. Yu. Sadov, K. A. McGreer, “Legendre polynomials as finite elements in boundary integral equations for transmission problem with periodic piecewise-linear boundary,” in Direct and Inverse Problems of Electromagnetic and Acoustic Wave Theory, Proceedings of IVth International Seminar/Workshop, Lviv, Ukraine, 1999 (Institute for Applied Problems of Mechanics and Mathematics of the National Academy of Sciences of Ukraine, Lviv, Ukraine, 1999), pp. 55–58.
  10. K. E. Atkinson, The Numerical Solution of Integral Equa-tions of the Second Kind (Cambridge U. Press, Cambridge, UK, 1997).
  11. W. Niethammer, “Numerical application of Euler’s series transformation and its generalizations,” Numer. Math. 34, 271–283 (1980).
    [CrossRef]
  12. S. Yu. Sadov, “Computation of quasiperiodic fundamental solution of Helmholtz equation,” in Advances in Difference Equations (Gordon & Breach, Amsterdam, 1997), pp. 551–558.
  13. A. G. Voronovich, Wave Scattering from Rough Surfaces, Vol. 17 of Springer Series in Wave Phenomena (Springer-Verlag, Berlin, 1994).
    [CrossRef]
  14. P. Vincent, “Differential methods,” in Electromagnetic Theory of Gratings, R. Petit, ed., Vol. 22 of Topics in Current Physics (Springer-Verlag, Berlin, 1980), Chap. 4.
    [CrossRef]
  15. P. Vincent, “Integral equation computation of bump grating efficiencies in TE polarization,” J. Opt. Soc. Am. A 10, 444–451 (1993).
    [CrossRef]
  16. D. Maystre, M. Nevière, R. Petit, “Experimental verifications and applications of the theory,” in Electromagnetic Theory of Gratings, R. Petit, ed., Vol. 22 of Topics in Current Physics (Springer-Verlag, Berlin, 1980), Chap. 6.

1999

M. S. D. Smith, K. A. McGreer, “Diffraction gratings utilizing total internal reflection facets in Littrow configuration,” IEEE Photonics Technol. Lett. 11, 84–86 (1999).
[CrossRef]

1998

Z. J. Sun, K. A. McGreer, J. N. Broughton, “Demultiplexer with 120 channels and 0.29-nm channel spacing,” IEEE Photonics Technol. Lett. 10, 90–93 (1998).
[CrossRef]

J.-J. He, B. Lamontagne, A. Delage, L. Erikson, M. Davies, E. S. Koteles, “Monolithic integrated wavelength demultiplexer based on a waveguide Rowland circle grating in InGaAsP/InP,” J. Lightwave Technol. 16, 631–638 (1998).
[CrossRef]

K. A. McGreer, “Arrayed waveguide gratings for wavelength routing,” IEEE Commun. Mag. 36, 62–68 (1998).
[CrossRef]

1997

Z. J. Sun, K. A. McGreer, J. N. Broughton, “Integrated concave grating WDM demultiplexer with 0.144 nm channel spacing,” Electron. Lett. 33, 1140–1142 (1997).
[CrossRef]

1993

1992

C. Cremer, N. Emeis, M. Schier, G. Heise, G. Ebbinghaus, L. Stoll, “Grating spectrograph integrated with photodiode array in InGaAsP/InGaAs/InP,” IEEE Photon. Technol. Lett. 4, 108–110 (1992).
[CrossRef]

1991

J. B. D. Soole, A. Scherer, H. P. LeBlanc, N. C. Andreadakis, R. Bhat, M. A. Koza, “Monolithic InP/InGaAsP/InP grating spectrometer for the 1.48–1.56 μm wavelength range,” Appl. Phys. Lett. 58, 1949–1951 (1991).
[CrossRef]

1988

M. K. Smit, “New focusing and dispersive planar component based on an optical phased array,” Electron. Lett. 24, 385–386 (1988).
[CrossRef]

1980

W. Niethammer, “Numerical application of Euler’s series transformation and its generalizations,” Numer. Math. 34, 271–283 (1980).
[CrossRef]

Andreadakis, N. C.

J. B. D. Soole, A. Scherer, H. P. LeBlanc, N. C. Andreadakis, R. Bhat, M. A. Koza, “Monolithic InP/InGaAsP/InP grating spectrometer for the 1.48–1.56 μm wavelength range,” Appl. Phys. Lett. 58, 1949–1951 (1991).
[CrossRef]

Atkinson, K. E.

K. E. Atkinson, The Numerical Solution of Integral Equa-tions of the Second Kind (Cambridge U. Press, Cambridge, UK, 1997).

Bhat, R.

J. B. D. Soole, A. Scherer, H. P. LeBlanc, N. C. Andreadakis, R. Bhat, M. A. Koza, “Monolithic InP/InGaAsP/InP grating spectrometer for the 1.48–1.56 μm wavelength range,” Appl. Phys. Lett. 58, 1949–1951 (1991).
[CrossRef]

Broughton, J. N.

Z. J. Sun, K. A. McGreer, J. N. Broughton, “Demultiplexer with 120 channels and 0.29-nm channel spacing,” IEEE Photonics Technol. Lett. 10, 90–93 (1998).
[CrossRef]

Z. J. Sun, K. A. McGreer, J. N. Broughton, “Integrated concave grating WDM demultiplexer with 0.144 nm channel spacing,” Electron. Lett. 33, 1140–1142 (1997).
[CrossRef]

Cremer, C.

C. Cremer, N. Emeis, M. Schier, G. Heise, G. Ebbinghaus, L. Stoll, “Grating spectrograph integrated with photodiode array in InGaAsP/InGaAs/InP,” IEEE Photon. Technol. Lett. 4, 108–110 (1992).
[CrossRef]

Davies, M.

Delage, A.

Ebbinghaus, G.

C. Cremer, N. Emeis, M. Schier, G. Heise, G. Ebbinghaus, L. Stoll, “Grating spectrograph integrated with photodiode array in InGaAsP/InGaAs/InP,” IEEE Photon. Technol. Lett. 4, 108–110 (1992).
[CrossRef]

Emeis, N.

C. Cremer, N. Emeis, M. Schier, G. Heise, G. Ebbinghaus, L. Stoll, “Grating spectrograph integrated with photodiode array in InGaAsP/InGaAs/InP,” IEEE Photon. Technol. Lett. 4, 108–110 (1992).
[CrossRef]

Erikson, L.

He, J.-J.

Heise, G.

C. Cremer, N. Emeis, M. Schier, G. Heise, G. Ebbinghaus, L. Stoll, “Grating spectrograph integrated with photodiode array in InGaAsP/InGaAs/InP,” IEEE Photon. Technol. Lett. 4, 108–110 (1992).
[CrossRef]

Koteles, E. S.

Koza, M. A.

J. B. D. Soole, A. Scherer, H. P. LeBlanc, N. C. Andreadakis, R. Bhat, M. A. Koza, “Monolithic InP/InGaAsP/InP grating spectrometer for the 1.48–1.56 μm wavelength range,” Appl. Phys. Lett. 58, 1949–1951 (1991).
[CrossRef]

Lamontagne, B.

LeBlanc, H. P.

J. B. D. Soole, A. Scherer, H. P. LeBlanc, N. C. Andreadakis, R. Bhat, M. A. Koza, “Monolithic InP/InGaAsP/InP grating spectrometer for the 1.48–1.56 μm wavelength range,” Appl. Phys. Lett. 58, 1949–1951 (1991).
[CrossRef]

Maystre, D.

D. Maystre, M. Nevière, R. Petit, “Experimental verifications and applications of the theory,” in Electromagnetic Theory of Gratings, R. Petit, ed., Vol. 22 of Topics in Current Physics (Springer-Verlag, Berlin, 1980), Chap. 6.

McGreer, K. A.

M. S. D. Smith, K. A. McGreer, “Diffraction gratings utilizing total internal reflection facets in Littrow configuration,” IEEE Photonics Technol. Lett. 11, 84–86 (1999).
[CrossRef]

Z. J. Sun, K. A. McGreer, J. N. Broughton, “Demultiplexer with 120 channels and 0.29-nm channel spacing,” IEEE Photonics Technol. Lett. 10, 90–93 (1998).
[CrossRef]

K. A. McGreer, “Arrayed waveguide gratings for wavelength routing,” IEEE Commun. Mag. 36, 62–68 (1998).
[CrossRef]

Z. J. Sun, K. A. McGreer, J. N. Broughton, “Integrated concave grating WDM demultiplexer with 0.144 nm channel spacing,” Electron. Lett. 33, 1140–1142 (1997).
[CrossRef]

S. Yu. Sadov, K. A. McGreer, “Legendre polynomials as finite elements in boundary integral equations for transmission problem with periodic piecewise-linear boundary,” in Direct and Inverse Problems of Electromagnetic and Acoustic Wave Theory, Proceedings of IVth International Seminar/Workshop, Lviv, Ukraine, 1999 (Institute for Applied Problems of Mechanics and Mathematics of the National Academy of Sciences of Ukraine, Lviv, Ukraine, 1999), pp. 55–58.

Nevière, M.

D. Maystre, M. Nevière, R. Petit, “Experimental verifications and applications of the theory,” in Electromagnetic Theory of Gratings, R. Petit, ed., Vol. 22 of Topics in Current Physics (Springer-Verlag, Berlin, 1980), Chap. 6.

Niethammer, W.

W. Niethammer, “Numerical application of Euler’s series transformation and its generalizations,” Numer. Math. 34, 271–283 (1980).
[CrossRef]

Petit, R.

D. Maystre, M. Nevière, R. Petit, “Experimental verifications and applications of the theory,” in Electromagnetic Theory of Gratings, R. Petit, ed., Vol. 22 of Topics in Current Physics (Springer-Verlag, Berlin, 1980), Chap. 6.

Sadov, S. Yu.

S. Yu. Sadov, “Computation of quasiperiodic fundamental solution of Helmholtz equation,” in Advances in Difference Equations (Gordon & Breach, Amsterdam, 1997), pp. 551–558.

S. Yu. Sadov, K. A. McGreer, “Legendre polynomials as finite elements in boundary integral equations for transmission problem with periodic piecewise-linear boundary,” in Direct and Inverse Problems of Electromagnetic and Acoustic Wave Theory, Proceedings of IVth International Seminar/Workshop, Lviv, Ukraine, 1999 (Institute for Applied Problems of Mechanics and Mathematics of the National Academy of Sciences of Ukraine, Lviv, Ukraine, 1999), pp. 55–58.

Scherer, A.

J. B. D. Soole, A. Scherer, H. P. LeBlanc, N. C. Andreadakis, R. Bhat, M. A. Koza, “Monolithic InP/InGaAsP/InP grating spectrometer for the 1.48–1.56 μm wavelength range,” Appl. Phys. Lett. 58, 1949–1951 (1991).
[CrossRef]

Schier, M.

C. Cremer, N. Emeis, M. Schier, G. Heise, G. Ebbinghaus, L. Stoll, “Grating spectrograph integrated with photodiode array in InGaAsP/InGaAs/InP,” IEEE Photon. Technol. Lett. 4, 108–110 (1992).
[CrossRef]

Smit, M. K.

M. K. Smit, “New focusing and dispersive planar component based on an optical phased array,” Electron. Lett. 24, 385–386 (1988).
[CrossRef]

Smith, M. S. D.

M. S. D. Smith, K. A. McGreer, “Diffraction gratings utilizing total internal reflection facets in Littrow configuration,” IEEE Photonics Technol. Lett. 11, 84–86 (1999).
[CrossRef]

Soole, J. B. D.

J. B. D. Soole, A. Scherer, H. P. LeBlanc, N. C. Andreadakis, R. Bhat, M. A. Koza, “Monolithic InP/InGaAsP/InP grating spectrometer for the 1.48–1.56 μm wavelength range,” Appl. Phys. Lett. 58, 1949–1951 (1991).
[CrossRef]

Stoll, L.

C. Cremer, N. Emeis, M. Schier, G. Heise, G. Ebbinghaus, L. Stoll, “Grating spectrograph integrated with photodiode array in InGaAsP/InGaAs/InP,” IEEE Photon. Technol. Lett. 4, 108–110 (1992).
[CrossRef]

Sun, Z. J.

Z. J. Sun, K. A. McGreer, J. N. Broughton, “Demultiplexer with 120 channels and 0.29-nm channel spacing,” IEEE Photonics Technol. Lett. 10, 90–93 (1998).
[CrossRef]

Z. J. Sun, K. A. McGreer, J. N. Broughton, “Integrated concave grating WDM demultiplexer with 0.144 nm channel spacing,” Electron. Lett. 33, 1140–1142 (1997).
[CrossRef]

Vincent, P.

P. Vincent, “Integral equation computation of bump grating efficiencies in TE polarization,” J. Opt. Soc. Am. A 10, 444–451 (1993).
[CrossRef]

P. Vincent, “Differential methods,” in Electromagnetic Theory of Gratings, R. Petit, ed., Vol. 22 of Topics in Current Physics (Springer-Verlag, Berlin, 1980), Chap. 4.
[CrossRef]

Voronovich, A. G.

A. G. Voronovich, Wave Scattering from Rough Surfaces, Vol. 17 of Springer Series in Wave Phenomena (Springer-Verlag, Berlin, 1994).
[CrossRef]

Appl. Phys. Lett.

J. B. D. Soole, A. Scherer, H. P. LeBlanc, N. C. Andreadakis, R. Bhat, M. A. Koza, “Monolithic InP/InGaAsP/InP grating spectrometer for the 1.48–1.56 μm wavelength range,” Appl. Phys. Lett. 58, 1949–1951 (1991).
[CrossRef]

Electron. Lett.

M. K. Smit, “New focusing and dispersive planar component based on an optical phased array,” Electron. Lett. 24, 385–386 (1988).
[CrossRef]

Z. J. Sun, K. A. McGreer, J. N. Broughton, “Integrated concave grating WDM demultiplexer with 0.144 nm channel spacing,” Electron. Lett. 33, 1140–1142 (1997).
[CrossRef]

IEEE Commun. Mag.

K. A. McGreer, “Arrayed waveguide gratings for wavelength routing,” IEEE Commun. Mag. 36, 62–68 (1998).
[CrossRef]

IEEE Photon. Technol. Lett.

C. Cremer, N. Emeis, M. Schier, G. Heise, G. Ebbinghaus, L. Stoll, “Grating spectrograph integrated with photodiode array in InGaAsP/InGaAs/InP,” IEEE Photon. Technol. Lett. 4, 108–110 (1992).
[CrossRef]

IEEE Photonics Technol. Lett.

Z. J. Sun, K. A. McGreer, J. N. Broughton, “Demultiplexer with 120 channels and 0.29-nm channel spacing,” IEEE Photonics Technol. Lett. 10, 90–93 (1998).
[CrossRef]

M. S. D. Smith, K. A. McGreer, “Diffraction gratings utilizing total internal reflection facets in Littrow configuration,” IEEE Photonics Technol. Lett. 11, 84–86 (1999).
[CrossRef]

J. Lightwave Technol.

J. Opt. Soc. Am. A

Numer. Math.

W. Niethammer, “Numerical application of Euler’s series transformation and its generalizations,” Numer. Math. 34, 271–283 (1980).
[CrossRef]

Other

S. Yu. Sadov, “Computation of quasiperiodic fundamental solution of Helmholtz equation,” in Advances in Difference Equations (Gordon & Breach, Amsterdam, 1997), pp. 551–558.

A. G. Voronovich, Wave Scattering from Rough Surfaces, Vol. 17 of Springer Series in Wave Phenomena (Springer-Verlag, Berlin, 1994).
[CrossRef]

P. Vincent, “Differential methods,” in Electromagnetic Theory of Gratings, R. Petit, ed., Vol. 22 of Topics in Current Physics (Springer-Verlag, Berlin, 1980), Chap. 4.
[CrossRef]

D. Maystre, M. Nevière, R. Petit, “Experimental verifications and applications of the theory,” in Electromagnetic Theory of Gratings, R. Petit, ed., Vol. 22 of Topics in Current Physics (Springer-Verlag, Berlin, 1980), Chap. 6.

S. Yu. Sadov, K. A. McGreer, “Legendre polynomials as finite elements in boundary integral equations for transmission problem with periodic piecewise-linear boundary,” in Direct and Inverse Problems of Electromagnetic and Acoustic Wave Theory, Proceedings of IVth International Seminar/Workshop, Lviv, Ukraine, 1999 (Institute for Applied Problems of Mechanics and Mathematics of the National Academy of Sciences of Ukraine, Lviv, Ukraine, 1999), pp. 55–58.

K. E. Atkinson, The Numerical Solution of Integral Equa-tions of the Second Kind (Cambridge U. Press, Cambridge, UK, 1997).

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Figures (4)

Fig. 1
Fig. 1

Dielectric echelle grating (left) and TIR grating (right). Each grating constitutes the boundary between the medium of refractive index n (top) and air (bottom). In this cross-section view, the facets are viewed as line segments and the edges are viewed as points. The incident ray (i) is partially diffracted into the retrodiffracted ray (r). In the ray approximation, the shaded facet, AB, is not active and only the illuminated facets, BC and CD, contribute to retrodiffraction. For the TIR grating the facets along BC and CD reflect by total internal reflection. In the electromagnetic theory, all facets play a role in diffraction.

Fig. 2
Fig. 2

Numerically calculated retrodiffraction efficiencies for (a) fourteenth-order gratings and (b) second-order gratings. Data are symbolized as circles for TIR gratings, squares for echelle gratings, and open/solid symbols for TE/TM polarization. The solid curve in (a) is the reflectivity, R, for light at normal incidence to a boundary between medium with refractive index n and air. The numerical data for the fourteenth-order grating qualitatively follow the reflectivity curves, but the data for the second-order grating do not.

Fig. 3
Fig. 3

Numerically calculated retrodiffraction efficiencies for (a) TIR gratings and (b) echelle gratings. Data are symbolized as circles for n=1.5, triangles for n=3.2, asterisks for perfectly conducting gratings, and open/solid symbols for TE/TM polarization. Data for n=2.2 (not shown) are similar to data for n=3.2. When n=1.5, a large PDL is observed for TIR gratings of all orders.

Fig. 4
Fig. 4

Numerically calculated retrodiffraction efficiencies for (a) TIR gratings and (b) echelle gratings. Data are symbolized as circles for n=1.5, triangles for n=3.2, asterisks for TE polarized light incident on perfectly conducting gratings, and open/solid symbols for TE/TM polarization. Data for n=2.2 (not shown) is similar to data for n=3.2. In (b), the data for n=1.5 overlaps the data for n=3.2 but has much larger variance relative to its linear fit. The solid lines are given by e=a-b/m, where a and b are the fitting parameters for the dielectric gratings listed in Table 1.

Tables (1)

Tables Icon

Table 1 Fitting Parameters for Diffraction Efficiency According to e=a+b/m, When Used in the mth-Order Littrow Mount for m>13.

Equations (1)

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e=a-b/m,

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