Abstract

The method that we have developed [ P-P. Borsboom, Ph.D. dissertation ( Delft University of Technology, Delft, The Netherlands); P-P. Borsboom and H. J. Frankena, J. Opt. Soc. Am. A 12, 1134– 1141 ( 1995)] is successfully applied to a two-dimensional focusing grating coupler. The field in the focal region has been determined for symmetrical chirped gratings consisting of as many as 124 corrugations. The intensity distribution in the focal region agrees well with the approximate predictions of geometrical optics. In the case of short focusing grating couplers high-frequency intensity variations are observed in the focal region.

© 1995 Optical Society of America

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References

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  1. S. Ura, T. Suhara, H. Nishihara, J. Koyama, “Focusing grating for integrated optical disc pickup device,” Trans. Inst. Electron. Commun. Eng. Jpn. Part C (Japan) 68-C, 803–811 (1985).
  2. S. Ura, T. Suhura, H. Nishihara, “Aberration characterizations of a focusing grating coupler in an integrated-optic disc pickup device,” Appl. Opt. 26, 4777–4782 (1987).
    [CrossRef] [PubMed]
  3. J. J. M. Braat, M. O. E. Laurijs, “Geometrical optics design and aberration analysis of a focusing grating coupler,” Opt. Eng. 33, 1037–1043 (1994).
    [CrossRef]
  4. Q.-H. Liu, W. C. Chew, “Analysis of discontinuities in planar dielectric waveguides: an eigenmode propagation method,” IEEE Trans. Microwave Theory Tech. 39, 422–430 (1991).
    [CrossRef]
  5. H. Shigesawa, M. Tsuji, “A new equivalent network method for analyzing discontinuity properties of open dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 37, 3–14 (1989).
    [CrossRef]
  6. P-P. Borsboom, H. J. Frankena, “Field analysis of two-dimensional integrated optical gratings,” J. Opt. Soc. Am. A 12, 1134–1141 (1995).
    [CrossRef]
  7. P-P. Borsboom, “Field analysis of two-dimensional integrated-optical gratings,” Ph.D. dissertation (Delft University of Technology, Delft, The Netherlands, 1994).
  8. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974), Chap. 1, pp. 1–30.
  9. C. Vassallo, Théorie des Guides D’Ondes Électromagnétiques (Eyrolles, Paris, 1985), Chap. 3, pp. 151–248.
  10. C. Vassallo, “Radiating normal modes of lossy planar waveguides,” J. Opt. Soc. Am. 69, 311–316 (1979).
    [CrossRef]
  11. S. F. Mahmoud, J. C. Beal, “Scattering of surface waves at a dielectric discontinuity on a planar waveguide,” IEEE Trans. Microwave Theory Tech. MTT-23, 193–198 (1975).
    [CrossRef]
  12. W. Biehlig, K. Hehl, U. Langbein, F. Lederer, “Light propagation in a planar dielectric slab waveguide with step discontinuities. Part I. Operator formalism,” Opt. Quantum Electron. 18, 219–228 (1986).
    [CrossRef]
  13. W. Biehlig, “Light propagation in a planar dielectric slab waveguide with step discontinuities. Part 2. Numerical analysis of TE-polarized fields,” Opt. Quantum Electron. 18, 229–238 (1986).
    [CrossRef]
  14. H. Shigesawa, M. Tsuji, “Mode propagation through a step discontinuity planar waveguide,” IEEE Trans. Microwave Theory Tech. MTT-34, 205–212 (1986).
    [CrossRef]
  15. R. Redheffer, “Difference equations and functional equations in transmission-line theory,” in Modern Mathematics for the Engineer, 2nd ed., E. F. Beckenbach, ed. University of California Engineering Extension Series (McGraw-Hill, New York, 1961), pp. 282–337.
  16. J. J. M. Braat, “Microscope objectives for optical disc systems,” in Huygens’ Principle 1690–1990: Theory and Applications, H. Blok, H. A. Ferwerda, H. K. Kuiken, (Elsevier Science B. V., Amsterdam, 1990), pp. 33–63.
  17. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), Chap. 8, pp. 439–441.

1995 (1)

1994 (1)

J. J. M. Braat, M. O. E. Laurijs, “Geometrical optics design and aberration analysis of a focusing grating coupler,” Opt. Eng. 33, 1037–1043 (1994).
[CrossRef]

1991 (1)

Q.-H. Liu, W. C. Chew, “Analysis of discontinuities in planar dielectric waveguides: an eigenmode propagation method,” IEEE Trans. Microwave Theory Tech. 39, 422–430 (1991).
[CrossRef]

1989 (1)

H. Shigesawa, M. Tsuji, “A new equivalent network method for analyzing discontinuity properties of open dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 37, 3–14 (1989).
[CrossRef]

1987 (1)

1986 (3)

W. Biehlig, K. Hehl, U. Langbein, F. Lederer, “Light propagation in a planar dielectric slab waveguide with step discontinuities. Part I. Operator formalism,” Opt. Quantum Electron. 18, 219–228 (1986).
[CrossRef]

W. Biehlig, “Light propagation in a planar dielectric slab waveguide with step discontinuities. Part 2. Numerical analysis of TE-polarized fields,” Opt. Quantum Electron. 18, 229–238 (1986).
[CrossRef]

H. Shigesawa, M. Tsuji, “Mode propagation through a step discontinuity planar waveguide,” IEEE Trans. Microwave Theory Tech. MTT-34, 205–212 (1986).
[CrossRef]

1985 (1)

S. Ura, T. Suhara, H. Nishihara, J. Koyama, “Focusing grating for integrated optical disc pickup device,” Trans. Inst. Electron. Commun. Eng. Jpn. Part C (Japan) 68-C, 803–811 (1985).

1979 (1)

1975 (1)

S. F. Mahmoud, J. C. Beal, “Scattering of surface waves at a dielectric discontinuity on a planar waveguide,” IEEE Trans. Microwave Theory Tech. MTT-23, 193–198 (1975).
[CrossRef]

Beal, J. C.

S. F. Mahmoud, J. C. Beal, “Scattering of surface waves at a dielectric discontinuity on a planar waveguide,” IEEE Trans. Microwave Theory Tech. MTT-23, 193–198 (1975).
[CrossRef]

Biehlig, W.

W. Biehlig, K. Hehl, U. Langbein, F. Lederer, “Light propagation in a planar dielectric slab waveguide with step discontinuities. Part I. Operator formalism,” Opt. Quantum Electron. 18, 219–228 (1986).
[CrossRef]

W. Biehlig, “Light propagation in a planar dielectric slab waveguide with step discontinuities. Part 2. Numerical analysis of TE-polarized fields,” Opt. Quantum Electron. 18, 229–238 (1986).
[CrossRef]

Blok, H.

J. J. M. Braat, “Microscope objectives for optical disc systems,” in Huygens’ Principle 1690–1990: Theory and Applications, H. Blok, H. A. Ferwerda, H. K. Kuiken, (Elsevier Science B. V., Amsterdam, 1990), pp. 33–63.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), Chap. 8, pp. 439–441.

Borsboom, P-P.

P-P. Borsboom, H. J. Frankena, “Field analysis of two-dimensional integrated optical gratings,” J. Opt. Soc. Am. A 12, 1134–1141 (1995).
[CrossRef]

P-P. Borsboom, “Field analysis of two-dimensional integrated-optical gratings,” Ph.D. dissertation (Delft University of Technology, Delft, The Netherlands, 1994).

Braat, J. J. M.

J. J. M. Braat, M. O. E. Laurijs, “Geometrical optics design and aberration analysis of a focusing grating coupler,” Opt. Eng. 33, 1037–1043 (1994).
[CrossRef]

J. J. M. Braat, “Microscope objectives for optical disc systems,” in Huygens’ Principle 1690–1990: Theory and Applications, H. Blok, H. A. Ferwerda, H. K. Kuiken, (Elsevier Science B. V., Amsterdam, 1990), pp. 33–63.

Chew, W. C.

Q.-H. Liu, W. C. Chew, “Analysis of discontinuities in planar dielectric waveguides: an eigenmode propagation method,” IEEE Trans. Microwave Theory Tech. 39, 422–430 (1991).
[CrossRef]

Ferwerda, H. A.

J. J. M. Braat, “Microscope objectives for optical disc systems,” in Huygens’ Principle 1690–1990: Theory and Applications, H. Blok, H. A. Ferwerda, H. K. Kuiken, (Elsevier Science B. V., Amsterdam, 1990), pp. 33–63.

Frankena, H. J.

Hehl, K.

W. Biehlig, K. Hehl, U. Langbein, F. Lederer, “Light propagation in a planar dielectric slab waveguide with step discontinuities. Part I. Operator formalism,” Opt. Quantum Electron. 18, 219–228 (1986).
[CrossRef]

Koyama, J.

S. Ura, T. Suhara, H. Nishihara, J. Koyama, “Focusing grating for integrated optical disc pickup device,” Trans. Inst. Electron. Commun. Eng. Jpn. Part C (Japan) 68-C, 803–811 (1985).

Kuiken, H. K.

J. J. M. Braat, “Microscope objectives for optical disc systems,” in Huygens’ Principle 1690–1990: Theory and Applications, H. Blok, H. A. Ferwerda, H. K. Kuiken, (Elsevier Science B. V., Amsterdam, 1990), pp. 33–63.

Langbein, U.

W. Biehlig, K. Hehl, U. Langbein, F. Lederer, “Light propagation in a planar dielectric slab waveguide with step discontinuities. Part I. Operator formalism,” Opt. Quantum Electron. 18, 219–228 (1986).
[CrossRef]

Laurijs, M. O. E.

J. J. M. Braat, M. O. E. Laurijs, “Geometrical optics design and aberration analysis of a focusing grating coupler,” Opt. Eng. 33, 1037–1043 (1994).
[CrossRef]

Lederer, F.

W. Biehlig, K. Hehl, U. Langbein, F. Lederer, “Light propagation in a planar dielectric slab waveguide with step discontinuities. Part I. Operator formalism,” Opt. Quantum Electron. 18, 219–228 (1986).
[CrossRef]

Liu, Q.-H.

Q.-H. Liu, W. C. Chew, “Analysis of discontinuities in planar dielectric waveguides: an eigenmode propagation method,” IEEE Trans. Microwave Theory Tech. 39, 422–430 (1991).
[CrossRef]

Mahmoud, S. F.

S. F. Mahmoud, J. C. Beal, “Scattering of surface waves at a dielectric discontinuity on a planar waveguide,” IEEE Trans. Microwave Theory Tech. MTT-23, 193–198 (1975).
[CrossRef]

Marcuse, D.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974), Chap. 1, pp. 1–30.

Nishihara, H.

S. Ura, T. Suhura, H. Nishihara, “Aberration characterizations of a focusing grating coupler in an integrated-optic disc pickup device,” Appl. Opt. 26, 4777–4782 (1987).
[CrossRef] [PubMed]

S. Ura, T. Suhara, H. Nishihara, J. Koyama, “Focusing grating for integrated optical disc pickup device,” Trans. Inst. Electron. Commun. Eng. Jpn. Part C (Japan) 68-C, 803–811 (1985).

Redheffer, R.

R. Redheffer, “Difference equations and functional equations in transmission-line theory,” in Modern Mathematics for the Engineer, 2nd ed., E. F. Beckenbach, ed. University of California Engineering Extension Series (McGraw-Hill, New York, 1961), pp. 282–337.

Shigesawa, H.

H. Shigesawa, M. Tsuji, “A new equivalent network method for analyzing discontinuity properties of open dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 37, 3–14 (1989).
[CrossRef]

H. Shigesawa, M. Tsuji, “Mode propagation through a step discontinuity planar waveguide,” IEEE Trans. Microwave Theory Tech. MTT-34, 205–212 (1986).
[CrossRef]

Suhara, T.

S. Ura, T. Suhara, H. Nishihara, J. Koyama, “Focusing grating for integrated optical disc pickup device,” Trans. Inst. Electron. Commun. Eng. Jpn. Part C (Japan) 68-C, 803–811 (1985).

Suhura, T.

Tsuji, M.

H. Shigesawa, M. Tsuji, “A new equivalent network method for analyzing discontinuity properties of open dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 37, 3–14 (1989).
[CrossRef]

H. Shigesawa, M. Tsuji, “Mode propagation through a step discontinuity planar waveguide,” IEEE Trans. Microwave Theory Tech. MTT-34, 205–212 (1986).
[CrossRef]

Ura, S.

S. Ura, T. Suhura, H. Nishihara, “Aberration characterizations of a focusing grating coupler in an integrated-optic disc pickup device,” Appl. Opt. 26, 4777–4782 (1987).
[CrossRef] [PubMed]

S. Ura, T. Suhara, H. Nishihara, J. Koyama, “Focusing grating for integrated optical disc pickup device,” Trans. Inst. Electron. Commun. Eng. Jpn. Part C (Japan) 68-C, 803–811 (1985).

Vassallo, C.

C. Vassallo, “Radiating normal modes of lossy planar waveguides,” J. Opt. Soc. Am. 69, 311–316 (1979).
[CrossRef]

C. Vassallo, Théorie des Guides D’Ondes Électromagnétiques (Eyrolles, Paris, 1985), Chap. 3, pp. 151–248.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), Chap. 8, pp. 439–441.

Appl. Opt. (1)

IEEE Trans. Microwave Theory Tech. (4)

Q.-H. Liu, W. C. Chew, “Analysis of discontinuities in planar dielectric waveguides: an eigenmode propagation method,” IEEE Trans. Microwave Theory Tech. 39, 422–430 (1991).
[CrossRef]

H. Shigesawa, M. Tsuji, “A new equivalent network method for analyzing discontinuity properties of open dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 37, 3–14 (1989).
[CrossRef]

H. Shigesawa, M. Tsuji, “Mode propagation through a step discontinuity planar waveguide,” IEEE Trans. Microwave Theory Tech. MTT-34, 205–212 (1986).
[CrossRef]

S. F. Mahmoud, J. C. Beal, “Scattering of surface waves at a dielectric discontinuity on a planar waveguide,” IEEE Trans. Microwave Theory Tech. MTT-23, 193–198 (1975).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Opt. Eng. (1)

J. J. M. Braat, M. O. E. Laurijs, “Geometrical optics design and aberration analysis of a focusing grating coupler,” Opt. Eng. 33, 1037–1043 (1994).
[CrossRef]

Opt. Quantum Electron. (2)

W. Biehlig, K. Hehl, U. Langbein, F. Lederer, “Light propagation in a planar dielectric slab waveguide with step discontinuities. Part I. Operator formalism,” Opt. Quantum Electron. 18, 219–228 (1986).
[CrossRef]

W. Biehlig, “Light propagation in a planar dielectric slab waveguide with step discontinuities. Part 2. Numerical analysis of TE-polarized fields,” Opt. Quantum Electron. 18, 229–238 (1986).
[CrossRef]

Trans. Inst. Electron. Commun. Eng. Jpn. Part C (Japan) (1)

S. Ura, T. Suhara, H. Nishihara, J. Koyama, “Focusing grating for integrated optical disc pickup device,” Trans. Inst. Electron. Commun. Eng. Jpn. Part C (Japan) 68-C, 803–811 (1985).

Other (6)

P-P. Borsboom, “Field analysis of two-dimensional integrated-optical gratings,” Ph.D. dissertation (Delft University of Technology, Delft, The Netherlands, 1994).

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974), Chap. 1, pp. 1–30.

C. Vassallo, Théorie des Guides D’Ondes Électromagnétiques (Eyrolles, Paris, 1985), Chap. 3, pp. 151–248.

R. Redheffer, “Difference equations and functional equations in transmission-line theory,” in Modern Mathematics for the Engineer, 2nd ed., E. F. Beckenbach, ed. University of California Engineering Extension Series (McGraw-Hill, New York, 1961), pp. 282–337.

J. J. M. Braat, “Microscope objectives for optical disc systems,” in Huygens’ Principle 1690–1990: Theory and Applications, H. Blok, H. A. Ferwerda, H. K. Kuiken, (Elsevier Science B. V., Amsterdam, 1990), pp. 33–63.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), Chap. 8, pp. 439–441.

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Figures (7)

Fig. 1
Fig. 1

Relief-type grating.

Fig. 2
Fig. 2

Geometry of the grating.

Fig. 3
Fig. 3

Focal region of a 19-corrugation grating with β = 45°, h = 30 μm.

Fig. 4
Fig. 4

Focal region of a 50-corrugation grating with β = 45°, h = 80 μm.

Fig. 5
Fig. 5

Focal region of a 124-corrugation grating with β = 45°, h = 200 μm.

Fig. 6
Fig. 6

Amplitude coefficient b1 (V) corresponding to the reflected radiation field of the FGC comprising 19 corrugations: (a) real part, (b) imaginary part.

Fig. 7
Fig. 7

Amplitude coefficient a1 (V) corresponding to the transmitted radiation field of the FGC comprising 19 corrugations: (a) real part, (b) imaginary part.

Equations (7)

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E y ( x , z ) = m = 0 M 1 [ A m exp ( j κ m z ) + B m exp ( j κ m z ) ] ψ m ( x ) 0 k x 1 k z 1 [ a 1 ( k x 1 ) exp ( j k z 1 z ) + b 1 ( k x 1 ) exp ( j k z 1 z ) ] ψ A ( k x 1 , x ) d k x 1 0 k x N k z N [ a N ( k x N ) exp ( j k z N z ) + b N ( k x N ) exp ( j k z N z ) ] ψ A ( k x N , x ) d k x N ,
ψ E = ψ A + ψ B ,
ψ O = ψ A ψ B ,
ψ 0 ( x ) = { A 0 , N exp [ j k x N ( x X N 1 ) ] for x > X N 1 B 0 , p exp [ j k x p ( x X p ) ] + C 0 , p exp [ + j k x p ( x X p ) ] for X p 1 < x < X p D 0 , 1 exp [ + j k x 1 ( x X 1 ) ] for x < X 1 ,
( W 2 + W 1 ) = ( S 11 S 12 S 21 S 22 ) ( W 1 + W 2 ) ,
( W k + 1 + W k ) = ( P 0 0 P ) ( W k + W k + 1 ) ,
z N s = n g n f h n g 2 n f 2 × [ 1 N s λ 0 2 n f h ( 1 n f N s λ 0 n g 2 h + N s 2 λ 0 2 4 n g 2 h 2 ) 1 / 2 ] ,

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