Abstract

A method for calculating arbitrary-order diffraction efficiencies of thick, lossless transmission gratings with arbitrary periodic grating shapes has been developed. This represents an extension of previous work to nonsinusoidal gratings and to higher-order Bragg angles. A Fourier-series representation of the grating is employed, along with a coupled-mode theory of diffraction. For illustration, numerical values of the diffraction efficiencies at the first three Bragg angles are calculated for sinusoidal, square-wave, triangular, and saw-tooth gratings. Numerical results for the same grating shapes with the same parameters are also calculated for comparison, by extending Burckhardt’s numerical method for analyzing thick sinusoidal gratings. The comparison shows that the coupled-mode theory provides results with relative computational ease and results that are in agreement with calculations obtained by extending the more-rigorous Burckhardt theory to nonsinusoidal grating shapes and to higher-order Bragg angles.

© 1975 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. U. Kogelnik, Bell Syst. Tech. J. 48, 2909 (1969).
    [Crossref]
  2. M. R. B. Forshaw, Opt. Laser Technol. 6, 28 (1974).
    [Crossref]
  3. B. H. Crawford, J. Sci. Instrum. 31, 333 (1954).
    [Crossref]
  4. J. N. Latta and R. C. Fairchild, J. Opt. Soc. Am. 63, 487 (1973).
  5. R. Shubert and J. H. Harris, J. Opt. Soc. Am. 61, 154 (1971).
    [Crossref]
  6. H. Kogelnik and C. V. Shank, Appl. Phys. Lett. 18, 152 (1971).
    [Crossref]
  7. I. P. Kaminow, H. P. Weber, and E. A. Chandross, Appl. Phys. Lett. 18, 497 (1971).
    [Crossref]
  8. H. Kogelnik and T. P. Sosnowski, Bell Syst. Tech. J. 49, 1602 (1970).
    [Crossref]
  9. M. L. Dakss, L. Kuhn, P. F. Heidrich, and B. A. Scott, Appl. Phys. Lett. 16, 523 (1970).
    [Crossref]
  10. J. M. Hammer, Appl. Phys. Lett. 18, 147 (1971).
    [Crossref]
  11. P. J. Van Heerden, Appl. Opt. 2, 393 (1963).
    [Crossref]
  12. K. S. Pennington and L. H. Lin, Appl. Phys. Lett. 7, 56 (1965).
    [Crossref]
  13. G. W. Stroke and A. E. Labeyrie, Phys. Lett. 20, 368 (1966).
    [Crossref]
  14. C. B. Burckhardt, J. Opt. Soc. Am. 56, 1502 (1966).
    [Crossref]
  15. P. Phariseau, Proc. Indian Acad. Sci. A 44, 165 (1956).
  16. C. F. Quate, C. D. W. Wilkinson, and D. K. Winslow, Proc. IEEE 53, 1604 (1965).
    [Crossref]
  17. R. S. Chu and T. Tamir, IEEE Tran. Micro. Thry. Tech. 18, 486 (1970).
    [Crossref]
  18. H. Kogelnik, J. Opt. Soc. Am. 57, 431 (1967).
    [Crossref]
  19. F. G. Kaspar, J. Opt. Soc. Am. 63, 37 (1973).
    [Crossref]
  20. T. Tamir and H. C. Wang, Can. J. Phys. 44, 2073 (1966).
    [Crossref]
  21. T. Tamir, Can. J. Phys. 44, 2461 (1966).
    [Crossref]
  22. D. A. Watkins, Topics in Electromagnetic Theory (Wiley, New York, 1958).
  23. M. G. Cohen and E. I. Gordon, Bell Syst. Tech. J. 45, 945 (1966).
    [Crossref]
  24. D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972), p. 71.
  25. W. W. Rigrod, J. Opt. Soc. Am. 64, 97 (1974); J. Opt. Soc. Am. 64, 895E (1974).
    [Crossref]
  26. F. S. Chen, J. T. LaMacchia, and D. B. Fraser, Appl. Phys. Lett. 13, 223 (1968).
    [Crossref]
  27. J. J. Amodei, RCA Rev. 32, 185 (1971).

1974 (2)

1973 (2)

J. N. Latta and R. C. Fairchild, J. Opt. Soc. Am. 63, 487 (1973).

F. G. Kaspar, J. Opt. Soc. Am. 63, 37 (1973).
[Crossref]

1971 (5)

R. Shubert and J. H. Harris, J. Opt. Soc. Am. 61, 154 (1971).
[Crossref]

H. Kogelnik and C. V. Shank, Appl. Phys. Lett. 18, 152 (1971).
[Crossref]

I. P. Kaminow, H. P. Weber, and E. A. Chandross, Appl. Phys. Lett. 18, 497 (1971).
[Crossref]

J. M. Hammer, Appl. Phys. Lett. 18, 147 (1971).
[Crossref]

J. J. Amodei, RCA Rev. 32, 185 (1971).

1970 (3)

R. S. Chu and T. Tamir, IEEE Tran. Micro. Thry. Tech. 18, 486 (1970).
[Crossref]

H. Kogelnik and T. P. Sosnowski, Bell Syst. Tech. J. 49, 1602 (1970).
[Crossref]

M. L. Dakss, L. Kuhn, P. F. Heidrich, and B. A. Scott, Appl. Phys. Lett. 16, 523 (1970).
[Crossref]

1969 (1)

U. Kogelnik, Bell Syst. Tech. J. 48, 2909 (1969).
[Crossref]

1968 (1)

F. S. Chen, J. T. LaMacchia, and D. B. Fraser, Appl. Phys. Lett. 13, 223 (1968).
[Crossref]

1967 (1)

1966 (5)

G. W. Stroke and A. E. Labeyrie, Phys. Lett. 20, 368 (1966).
[Crossref]

C. B. Burckhardt, J. Opt. Soc. Am. 56, 1502 (1966).
[Crossref]

T. Tamir and H. C. Wang, Can. J. Phys. 44, 2073 (1966).
[Crossref]

T. Tamir, Can. J. Phys. 44, 2461 (1966).
[Crossref]

M. G. Cohen and E. I. Gordon, Bell Syst. Tech. J. 45, 945 (1966).
[Crossref]

1965 (2)

K. S. Pennington and L. H. Lin, Appl. Phys. Lett. 7, 56 (1965).
[Crossref]

C. F. Quate, C. D. W. Wilkinson, and D. K. Winslow, Proc. IEEE 53, 1604 (1965).
[Crossref]

1963 (1)

1956 (1)

P. Phariseau, Proc. Indian Acad. Sci. A 44, 165 (1956).

1954 (1)

B. H. Crawford, J. Sci. Instrum. 31, 333 (1954).
[Crossref]

Amodei, J. J.

J. J. Amodei, RCA Rev. 32, 185 (1971).

Burckhardt, C. B.

Chandross, E. A.

I. P. Kaminow, H. P. Weber, and E. A. Chandross, Appl. Phys. Lett. 18, 497 (1971).
[Crossref]

Chen, F. S.

F. S. Chen, J. T. LaMacchia, and D. B. Fraser, Appl. Phys. Lett. 13, 223 (1968).
[Crossref]

Chu, R. S.

R. S. Chu and T. Tamir, IEEE Tran. Micro. Thry. Tech. 18, 486 (1970).
[Crossref]

Cohen, M. G.

M. G. Cohen and E. I. Gordon, Bell Syst. Tech. J. 45, 945 (1966).
[Crossref]

Crawford, B. H.

B. H. Crawford, J. Sci. Instrum. 31, 333 (1954).
[Crossref]

Dakss, M. L.

M. L. Dakss, L. Kuhn, P. F. Heidrich, and B. A. Scott, Appl. Phys. Lett. 16, 523 (1970).
[Crossref]

Fairchild, R. C.

J. N. Latta and R. C. Fairchild, J. Opt. Soc. Am. 63, 487 (1973).

Forshaw, M. R. B.

M. R. B. Forshaw, Opt. Laser Technol. 6, 28 (1974).
[Crossref]

Fraser, D. B.

F. S. Chen, J. T. LaMacchia, and D. B. Fraser, Appl. Phys. Lett. 13, 223 (1968).
[Crossref]

Gordon, E. I.

M. G. Cohen and E. I. Gordon, Bell Syst. Tech. J. 45, 945 (1966).
[Crossref]

Hammer, J. M.

J. M. Hammer, Appl. Phys. Lett. 18, 147 (1971).
[Crossref]

Harris, J. H.

Heidrich, P. F.

M. L. Dakss, L. Kuhn, P. F. Heidrich, and B. A. Scott, Appl. Phys. Lett. 16, 523 (1970).
[Crossref]

Kaminow, I. P.

I. P. Kaminow, H. P. Weber, and E. A. Chandross, Appl. Phys. Lett. 18, 497 (1971).
[Crossref]

Kaspar, F. G.

Kogelnik, H.

H. Kogelnik and C. V. Shank, Appl. Phys. Lett. 18, 152 (1971).
[Crossref]

H. Kogelnik and T. P. Sosnowski, Bell Syst. Tech. J. 49, 1602 (1970).
[Crossref]

H. Kogelnik, J. Opt. Soc. Am. 57, 431 (1967).
[Crossref]

Kogelnik, U.

U. Kogelnik, Bell Syst. Tech. J. 48, 2909 (1969).
[Crossref]

Kuhn, L.

M. L. Dakss, L. Kuhn, P. F. Heidrich, and B. A. Scott, Appl. Phys. Lett. 16, 523 (1970).
[Crossref]

Labeyrie, A. E.

G. W. Stroke and A. E. Labeyrie, Phys. Lett. 20, 368 (1966).
[Crossref]

LaMacchia, J. T.

F. S. Chen, J. T. LaMacchia, and D. B. Fraser, Appl. Phys. Lett. 13, 223 (1968).
[Crossref]

Latta, J. N.

J. N. Latta and R. C. Fairchild, J. Opt. Soc. Am. 63, 487 (1973).

Lin, L. H.

K. S. Pennington and L. H. Lin, Appl. Phys. Lett. 7, 56 (1965).
[Crossref]

Marcuse, D.

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972), p. 71.

Pennington, K. S.

K. S. Pennington and L. H. Lin, Appl. Phys. Lett. 7, 56 (1965).
[Crossref]

Phariseau, P.

P. Phariseau, Proc. Indian Acad. Sci. A 44, 165 (1956).

Quate, C. F.

C. F. Quate, C. D. W. Wilkinson, and D. K. Winslow, Proc. IEEE 53, 1604 (1965).
[Crossref]

Rigrod, W. W.

Scott, B. A.

M. L. Dakss, L. Kuhn, P. F. Heidrich, and B. A. Scott, Appl. Phys. Lett. 16, 523 (1970).
[Crossref]

Shank, C. V.

H. Kogelnik and C. V. Shank, Appl. Phys. Lett. 18, 152 (1971).
[Crossref]

Shubert, R.

Sosnowski, T. P.

H. Kogelnik and T. P. Sosnowski, Bell Syst. Tech. J. 49, 1602 (1970).
[Crossref]

Stroke, G. W.

G. W. Stroke and A. E. Labeyrie, Phys. Lett. 20, 368 (1966).
[Crossref]

Tamir, T.

R. S. Chu and T. Tamir, IEEE Tran. Micro. Thry. Tech. 18, 486 (1970).
[Crossref]

T. Tamir, Can. J. Phys. 44, 2461 (1966).
[Crossref]

T. Tamir and H. C. Wang, Can. J. Phys. 44, 2073 (1966).
[Crossref]

Van Heerden, P. J.

Wang, H. C.

T. Tamir and H. C. Wang, Can. J. Phys. 44, 2073 (1966).
[Crossref]

Watkins, D. A.

D. A. Watkins, Topics in Electromagnetic Theory (Wiley, New York, 1958).

Weber, H. P.

I. P. Kaminow, H. P. Weber, and E. A. Chandross, Appl. Phys. Lett. 18, 497 (1971).
[Crossref]

Wilkinson, C. D. W.

C. F. Quate, C. D. W. Wilkinson, and D. K. Winslow, Proc. IEEE 53, 1604 (1965).
[Crossref]

Winslow, D. K.

C. F. Quate, C. D. W. Wilkinson, and D. K. Winslow, Proc. IEEE 53, 1604 (1965).
[Crossref]

Appl. Opt. (1)

Appl. Phys. Lett. (6)

K. S. Pennington and L. H. Lin, Appl. Phys. Lett. 7, 56 (1965).
[Crossref]

H. Kogelnik and C. V. Shank, Appl. Phys. Lett. 18, 152 (1971).
[Crossref]

I. P. Kaminow, H. P. Weber, and E. A. Chandross, Appl. Phys. Lett. 18, 497 (1971).
[Crossref]

M. L. Dakss, L. Kuhn, P. F. Heidrich, and B. A. Scott, Appl. Phys. Lett. 16, 523 (1970).
[Crossref]

J. M. Hammer, Appl. Phys. Lett. 18, 147 (1971).
[Crossref]

F. S. Chen, J. T. LaMacchia, and D. B. Fraser, Appl. Phys. Lett. 13, 223 (1968).
[Crossref]

Bell Syst. Tech. J. (3)

M. G. Cohen and E. I. Gordon, Bell Syst. Tech. J. 45, 945 (1966).
[Crossref]

H. Kogelnik and T. P. Sosnowski, Bell Syst. Tech. J. 49, 1602 (1970).
[Crossref]

U. Kogelnik, Bell Syst. Tech. J. 48, 2909 (1969).
[Crossref]

Can. J. Phys. (2)

T. Tamir and H. C. Wang, Can. J. Phys. 44, 2073 (1966).
[Crossref]

T. Tamir, Can. J. Phys. 44, 2461 (1966).
[Crossref]

IEEE Tran. Micro. Thry. Tech. (1)

R. S. Chu and T. Tamir, IEEE Tran. Micro. Thry. Tech. 18, 486 (1970).
[Crossref]

J. Opt. Soc. Am. (6)

J. Sci. Instrum. (1)

B. H. Crawford, J. Sci. Instrum. 31, 333 (1954).
[Crossref]

Opt. Laser Technol. (1)

M. R. B. Forshaw, Opt. Laser Technol. 6, 28 (1974).
[Crossref]

Phys. Lett. (1)

G. W. Stroke and A. E. Labeyrie, Phys. Lett. 20, 368 (1966).
[Crossref]

Proc. IEEE (1)

C. F. Quate, C. D. W. Wilkinson, and D. K. Winslow, Proc. IEEE 53, 1604 (1965).
[Crossref]

Proc. Indian Acad. Sci. A (1)

P. Phariseau, Proc. Indian Acad. Sci. A 44, 165 (1956).

RCA Rev. (1)

J. J. Amodei, RCA Rev. 32, 185 (1971).

Other (2)

D. A. Watkins, Topics in Electromagnetic Theory (Wiley, New York, 1958).

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972), p. 71.

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 (1)

FIG. 1
FIG. 1

Geometry of a thick grating with unslated fringes. The spatial modulation of is indicated by the line pattern.

Tables (2)

Tables Icon

TABLE I First three Fourier components for various relative dielectric constant profiles (grating shapes). Components are normalized to the amplitude of the fundamental grating s1.

Tables Icon

TABLE II Comparison of diffraction efficiency in percent at the first-, second-, and third-order Bragg angles for transmission gratings with boundary reflections and with the same average and fundamental Fourier grating components. The grating parameters are r0 = 2.3225 (value used in Refs. 14 and 18), s1 = 10−4r0, L = 3.6303 μm, and the wavelength λ = 0.6328 μm. Diffraction efficiencies of less than 5 × 10−8% are listed as 0.00(−5).

Equations (35)

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

r ( x ) = ( x ) 0 = r 0 + h = 1 [ c h cos ( h K x ) + s h sin ( h K x ) ] ,
n ( x ) = n 0 + h = 1 [ n c h cos ( h K x ) + n s h sin ( h K x ) ] ,
[ 2 + k 2 r ( x ) ] E ( x , z ) = 0 ,
q c h = 2 ( L / h λ 2 ) c h , h = 1 , 2 , 3 ,
q s h = 2 ( L / h λ 2 ) s h , h = 1 , 2 , 3 ,
E ˜ ( x , z ) = S ˜ 0 ( z ) exp ( j η ˜ 0 x ) + S ˜ 0 ( z ) exp ( j η ˜ m x ) ,
θ m = sin 1 ( η m / k ) = sin 1 ( sin θ 0 m λ / L ) .
d S ˜ 0 d z j ξ ¯ 0 S ˜ 0 j ( h m ( C c h m h + j C s h m h ) ) S ˜ m = 0 ,
d S ˜ m d z j ξ ¯ m S ˜ m j ( h m ( C c h m h j C s h m h ) ) S ˜ 0 = 0 ,
ξ ¯ m = { k 2 r 0 [ k ( r 0 ) 1 / 2 sin φ ( 2 m π / L ) ] 2 } 1 / 2 ,
C c h m h 1 ξ ¯ 0 + ξ ¯ m ( 1 ( 2 ) ( m h 1 ) h ( m h 1 ) ! ) 2 ( π L / h ) 2 { [ ( q c h ) 2 + ( q s h ) 2 ] 1 / 2 } m h × cos [ m h tan 1 ( s h / c h ) ] ,
C s h m h 1 ξ ¯ 0 + ξ ¯ m ( 1 ( 2 ) ( m h 1 ) ( m h 1 ) ! ) 2 ( π L / h ) 2 { [ ( q c h ) 2 + ( q s h ) 2 ] 1 / 2 } m h × sin [ m h tan 1 ( s h / c h ) ] .
S ˜ 0 ( z ) = A 0 exp ( j ξ ˜ 0 z ) + B 0 exp ( j ξ ˜ m z ) ,
S ˜ m ( z ) = A m exp ( j ξ ˜ 0 z ) + B m exp ( j ξ ˜ m z ) .
ξ ˜ 0 , m = ξ ¯ 0 + ξ ¯ m 2 ± [ ( ξ ¯ 0 ξ ¯ m 2 ) 2 + ( h m C c h m h ) 2 + ( h m C s h m h ) 2 ] 1 / 2 ,
( h m ( C c h m h j C s h n h ) ) A 0 ( ξ ˜ 0 ξ ¯ m ) A m = 0 ,
( h m ( C c h m h j C s h n h ) ) B 0 ( ξ ˜ m ξ ¯ m ) B m = 0 ,
S ˜ 0 ( 0 ) = A 0 + B 0 = 1.
S ˜ m ( 0 ) = A m + B m = 0.
A 0 = ξ ¯ 0 ξ ¯ m ξ ˜ 0 ξ ˜ m = 1 2 ( ( ξ ¯ 0 ξ ¯ m ) + { ( ξ ¯ 0 ξ ¯ m ) 2 + 4 [ ( h m C c h m h ) 2 + ( h m C s h m h ) 2 ] } 1 / 2 ) / ( { ( ξ ¯ 0 ξ ¯ m ) 2 + 4 [ ( h m C c h m h ) 2 + ( h m C s h m h ) 2 ] } 1 / 2 ) ,
B 0 = ξ ¯ m ξ ¯ m ξ ˜ 0 ξ ˜ m = 1 2 ( ( ξ ¯ m ξ ¯ 0 ) + { ( ξ ¯ 0 ξ ¯ m ) 2 + 4 [ ( h m C c h m h ) 2 + ( h m C s h m h ) 2 ] } 1 / 2 ) / ( { ( ξ ¯ 0 ξ ¯ m ) 2 + 4 [ ( h m C c h m h ) 2 + ( h m C s h m h ) 2 ] } 1 / 2 ) ,
A m = B m = [ ( h m C c h m h ) j ( h m C s h m h ) ] / ( { ( ξ ¯ 0 ξ ¯ m ) 2 + 4 [ ( h m C c h m h ) 2 + ( h m C s h m h ) 2 ] } 1 / 2 ) .
ξ ˜ 0 , m = ξ ¯ 0 ± [ ( h m C c h m h ) 2 + ( h m C s h m h ) 2 ] 1 / 2 ,
A 0 = 1 2 = B 0 ,
A m = B m = 1 2 [ ( h m C c h m h ) j ( h m C s h m h ) ] / { [ ( h m C c h m h ) 2 + ( h m C s h m h ) 2 ] 1 / 2 } .
S ˜ 0 ( z ) = exp ( j ξ ¯ 0 z ) cos { [ ( h m C c h m h ) 2 + ( h m C s h m h ) 2 ] 1 / 2 z } ,
S ˜ m ( z ) = j 2 A m exp ( j ξ ¯ 0 z ) × sin { [ ( h m C c h m h ) 2 + ( h m C s h m h ) 2 ] 1 / 2 z } ,
D E m S ˜ m ( d ) S ˜ m * ( d ) / S ˜ 0 ( 0 ) S ˜ 0 * ( 0 ) ,
D E m = sin 2 { [ ( h m C c h m h ) 2 + ( h m C s h m h ) 2 ] 1 / 2 d } ,
D E m = sin 2 ( { [ h m π ( 2 ) m h ( L ( m h 1 ) ( m h 1 ) ! ( h ) ( m h 1 ) ) 2 × { [ ( c h ) 2 + ( s h ) 2 ] 1 / 2 } m h λ ( 2 m h 1 ) cos [ m h tan 1 ( s h / c h ) ] ( r 0 ) 1 / 2 cos φ ] 2 + [ h m π ( 2 ) m h ( L ( m h 1 ) ( m h 1 ) ! ( h ) ( m h 1 ) ) 2 { [ ( c h ) 2 + ( s h ) 2 ] 1 / 2 } m h λ ( 2 m h 1 ) × sin [ m h tan 1 ( s h / c h ) ] ( r 0 ) 1 / 2 cos φ ] 2 } 1 / 2 d ) .
D E 1 = sin 2 ( s 1 π d 2 λ ( r 0 ) 1 / 2 cos φ ) ,
D E 2 = sin 2 [ ( L 4 ( s 1 ) 4 4 λ 4 + ( s 2 ) 2 ) 1 / 2 π d 2 λ ( r 0 ) 1 / 2 cos φ ] ,
D E 3 = sin 2 [ ( s 3 L 4 ( s 1 ) 3 16 λ 4 ) π d 2 λ ( r 0 ) 1 / 2 cos φ ] .
D E m = sin 2 ( { [ h m ( L ( m h 1 ) ( m h 1 ) ! ( h ) ( m h 1 ) ) 2 π { [ ( n c h ) 2 + ( n s h ) 2 ] 1 / 2 } m h λ ( 2 m h 1 ) × ( n 0 ) ( m h 1 ) cos [ m h tan 1 ( n s h / n c h ) ] cos φ ] 2 + [ h m ( L ( m h 1 ) ( m h 1 ) ! ( h ) ( m h 1 ) ) 2 × π { [ ( n c h ) 2 + ( n s h ) 2 ] 1 / 2 } m h λ ( 2 m h 1 ) ( n 0 ) ( m h 1 ) sin [ m h tan 1 ( n s h / n c h ) ] cos φ ] 2 } 1 / 2 d ) .
τ m = ( 1 R ) 2 [ 1 + 2 R cos ( 2 β d ) + R 2 ] / { ( 1 R 2 ) 2 + 4 R 2 [ cos 2 ( 2 ν m d ) + cos 2 ( 2 β d ) ] 4 R ( 1 R 2 ) cos ( 2 ν m d ) cos ( 2 β d ) } ,