Abstract

An investigation into methods for achromatizing the coupling angle characteristics of waveguide input/output couplers is described. The basic approach involves correcting the inherent angular dispersion of conventional waveguide couplers with a diffraction grating. Two configurations are analyzed in detail: a hybrid prism/grating coupler and a double grating coupler. Expressions are derived for values of the grating parameters that produce achromatic coupling. A method is also presented to predict the achromatic wavelength range and maximize it with the available degrees of freedom. For a coupling angle tolerance of 0.005°, it is found that with double grating couplers achromatic wavelength ranges of the order of 10 nm can be obtained, and that with prism/grating couplers this range can be as large as 200 nm.

© 1991 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. Shubert, J. H. Harris, “Optical Surface Waves on Thin Films and Their Application to Integrated Data Processors,” IEEE Trans. Microwave Theory Tech. MTT-16, 1048–1054 (1968).
    [CrossRef]
  2. D. Marcuse, E. A. J. Marcatili, “Excitation of Waveguides for Integrated Optics with Laser Beams,” Bell Syst. Tech. J. 50, 43–57 (1971):
  3. R. G. Hunsperger, A. Lee, A. Yariv, “Parallel End-Butt Coupling for Optical Integrated Circuits,” Appl. Opt. 16, 1026–1032 (1977).
    [CrossRef] [PubMed]
  4. P. K. Tien, R. J. Martin, “Experiments on Light Waves in a Thin Tapered Film and a New Light-Wave Coupler,” Appl. Phys. Lett. 18, 398–401 (1971).
    [CrossRef]
  5. P. K. Tien, R. J. Martin, G. Smolinsky, “Formation of Light-Guiding Interconnections in an Integrated Optical Circuit by Composite Tapered-Film Coupling,” Appl. Opt. 12, 1909–1916 (1973).
    [CrossRef] [PubMed]
  6. J. H. Harris, R. Shubert, J. N. Polky, “Beam Coupling to Films,” J. Opt. Soc. Am. 60, 1007–1016 (1970).
    [CrossRef]
  7. P. K. Tien, R. Ulrich, “Theory of Prism-Film Coupler and Thin-Film Light Guides,” J. Opt. Soc. Am. 60, 1325–1337 (1970).
    [CrossRef]
  8. R. Ulrich, “Theory of Prism-Film Coupler by Plane-Wave Analysis,” J. Opt. Soc. Am. 60, 1337–1350 (1970).
    [CrossRef]
  9. M. L. Dakss, L. Kuhn, P. F. Heidrich, B. A. Scott, “Grating Coupler for Efficient Excitation of Optical Guided Waves in Thin Films,” Appl. Phys. Lett. 16, 523–525 (1970).
    [CrossRef]
  10. H. Kogelnik, T. P. Sosnowski, “Holographic Thin Film Couplers,” Bell Syst. Tech. J. 49, 1602–1608 (1970).
  11. R. Ulrich, “Efficiency of Optical-Grating Couplers,” J. Opt. Soc. Am. 63, 1419–1431 (1973).
    [CrossRef]
  12. M. Miler, M. Skalsky, “Stigmatically Focusing Grating Coupler,” Electron. Lett. 15, 275–276 (1979).
    [CrossRef]
  13. D. Heitmann, C. Ortiz, “Calculation and Experimental Verification of Two-Dimensional Focusing Grating Couplers,” IEEE J. Quantum Electron. QE-17, 1257–1263 (1981).
    [CrossRef]
  14. For example, see H. Nishihara, M. Haruna, T. Suhara, Optical Integrated Circuits (McGraw-Hill, New York, 1989), pp. 11–15.
  15. R. Ulrich, R. Torge, “Measurement of Thin Film Parameters with a Prism Coupler,” Appl. Opt. 12, 2901–2908 (1973).
    [CrossRef] [PubMed]
  16. J. M. White, P. F. Heidrich, “Optical Waveguide Refractive Index Profiles Determined from Measurements of Mode Indices: a Simple Analysis,” Appl. Opt. 15, 151–155 (1976).
    [CrossRef] [PubMed]
  17. S. Ura, T. Suhara, H. Nishihara, “Laser Diode Spectrum in Integrated-Optic Pickup Device,” in Technical Digest, First Optoelectronics Conference (OEC ’86), Tokyo (1986), pp. 64–65.
  18. T. Suhara, H. Nishihara, “Integrated-Optic Pickup Devices Using Waveguide Holographic Components (Extended Paper),” Proc. Soc. Photo-Opt. Instrum. Eng. 1136, 92–99 (1989).
  19. T. Suhara, H. Ishimara, S. Ura, H. Nishihara, “Integration of Detection Optics for Magnetooptical Disk Pickup,” in Technical Digest, Seventh International Conference on Integrated Optics and Optical Fiber Communication, Kobe, Japan (1989), Vol. 4, pp. 80–81.
  20. Schott glass characteristics: Optical Glass, Catalog 3111e, Schott Glass Technologies (1984). Corning 7059 characteristics: Corning Material Information Data Sheet MI-7059-89. S1O2 characteristics:Handbook of Laser Science and Technology, Vol. IV: Optical Materials Part 2: Properties, M. J. Weber, Ed. (CRC Press, Boca Raton, FL, 1986), p. 31.
  21. K. Ogawa, W. S. C. Chang, B. L. Sopori, F. J. Rosenbaum, “A Theoretical Analysis of Etched Grating Couplers for Integrated Optics,” IEEE J. Quantum Electron. QE-9, 29–42 (1973).
    [CrossRef]
  22. P. K. Tien, R. Ulrich, “Theory of Prism-Film Coupler and Thin-Film Light Guides,” J. Opt. Soc. Am. 60, 1325–1337 (1970).
    [CrossRef]

1989 (1)

T. Suhara, H. Nishihara, “Integrated-Optic Pickup Devices Using Waveguide Holographic Components (Extended Paper),” Proc. Soc. Photo-Opt. Instrum. Eng. 1136, 92–99 (1989).

1981 (1)

D. Heitmann, C. Ortiz, “Calculation and Experimental Verification of Two-Dimensional Focusing Grating Couplers,” IEEE J. Quantum Electron. QE-17, 1257–1263 (1981).
[CrossRef]

1979 (1)

M. Miler, M. Skalsky, “Stigmatically Focusing Grating Coupler,” Electron. Lett. 15, 275–276 (1979).
[CrossRef]

1977 (1)

1976 (1)

1973 (4)

1971 (2)

P. K. Tien, R. J. Martin, “Experiments on Light Waves in a Thin Tapered Film and a New Light-Wave Coupler,” Appl. Phys. Lett. 18, 398–401 (1971).
[CrossRef]

D. Marcuse, E. A. J. Marcatili, “Excitation of Waveguides for Integrated Optics with Laser Beams,” Bell Syst. Tech. J. 50, 43–57 (1971):

1970 (6)

1968 (1)

R. Shubert, J. H. Harris, “Optical Surface Waves on Thin Films and Their Application to Integrated Data Processors,” IEEE Trans. Microwave Theory Tech. MTT-16, 1048–1054 (1968).
[CrossRef]

Chang, W. S. C.

K. Ogawa, W. S. C. Chang, B. L. Sopori, F. J. Rosenbaum, “A Theoretical Analysis of Etched Grating Couplers for Integrated Optics,” IEEE J. Quantum Electron. QE-9, 29–42 (1973).
[CrossRef]

Dakss, M. L.

M. L. Dakss, L. Kuhn, P. F. Heidrich, B. A. Scott, “Grating Coupler for Efficient Excitation of Optical Guided Waves in Thin Films,” Appl. Phys. Lett. 16, 523–525 (1970).
[CrossRef]

Harris, J. H.

J. H. Harris, R. Shubert, J. N. Polky, “Beam Coupling to Films,” J. Opt. Soc. Am. 60, 1007–1016 (1970).
[CrossRef]

R. Shubert, J. H. Harris, “Optical Surface Waves on Thin Films and Their Application to Integrated Data Processors,” IEEE Trans. Microwave Theory Tech. MTT-16, 1048–1054 (1968).
[CrossRef]

Haruna, M.

For example, see H. Nishihara, M. Haruna, T. Suhara, Optical Integrated Circuits (McGraw-Hill, New York, 1989), pp. 11–15.

Heidrich, P. F.

J. M. White, P. F. Heidrich, “Optical Waveguide Refractive Index Profiles Determined from Measurements of Mode Indices: a Simple Analysis,” Appl. Opt. 15, 151–155 (1976).
[CrossRef] [PubMed]

M. L. Dakss, L. Kuhn, P. F. Heidrich, B. A. Scott, “Grating Coupler for Efficient Excitation of Optical Guided Waves in Thin Films,” Appl. Phys. Lett. 16, 523–525 (1970).
[CrossRef]

Heitmann, D.

D. Heitmann, C. Ortiz, “Calculation and Experimental Verification of Two-Dimensional Focusing Grating Couplers,” IEEE J. Quantum Electron. QE-17, 1257–1263 (1981).
[CrossRef]

Hunsperger, R. G.

Ishimara, H.

T. Suhara, H. Ishimara, S. Ura, H. Nishihara, “Integration of Detection Optics for Magnetooptical Disk Pickup,” in Technical Digest, Seventh International Conference on Integrated Optics and Optical Fiber Communication, Kobe, Japan (1989), Vol. 4, pp. 80–81.

Kogelnik, H.

H. Kogelnik, T. P. Sosnowski, “Holographic Thin Film Couplers,” Bell Syst. Tech. J. 49, 1602–1608 (1970).

Kuhn, L.

M. L. Dakss, L. Kuhn, P. F. Heidrich, B. A. Scott, “Grating Coupler for Efficient Excitation of Optical Guided Waves in Thin Films,” Appl. Phys. Lett. 16, 523–525 (1970).
[CrossRef]

Lee, A.

Marcatili, E. A. J.

D. Marcuse, E. A. J. Marcatili, “Excitation of Waveguides for Integrated Optics with Laser Beams,” Bell Syst. Tech. J. 50, 43–57 (1971):

Marcuse, D.

D. Marcuse, E. A. J. Marcatili, “Excitation of Waveguides for Integrated Optics with Laser Beams,” Bell Syst. Tech. J. 50, 43–57 (1971):

Martin, R. J.

P. K. Tien, R. J. Martin, G. Smolinsky, “Formation of Light-Guiding Interconnections in an Integrated Optical Circuit by Composite Tapered-Film Coupling,” Appl. Opt. 12, 1909–1916 (1973).
[CrossRef] [PubMed]

P. K. Tien, R. J. Martin, “Experiments on Light Waves in a Thin Tapered Film and a New Light-Wave Coupler,” Appl. Phys. Lett. 18, 398–401 (1971).
[CrossRef]

Miler, M.

M. Miler, M. Skalsky, “Stigmatically Focusing Grating Coupler,” Electron. Lett. 15, 275–276 (1979).
[CrossRef]

Nishihara, H.

T. Suhara, H. Nishihara, “Integrated-Optic Pickup Devices Using Waveguide Holographic Components (Extended Paper),” Proc. Soc. Photo-Opt. Instrum. Eng. 1136, 92–99 (1989).

S. Ura, T. Suhara, H. Nishihara, “Laser Diode Spectrum in Integrated-Optic Pickup Device,” in Technical Digest, First Optoelectronics Conference (OEC ’86), Tokyo (1986), pp. 64–65.

T. Suhara, H. Ishimara, S. Ura, H. Nishihara, “Integration of Detection Optics for Magnetooptical Disk Pickup,” in Technical Digest, Seventh International Conference on Integrated Optics and Optical Fiber Communication, Kobe, Japan (1989), Vol. 4, pp. 80–81.

For example, see H. Nishihara, M. Haruna, T. Suhara, Optical Integrated Circuits (McGraw-Hill, New York, 1989), pp. 11–15.

Ogawa, K.

K. Ogawa, W. S. C. Chang, B. L. Sopori, F. J. Rosenbaum, “A Theoretical Analysis of Etched Grating Couplers for Integrated Optics,” IEEE J. Quantum Electron. QE-9, 29–42 (1973).
[CrossRef]

Ortiz, C.

D. Heitmann, C. Ortiz, “Calculation and Experimental Verification of Two-Dimensional Focusing Grating Couplers,” IEEE J. Quantum Electron. QE-17, 1257–1263 (1981).
[CrossRef]

Polky, J. N.

Rosenbaum, F. J.

K. Ogawa, W. S. C. Chang, B. L. Sopori, F. J. Rosenbaum, “A Theoretical Analysis of Etched Grating Couplers for Integrated Optics,” IEEE J. Quantum Electron. QE-9, 29–42 (1973).
[CrossRef]

Scott, B. A.

M. L. Dakss, L. Kuhn, P. F. Heidrich, B. A. Scott, “Grating Coupler for Efficient Excitation of Optical Guided Waves in Thin Films,” Appl. Phys. Lett. 16, 523–525 (1970).
[CrossRef]

Shubert, R.

J. H. Harris, R. Shubert, J. N. Polky, “Beam Coupling to Films,” J. Opt. Soc. Am. 60, 1007–1016 (1970).
[CrossRef]

R. Shubert, J. H. Harris, “Optical Surface Waves on Thin Films and Their Application to Integrated Data Processors,” IEEE Trans. Microwave Theory Tech. MTT-16, 1048–1054 (1968).
[CrossRef]

Skalsky, M.

M. Miler, M. Skalsky, “Stigmatically Focusing Grating Coupler,” Electron. Lett. 15, 275–276 (1979).
[CrossRef]

Smolinsky, G.

Sopori, B. L.

K. Ogawa, W. S. C. Chang, B. L. Sopori, F. J. Rosenbaum, “A Theoretical Analysis of Etched Grating Couplers for Integrated Optics,” IEEE J. Quantum Electron. QE-9, 29–42 (1973).
[CrossRef]

Sosnowski, T. P.

H. Kogelnik, T. P. Sosnowski, “Holographic Thin Film Couplers,” Bell Syst. Tech. J. 49, 1602–1608 (1970).

Suhara, T.

T. Suhara, H. Nishihara, “Integrated-Optic Pickup Devices Using Waveguide Holographic Components (Extended Paper),” Proc. Soc. Photo-Opt. Instrum. Eng. 1136, 92–99 (1989).

T. Suhara, H. Ishimara, S. Ura, H. Nishihara, “Integration of Detection Optics for Magnetooptical Disk Pickup,” in Technical Digest, Seventh International Conference on Integrated Optics and Optical Fiber Communication, Kobe, Japan (1989), Vol. 4, pp. 80–81.

S. Ura, T. Suhara, H. Nishihara, “Laser Diode Spectrum in Integrated-Optic Pickup Device,” in Technical Digest, First Optoelectronics Conference (OEC ’86), Tokyo (1986), pp. 64–65.

For example, see H. Nishihara, M. Haruna, T. Suhara, Optical Integrated Circuits (McGraw-Hill, New York, 1989), pp. 11–15.

Tien, P. K.

Torge, R.

Ulrich, R.

Ura, S.

T. Suhara, H. Ishimara, S. Ura, H. Nishihara, “Integration of Detection Optics for Magnetooptical Disk Pickup,” in Technical Digest, Seventh International Conference on Integrated Optics and Optical Fiber Communication, Kobe, Japan (1989), Vol. 4, pp. 80–81.

S. Ura, T. Suhara, H. Nishihara, “Laser Diode Spectrum in Integrated-Optic Pickup Device,” in Technical Digest, First Optoelectronics Conference (OEC ’86), Tokyo (1986), pp. 64–65.

White, J. M.

Yariv, A.

Appl. Opt. (4)

Appl. Phys. Lett. (2)

M. L. Dakss, L. Kuhn, P. F. Heidrich, B. A. Scott, “Grating Coupler for Efficient Excitation of Optical Guided Waves in Thin Films,” Appl. Phys. Lett. 16, 523–525 (1970).
[CrossRef]

P. K. Tien, R. J. Martin, “Experiments on Light Waves in a Thin Tapered Film and a New Light-Wave Coupler,” Appl. Phys. Lett. 18, 398–401 (1971).
[CrossRef]

Bell Syst. Tech. J. (2)

D. Marcuse, E. A. J. Marcatili, “Excitation of Waveguides for Integrated Optics with Laser Beams,” Bell Syst. Tech. J. 50, 43–57 (1971):

H. Kogelnik, T. P. Sosnowski, “Holographic Thin Film Couplers,” Bell Syst. Tech. J. 49, 1602–1608 (1970).

Electron. Lett. (1)

M. Miler, M. Skalsky, “Stigmatically Focusing Grating Coupler,” Electron. Lett. 15, 275–276 (1979).
[CrossRef]

IEEE J. Quantum Electron. (2)

D. Heitmann, C. Ortiz, “Calculation and Experimental Verification of Two-Dimensional Focusing Grating Couplers,” IEEE J. Quantum Electron. QE-17, 1257–1263 (1981).
[CrossRef]

K. Ogawa, W. S. C. Chang, B. L. Sopori, F. J. Rosenbaum, “A Theoretical Analysis of Etched Grating Couplers for Integrated Optics,” IEEE J. Quantum Electron. QE-9, 29–42 (1973).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

R. Shubert, J. H. Harris, “Optical Surface Waves on Thin Films and Their Application to Integrated Data Processors,” IEEE Trans. Microwave Theory Tech. MTT-16, 1048–1054 (1968).
[CrossRef]

J. Opt. Soc. Am. (5)

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

T. Suhara, H. Nishihara, “Integrated-Optic Pickup Devices Using Waveguide Holographic Components (Extended Paper),” Proc. Soc. Photo-Opt. Instrum. Eng. 1136, 92–99 (1989).

Other (4)

T. Suhara, H. Ishimara, S. Ura, H. Nishihara, “Integration of Detection Optics for Magnetooptical Disk Pickup,” in Technical Digest, Seventh International Conference on Integrated Optics and Optical Fiber Communication, Kobe, Japan (1989), Vol. 4, pp. 80–81.

Schott glass characteristics: Optical Glass, Catalog 3111e, Schott Glass Technologies (1984). Corning 7059 characteristics: Corning Material Information Data Sheet MI-7059-89. S1O2 characteristics:Handbook of Laser Science and Technology, Vol. IV: Optical Materials Part 2: Properties, M. J. Weber, Ed. (CRC Press, Boca Raton, FL, 1986), p. 31.

S. Ura, T. Suhara, H. Nishihara, “Laser Diode Spectrum in Integrated-Optic Pickup Device,” in Technical Digest, First Optoelectronics Conference (OEC ’86), Tokyo (1986), pp. 64–65.

For example, see H. Nishihara, M. Haruna, T. Suhara, Optical Integrated Circuits (McGraw-Hill, New York, 1989), pp. 11–15.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (20)

Fig. 1
Fig. 1

Conventional types of waveguide input/output coupler: (a) prism; (b) grating.

Fig. 2
Fig. 2

Achromatic input/output coupler configurations: (a) hybrid prism/grating coupler; (b) double grating coupler.

Fig. 3
Fig. 3

Total coupling angle (solid curve) and grating period (dashed curve) for the hybrid prism/grating achromatic coupler as a function of the prism angle. The waveguide used in the model consisted of a 0.6-μm layer of Corning 7059 glass on an oxidized silicon substrate, and the prism material was Schott SF6. The design wavelength was 632.8 nm.

Fig. 4
Fig. 4

Second derivative of the final coupling angle (dashed line) and achromatic wavelength range (solid line) of a hybrid prism/grating achromatic coupler as a function of the prism angle. The waveguide that was modeled consisted of a 0.6-μm layer of Corning 7059 glass on an oxidized silicon substrate, and the prism material was Schott SF6. The design wavelength was 632.8 nm, and the angular tolerance for the range calculations was 0.005°.

Fig. 5
Fig. 5

Second derivative of the final coupling angle (dashed line) and the achromatic wavelength range (solid line) of a hybrid prism/grating achromatic coupler as a function of the prism angle. The dotted line is the approximate range calculated by Eq. (13). The point labeled (B) corresponds to the maximum achromatic wavelength range, and the point labeled (C) corresponds to the zero of the second derivative. The waveguide that was modeled consisted of a 0.6-μm layer of Corning 7059 glass on an oxidized silicon substrate, and the prism material was Schott LaSF3. The design wavelength was 632.8 nm, and the angular tolerance for the range calculations was 0.005°.

Fig. 6
Fig. 6

Coupling angle error as a function of wavelength error for a hybrid prism/grating achromatic coupler for three different prism angles. The lines correspond to the prism angles labeled (A)–(C) on Fig. 5. Curve (A) corresponds to a prism angle of 60°, curve (B) corresponds to the prism angle which gives the maximum achromatic wavelength range (38°), and curve (C) corresponds to the prism angle which gives a zero of the second derivative (31°). The waveguide that was modeled consisted of a 0.6-μm layer of Corning 7059 glass on an oxidized silicon substrate, and the prism material was Schott LaSF3. The design wavelength was 632.8 nm. The dashed lines at ±0.005° correspond to the angular tolerance for the range calculations.

Fig. 7
Fig. 7

Achromatic wavelength range of a hybrid prism/grating achromatic coupler as a function of prism angle for three different prism materials: SF6 (solid curve); LaSF3 (dotted curve); and LaSFN30 (dashed curve). The waveguide used in the model was a 0.6-μm layer of Corning 7059 glass on an oxidized silicon substrate. The design wavelength was 632.8 nm, and the angular tolerance for the range calculations was 0.005°.

Fig. 8
Fig. 8

Total coupling angle (solid curve) and grating period (dashed curve) for a double grating achromatic coupler as a function of the grating angle. The waveguide used in thé model consisted of a 0.6-μm layer of Corning 7059 glass on an oxidized silicon substrate. The design wavelength was 632.8 nm, and the period of the grating coupler was chosen to give a nominal coupling angle of θ = 20°.

Fig. 9
Fig. 9

Second derivative of the final coupling angle (dashed line) and achromatic wavelength range (solid line) of a double grating achromatic coupler as a function of the grating angle. The waveguide that was modeled consisted of a 0.6-μm layer of Corning 7059 glass on an oxidized silicon substrate. The period of the grating coupler was chosen to give a nominal coupling angle of θ = 20°. The design wavelength was 632.8 nm, and the angular tolerance for the range calculations was 0.005°.

Fig. 10
Fig. 10

Achromatic wavelength range of a double grating achromatic coupler as a function of the nominal coupling angle. The grating angle at each nominal coupling angle was chosen to give the maximum possible achromatic range. The waveguide that was modeled consisted of a 0.6-μm layer of Corning 7059 glass on an oxidized silicon substrate. The design wavelength was 632.8 nm, and the angular tolerance for the range calculation was 0.005°.

Fig. 11
Fig. 11

Comparison of coupling angle shifts caused by wavelength changes for conventional prism and grating couplers and achromatic prism/grating and double grating couplers. The waveguide was a 0.6-μm layer of Corning 7059 glass on an oxidized silicon substrate. The prism was made of LaSF3 with a prism angle of 38°. The nominal coupling angle for the grating couplers was 20°, and the grating angle for the double grating coupler was 63°.

Fig. 12
Fig. 12

Relative coupling efficiency of a conventional grating coupler as a function of wavelength deviation. The waveguide was a 0.6-μm layer of Corning 7059 glass on an oxidized silicon substrate. The nominal coupling angle was 20°, and the grating coupler length was 1 mm.

Fig. 13
Fig. 13

Relative coupling efficiency of a conventional prism coupler as a function of wavelength deviation. The waveguide was a 0.6-μm layer of Corning 7059 glass on an oxidized silicon substrate. The prism was made of LaSF3 with a prism angle of 38°. A gap of 1 μm was assumed between the waveguide and prism.

Fig. 14
Fig. 14

Relative coupling efficiency of an achromatic double grating coupler as a function of wavelength deviation. The waveguide was a 0.6-μm layer of Corning 7059 glass on an oxidized silicon substrate. The nominal coupling angle was 20°, and the grating coupler length was 1 mm. The angle for the compensating grating was 63°.

Fig. 15
Fig. 15

Relative coupling efficiency of an achromatic prism/grating coupler as a function of wavelength deviation. The waveguide was a 0.6-μm layer of Corning 7059 glass on an oxidized silicon substrate. The prism was made of LaSF3 with a prism angle of 38°. A gap of 1 μm was assumed between the waveguide and prism.

Fig. 16
Fig. 16

Effect of error in the compensating grating period or angle on the coupling angle error of an achromatic coupler. The solid curve represents typical coupling angle characteristics at the achromatic coupling condition. The dotted curve represents the coupling angle characteristics with an error in the grating period or grating angle, which is just small enough so that the achromatic wavelength range is not significantly reduced.

Fig. 17
Fig. 17

Achromatic wavelength range of achromatic couplers as a function of errors in the compensating grating period. The waveguide was a 0.6-μm layer of Corning 7059 glass on an oxidized silicon substrate. The prism was made of SF6 with a prism angle of 38°. The grating angle for the double grating coupler was 63°. The coupling angle tolerance that was used for the range calculations was 0.005°.

Fig. 18
Fig. 18

Achromatic wavelength range of achromatic couplers as a function of errors in the compensating grating angle. The waveguide was a 0.6-μm layer of Corning 7059 glass on an oxidized silicon substrate. The prism was made of SF6 with a nominal prism angle of 38°. The nominal grating angle for the double grating coupler was 63°. The coupling angle tolerance that was used for the range calculations was 0.005°.

Fig. 19
Fig. 19

Achromatic wavelength range of two hybrid prism/grating couplers as a function of errors in the compensating grating period. The second derivative of the final coupling angle is small in both cases, so that the approximation made in Eq. (11) is invalid. The waveguide was a 0.6-μm layer of Corning 7059 glass on an oxidized silicon substrate. The prisms were made of LaSF3 and LaSFN30 with prism angles of 38 and 50°, respectively. The coupling angle tolerance that was used for the range calculations was 0.005°.

Fig. 20
Fig. 20

Achromatic wavelength range of two hybrid prism/grating couplers as a function of errors in the prism angle. The second derivative of the final coupling angle is small in both cases, so that the approximation made in Eq. (11) is invalid. The waveguide was a 0.6-μm layer of Corning 7059 glass on an oxidized silicon substrate. The prisms were made of LaSF3 and LaSFN30 with prism angles of 38 and 50°, respectively. The coupling angle tolerance that was used for the range calculations was 0.005°.

Tables (2)

Tables Icon

Table I Material Dispersion Characteristics Used In Calculations

Tables Icon

Table II Configurations Used for Coupler Performance Comparison

Equations (60)

Equations on this page are rendered with MathJax. Learn more.

N ( λ ) k = n p ( λ ) k sin θ ,
N ( λ ) k 2 π d g 1 m g 1 = k sin θ ,
d θ d λ = n p d N d λ N d n p d λ n p 2 cos θ .
m p g λ = d p g ( sin θ p g n p sin θ i ) ,
m p g = d p g ( cos θ p g d θ p g d λ n p cos θ i d θ i d λ d n p d λ sin θ i ) .
d θ i d λ = 1 n p cos θ i ( m p g d p g + d n p d d λ sin θ i ) .
θ i = θ α ,
d θ i d λ = d θ d λ .
d p g = m p g cos θ cos θ i ( d N d λ N n p d n p d λ ) + sin θ i cos θ d n p d λ .
θ p g ( λ ) = θ p g ( λ 0 ) + d θ p g ( λ 0 ) d λ ( λ λ 0 ) + d 2 θ p g ( λ 0 ) d λ 2 ( λ λ 0 ) 2 2 ! + O [ ( λ λ 0 ) 3 ] .
θ p g ( λ ) θ p g ( λ 0 ) + d 2 θ p g ( λ 0 ) d λ 2 ( λ λ 0 ) 2 2 .
Δ θ p g θ p g ( λ ) θ p g ( λ 0 ) d 2 θ p g ( λ 0 ) d λ 2 ( λ λ 0 ) 2 2 .
R λ 2 ( λ λ 0 ) 2 2 Δ θ tol | d 2 θ p g ( λ 0 ) d λ 2 | .
d 2 θ p g d λ 2 = cos θ i cos θ p g ( n p d 2 θ i d λ 2 + 2 d θ i d λ d n p d λ ) + sin θ i cos θ p g [ d 2 n p d λ 2 n p ( d θ i d λ ) 2 ] .
d θ i d λ = n p d N d λ N d n p d λ n p 2 cos θ ,
d 2 θ i d λ 2 = 1 n p 4 cos 3 θ { cos 2 θ [ n p 3 d 2 N d λ 2 n p 2 N d 2 n p d λ 2 2 n p 2 d N d λ d n p d λ + 2 N n p ( d n p d λ ) 2 ] + sin θ [ n p 2 ( d N d λ ) 2 2 N n p d N d λ d n p d λ + N 2 ( d n p d λ ) 2 ] } .
α o , 1 = θ tan 1 [ n p d 2 θ i d λ 2 + 2 d θ i d λ d n p d λ n p ( d θ i d λ ) 2 d 2 n p d λ 2 ] ,
α o , 2 = θ tan 1 ( c 1 A cos θ + c 2 B c 1 B cos θ + c 3 A ) .
d θ d λ = d N d λ m g 1 d g 1 cos θ .
m g 2 λ = d g 2 ( sin θ g g sin θ i ) ,
m g 2 = d g 2 ( cos θ g g d θ g g d λ cos θ i d θ i d λ ) .
d θ i d λ = m g 2 d g 2 cos θ i .
θ i = θ β ,
d θ i d λ = d θ d λ .
d g 2 = m g 2 cos θ cos θ i ( d N d λ m g 1 d g 1 ) .
d 2 θ g g d λ 2 = cos θ i cos θ g g d 2 θ i d λ 2 sin θ i cos θ g g ( d θ i d λ ) 2 .
d θ i d λ = d N d λ m g 1 d g 1 cos θ ,
d 2 θ i d λ 2 = 1 cos 3 θ [ cos 2 θ d 2 N d λ 2 + sin θ ( d N d λ m g 1 d g 1 ) 2 ] .
R λ 2 2 Δ θ tol | d 2 θ g g ( λ 0 ) d λ 2 | .
β o , 1 = θ tan 1 [ d 2 θ i d λ 2 ( d θ i d λ ) 2 ] .
β o , 2 = θ tan 1 { ( d θ i d λ ) 2 [ ( d N d λ m g 1 d g 1 ) λ 2 cos 2 θ ] d 2 θ i d λ 2 ( d N d λ m g 1 d g 1 ) λ cos θ ( d θ i ) d λ ) 2 ( d N d λ m g 1 d g 1 ) λ cos θ } .
η η 0 = sinc 2 ( L y Δ β 2 ) ,
Δ β = k ( sin θ sin θ λ ) .
η η 0 = 1 1 + F sin 2 ( δ / 2 ) ,
Δ θ d θ d λ Δ λ + d 2 θ d λ 2 ( Δ λ ) 2 2 .
Δ θ d θ d λ Δ λ + 4 Δ θ tol R λ 2 ( Δ λ ) 2 .
d θ d λ = 4 Δ θ tol R λ .
d θ p g d λ = 1 cos θ p g ( m p g d p g + n p cos θ i d θ i d λ + d n p d λ sin θ i ) .
d p g = d p g 0 ( 1 + f p g ) ,
f p g = | 1 R λ 4 Δ θ tol cos θ p g ( n p cos θ i d θ i d λ + d n p d λ sin θ i ) 1 | .
f g g = | 1 R λ cos θ i 4 Δ θ tol cos θ g g d θ i d λ 1 | .
θ i = θ i 0 + Δ θ i ,
Δ θ i p g = | 4 Δ θ tol cos θ p g R λ ( d n p d λ cos θ i 0 n p sin θ i 0 d θ i d λ ) | .
Δ θ i g g = | 4 Δ θ tol cos θ g g R λ sin θ i 0 d θ i d λ | .
d 2 θ p g d λ 2 = A cos θ i cos θ p g + B sin θ i cos θ p g ,
A = n p d 2 θ i d λ 2 + 2 d θ i d λ d n p d λ ,
B = d 2 n p d λ 2 n p ( d θ i d λ ) 2 .
d d θ i ( d 2 θ p g d λ 2 ) = A cos 2 θ p g ( cos θ i sin θ p g d θ p g d θ i cos θ p g sin θ i ) + B cos 2 θ p g ( sin θ i sin θ p g d θ p g d θ i + cos θ p g cos θ i ) .
sin θ p g n p sin θ i = λ cos θ [ cos θ i ( d N d λ N n p d n p d λ ) + sin θ i cos θ d n p d λ ] .
d θ p g d θ i = n p cos θ i cos θ p g + λ cos θ p g cos θ × [ sin θ i ( d N d λ N n p d n p d λ ) cos θ i cos θ d n p d λ ] .
θ i = tan 1 ( c 1 A cos θ + c 2 B c 1 B cos θ + c 3 A ) ,
c 1 = n p λ ( d N d λ n p d n p d λ N ) ( n p d n p d λ λ ) ,
c 2 = N 2 n p 2 + λ 2 ( d N d λ n p d n p d λ N ) 2 ,
c 3 = ( N 2 n p 2 ) [ 1 ( n p d n p d λ λ ) 2 ] .
α o , 2 = θ tan 1 ( c 1 A cos θ + c 2 B c 1 B cos θ + c 3 A ) .
d d θ i ( d 2 θ g g d λ 2 ) = 1 cos 2 θ g g [ ( d 2 θ i d λ 2 ) ( cos θ i sin θ g g d θ g g d θ i cos θ g g sin θ i ) ( d θ i d λ ) 2 ( sin θ i sin θ g g d θ g g d θ i + cos θ g g cos θ i ) ] .
sin θ g g sin θ i = λ c o s θ i cos θ ( d N d λ m g 1 d g 1 ) ,
d θ g g d θ i = cos θ i cos θ g g + λ sin θ i cos θ cos θ g g ( d N d λ m g 1 d g 1 ) .
θ i = tan 1 { ( d θ i d λ ) 2 [ ( d N d λ m g 1 d g 1 ) 2 λ 2 cos 2 θ ] d 2 θ i d λ 2 ( d N d λ m g 1 d g 1 ) λ cos θ ( d θ i d λ ) 2 ( d N d λ m g 1 d g 1 ) λ cos θ } .
β o , 2 = θ tan 1 { ( d θ i d λ ) 2 [ ( d N d λ m g 1 d g 1 ) 2 λ 2 cos 2 θ ] d 2 θ i d λ ( d N d λ m g 1 d g 1 ) λ cos θ ( d θ i d λ ) 2 ( d N d λ m g 1 d g 1 ) λ cos θ } .

Metrics