Abstract

Diffraction by a dielectric surface-relief grating is analyzed using rigorous coupled-wave theory. The analysis applies to arbitrary grating profiles, groove depths, angles of incidence, and wavelengths. Example results for a wide range of groove depths are presented for sinusoidal, square-wave, triangular, and sawtooth gratings. Diffraction efficiencies obtained from the present method of analysis are compared with previously published numerical results. To obtain large diffraction efficiencies (greater than 85%) for gratings with typical substrate permittivities, it is shown that the grating profile should possess even symmetry.

© 1982 Optical Society of America

Full Article  |  PDF Article

Corrections

M. G. Moharam and T. K. Gaylord, "Diffraction analysis of dielectric surface-relief gratings: erratum," J. Opt. Soc. Am. 73, 411-411 (1983)
https://www.osapublishing.org/josa/abstract.cfm?uri=josa-73-3-411

References

  • View by:
  • |
  • |
  • |

  1. R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980).
    [Crossref]
  2. M. Neviere, R. Petit, and M. Cadilhac, “About the theory of optical grating coupler-waveguide systems,” Opt. Commun. 8, 113–117 (1973).
    [Crossref]
  3. S. T. Peng, H. L. Bertoni, and T. Tamir, “Analysis of periodic thin-film structures with rectangular profiles,” Opt. Commun. 10, 91–94 (1974).
    [Crossref]
  4. S. T. Peng and T. Tamir, “Directional blazing of waves guided by asymmetrical dielectric gratings,” Opt. Commun. 11, 405–409 (1974).
    [Crossref]
  5. S. T. Peng, T. Tamir, and H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
    [Crossref]
  6. D. Marcuse, “Exact theory of TE-wave scattering from blazed dielectric gratings,” Bell Syst. Tech. J. 55, 1295–1317 (1976).
    [Crossref]
  7. K. Knop, “Rigorous diffraction theory for transmission phase gratings with deep rectangular grooves,” J. Opt. Soc. Am. 68, 1206–1210 (1978).
    [Crossref]
  8. K. C. Chang and T. Tamir, “Simplified approach to surface-wave scattering by blazed dielectric gratings,” Appl. Opt. 19, 282–288 (1980).
    [Crossref] [PubMed]
  9. W. Streifer, D. R. Scifres, and R. D. Burnham, “Analysis of grating-coupled radiation in GaAs:GaAlAs lasers and waveguides,” IEEE J. Quantum Electron. QE-12, 422–428 (1976).
    [Crossref]
  10. T. Tamir and S. T. Peng, “Analysis and design of grating couplers,” Appl. Phys. 14, 235–254 (1977).
    [Crossref]
  11. A. R. Neureuther and K. Zaki, “Numerical methods for the analysis of scattering from nonplanar and periodic structures,” Alta Freq. 38, 282–285 (1969).
  12. K. Zaki, “Numerical methods for the analysis of scattering from nonplanar periodic structures,” Ph.D. thesis (University of California, Berkeley, Calif., 1969) (unpublished).
  13. P. M. van den Berg, “Rigorous diffraction theory of optical reflection and transmission gratings,” Ph.D. thesis (Delft University of Technology, Delft, The Netherlands, 1971), rep. no. 1971-16 (unpublished).
  14. D. E. Tremain and K. K. Mei, “Application of the unimoment method to scattering from periodic dielectric structures,” J. Opt. Soc. Am. 68, 775–783 (1978).
    [Crossref]
  15. K. C. Chang, “Surface-wave scattering by dielectric gratings with arbitrary profiles,” Ph.D. thesis (Polytechnic Institute of New York, New York, 1979) (unpublished).
  16. K. C. Chang, V. Shah, and T. Tamir, “Scattering and guiding of waves by dielectric gratings with arbitrary profiles,” J. Opt. Soc. Am. 70, 804–813 (1980).
    [Crossref]
  17. M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of planar grating diffraction,” J. Opt. Soc. Am. 71, 811–818 (1981).
    [Crossref]
  18. L. R. Lewis and A. Hessel, “Propagation characteristics of periodic arrays of dielectric slabs,” IEEE Trans. Microwave Theory Tech. MTT-19, 276–286 (1971).
    [Crossref]
  19. E.g., C. L. Liu and J. W. S. Liu, Linear Systems Analysis (McGraw-Hill, New York, 1975).
  20. E.g., program EIGRF from the International Mathematics and Statistics Library (IMSL), Houston, Texas.
  21. E.g., B. Carnahan, H. A. Luther, and J. O. Wilkes, Applied Numerical Methods (Wiley, New York, 1969).

1981 (1)

1980 (2)

1978 (2)

1977 (1)

T. Tamir and S. T. Peng, “Analysis and design of grating couplers,” Appl. Phys. 14, 235–254 (1977).
[Crossref]

1976 (2)

D. Marcuse, “Exact theory of TE-wave scattering from blazed dielectric gratings,” Bell Syst. Tech. J. 55, 1295–1317 (1976).
[Crossref]

W. Streifer, D. R. Scifres, and R. D. Burnham, “Analysis of grating-coupled radiation in GaAs:GaAlAs lasers and waveguides,” IEEE J. Quantum Electron. QE-12, 422–428 (1976).
[Crossref]

1975 (1)

S. T. Peng, T. Tamir, and H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[Crossref]

1974 (2)

S. T. Peng, H. L. Bertoni, and T. Tamir, “Analysis of periodic thin-film structures with rectangular profiles,” Opt. Commun. 10, 91–94 (1974).
[Crossref]

S. T. Peng and T. Tamir, “Directional blazing of waves guided by asymmetrical dielectric gratings,” Opt. Commun. 11, 405–409 (1974).
[Crossref]

1973 (1)

M. Neviere, R. Petit, and M. Cadilhac, “About the theory of optical grating coupler-waveguide systems,” Opt. Commun. 8, 113–117 (1973).
[Crossref]

1971 (1)

L. R. Lewis and A. Hessel, “Propagation characteristics of periodic arrays of dielectric slabs,” IEEE Trans. Microwave Theory Tech. MTT-19, 276–286 (1971).
[Crossref]

1969 (1)

A. R. Neureuther and K. Zaki, “Numerical methods for the analysis of scattering from nonplanar and periodic structures,” Alta Freq. 38, 282–285 (1969).

Bertoni, H. L.

S. T. Peng, T. Tamir, and H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[Crossref]

S. T. Peng, H. L. Bertoni, and T. Tamir, “Analysis of periodic thin-film structures with rectangular profiles,” Opt. Commun. 10, 91–94 (1974).
[Crossref]

Burnham, R. D.

W. Streifer, D. R. Scifres, and R. D. Burnham, “Analysis of grating-coupled radiation in GaAs:GaAlAs lasers and waveguides,” IEEE J. Quantum Electron. QE-12, 422–428 (1976).
[Crossref]

Cadilhac, M.

M. Neviere, R. Petit, and M. Cadilhac, “About the theory of optical grating coupler-waveguide systems,” Opt. Commun. 8, 113–117 (1973).
[Crossref]

Carnahan, B.

E.g., B. Carnahan, H. A. Luther, and J. O. Wilkes, Applied Numerical Methods (Wiley, New York, 1969).

Chang, K. C.

Gaylord, T. K.

Hessel, A.

L. R. Lewis and A. Hessel, “Propagation characteristics of periodic arrays of dielectric slabs,” IEEE Trans. Microwave Theory Tech. MTT-19, 276–286 (1971).
[Crossref]

Knop, K.

Lewis, L. R.

L. R. Lewis and A. Hessel, “Propagation characteristics of periodic arrays of dielectric slabs,” IEEE Trans. Microwave Theory Tech. MTT-19, 276–286 (1971).
[Crossref]

Liu, C. L.

E.g., C. L. Liu and J. W. S. Liu, Linear Systems Analysis (McGraw-Hill, New York, 1975).

Liu, J. W. S.

E.g., C. L. Liu and J. W. S. Liu, Linear Systems Analysis (McGraw-Hill, New York, 1975).

Luther, H. A.

E.g., B. Carnahan, H. A. Luther, and J. O. Wilkes, Applied Numerical Methods (Wiley, New York, 1969).

Marcuse, D.

D. Marcuse, “Exact theory of TE-wave scattering from blazed dielectric gratings,” Bell Syst. Tech. J. 55, 1295–1317 (1976).
[Crossref]

Mei, K. K.

Moharam, M. G.

Neureuther, A. R.

A. R. Neureuther and K. Zaki, “Numerical methods for the analysis of scattering from nonplanar and periodic structures,” Alta Freq. 38, 282–285 (1969).

Neviere, M.

M. Neviere, R. Petit, and M. Cadilhac, “About the theory of optical grating coupler-waveguide systems,” Opt. Commun. 8, 113–117 (1973).
[Crossref]

Peng, S. T.

T. Tamir and S. T. Peng, “Analysis and design of grating couplers,” Appl. Phys. 14, 235–254 (1977).
[Crossref]

S. T. Peng, T. Tamir, and H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[Crossref]

S. T. Peng and T. Tamir, “Directional blazing of waves guided by asymmetrical dielectric gratings,” Opt. Commun. 11, 405–409 (1974).
[Crossref]

S. T. Peng, H. L. Bertoni, and T. Tamir, “Analysis of periodic thin-film structures with rectangular profiles,” Opt. Commun. 10, 91–94 (1974).
[Crossref]

Petit, R.

M. Neviere, R. Petit, and M. Cadilhac, “About the theory of optical grating coupler-waveguide systems,” Opt. Commun. 8, 113–117 (1973).
[Crossref]

Scifres, D. R.

W. Streifer, D. R. Scifres, and R. D. Burnham, “Analysis of grating-coupled radiation in GaAs:GaAlAs lasers and waveguides,” IEEE J. Quantum Electron. QE-12, 422–428 (1976).
[Crossref]

Shah, V.

Streifer, W.

W. Streifer, D. R. Scifres, and R. D. Burnham, “Analysis of grating-coupled radiation in GaAs:GaAlAs lasers and waveguides,” IEEE J. Quantum Electron. QE-12, 422–428 (1976).
[Crossref]

Tamir, T.

K. C. Chang, V. Shah, and T. Tamir, “Scattering and guiding of waves by dielectric gratings with arbitrary profiles,” J. Opt. Soc. Am. 70, 804–813 (1980).
[Crossref]

K. C. Chang and T. Tamir, “Simplified approach to surface-wave scattering by blazed dielectric gratings,” Appl. Opt. 19, 282–288 (1980).
[Crossref] [PubMed]

T. Tamir and S. T. Peng, “Analysis and design of grating couplers,” Appl. Phys. 14, 235–254 (1977).
[Crossref]

S. T. Peng, T. Tamir, and H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[Crossref]

S. T. Peng and T. Tamir, “Directional blazing of waves guided by asymmetrical dielectric gratings,” Opt. Commun. 11, 405–409 (1974).
[Crossref]

S. T. Peng, H. L. Bertoni, and T. Tamir, “Analysis of periodic thin-film structures with rectangular profiles,” Opt. Commun. 10, 91–94 (1974).
[Crossref]

Tremain, D. E.

van den Berg, P. M.

P. M. van den Berg, “Rigorous diffraction theory of optical reflection and transmission gratings,” Ph.D. thesis (Delft University of Technology, Delft, The Netherlands, 1971), rep. no. 1971-16 (unpublished).

Wilkes, J. O.

E.g., B. Carnahan, H. A. Luther, and J. O. Wilkes, Applied Numerical Methods (Wiley, New York, 1969).

Zaki, K.

A. R. Neureuther and K. Zaki, “Numerical methods for the analysis of scattering from nonplanar and periodic structures,” Alta Freq. 38, 282–285 (1969).

K. Zaki, “Numerical methods for the analysis of scattering from nonplanar periodic structures,” Ph.D. thesis (University of California, Berkeley, Calif., 1969) (unpublished).

Alta Freq. (1)

A. R. Neureuther and K. Zaki, “Numerical methods for the analysis of scattering from nonplanar and periodic structures,” Alta Freq. 38, 282–285 (1969).

Appl. Opt. (1)

Appl. Phys. (1)

T. Tamir and S. T. Peng, “Analysis and design of grating couplers,” Appl. Phys. 14, 235–254 (1977).
[Crossref]

Bell Syst. Tech. J. (1)

D. Marcuse, “Exact theory of TE-wave scattering from blazed dielectric gratings,” Bell Syst. Tech. J. 55, 1295–1317 (1976).
[Crossref]

IEEE J. Quantum Electron. (1)

W. Streifer, D. R. Scifres, and R. D. Burnham, “Analysis of grating-coupled radiation in GaAs:GaAlAs lasers and waveguides,” IEEE J. Quantum Electron. QE-12, 422–428 (1976).
[Crossref]

IEEE Trans. Microwave Theory Tech. (2)

S. T. Peng, T. Tamir, and H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[Crossref]

L. R. Lewis and A. Hessel, “Propagation characteristics of periodic arrays of dielectric slabs,” IEEE Trans. Microwave Theory Tech. MTT-19, 276–286 (1971).
[Crossref]

J. Opt. Soc. Am. (4)

Opt. Commun. (3)

M. Neviere, R. Petit, and M. Cadilhac, “About the theory of optical grating coupler-waveguide systems,” Opt. Commun. 8, 113–117 (1973).
[Crossref]

S. T. Peng, H. L. Bertoni, and T. Tamir, “Analysis of periodic thin-film structures with rectangular profiles,” Opt. Commun. 10, 91–94 (1974).
[Crossref]

S. T. Peng and T. Tamir, “Directional blazing of waves guided by asymmetrical dielectric gratings,” Opt. Commun. 11, 405–409 (1974).
[Crossref]

Other (7)

R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980).
[Crossref]

K. C. Chang, “Surface-wave scattering by dielectric gratings with arbitrary profiles,” Ph.D. thesis (Polytechnic Institute of New York, New York, 1979) (unpublished).

K. Zaki, “Numerical methods for the analysis of scattering from nonplanar periodic structures,” Ph.D. thesis (University of California, Berkeley, Calif., 1969) (unpublished).

P. M. van den Berg, “Rigorous diffraction theory of optical reflection and transmission gratings,” Ph.D. thesis (Delft University of Technology, Delft, The Netherlands, 1971), rep. no. 1971-16 (unpublished).

E.g., C. L. Liu and J. W. S. Liu, Linear Systems Analysis (McGraw-Hill, New York, 1975).

E.g., program EIGRF from the International Mathematics and Statistics Library (IMSL), Houston, Texas.

E.g., B. Carnahan, H. A. Luther, and J. O. Wilkes, Applied Numerical Methods (Wiley, New York, 1969).

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 (8)

Fig. 1
Fig. 1

Geometry of dielectric surface-relief grating.

Fig. 2
Fig. 2

The nth planar grating resulting from the decomposition of the surface-relief grating into N thin gratings.

Fig. 3
Fig. 3

Matrix-equation representation of 2(N + 1)s boundary-condition equations, where s is the total number of diffracted waves retained in the analysis. C1 represents the column vector Cq′1, where q′ = 1 to 2s and likewise for C2 through CN. The reflected and transmitted amplitudes Ri and Ti are column vectors of length s. The product output vector, before manipulation, is all zeros except for the two ones that are shown. These correspond to the normalized E and H values in the input wave.

Fig. 4
Fig. 4

Diffraction efficiency as a function of groove depth for a lossless sinusoidal surface-relief grating. Incidence is at first Bragg angle (m = 1), λ = Λ, I = 1.00, and III = 2.50.

Fig. 5
Fig. 5

Diffraction efficiency as a function of groove depth for a lossless square-wave surface-relief grating. Incidence is at first Bragg angle (m = 1), λ = Λ, I = 1.00, and III = 2.50.

Fig. 6
Fig. 6

Diffraction efficiency as a function of groove depth for a lossless triangular grating. Incidence is at first Bragg angle (m = 1), λ = Λ, I = 1.00, and III = 2.50.

Fig. 7
Fig. 7

Diffraction efficiency as a function of groove depth for a lossless sawtooth grating. Incidence is at first Bragg angle (m = 1) with the inclined surfaces of the sawtooth facing the incident beam. Also λ = Λ, I = 1.00, and III = 2.50.

Fig. 8
Fig. 8

Diffraction efficiency as a function of groove depth for a lossless sawtooth grating. Incidence is at first Bragg angle (m = 1) with the inclined surfaces of the sawtooth facing away from the incident beam. Also λ = Λ, I = 1.00, and III = 2.50.

Tables (3)

Tables Icon

Table 1 Maximum Transmitted First-Order (i = + 1) Diffraction Efficiencies for Various Grating Profilesa

Tables Icon

Table 2 Comparison of Diffraction Efficiencies Calculated by van den Berga and Chang et al.b with Values Calculated in the Present Workc

Tables Icon

Table 3 Comparison of Diffraction Efficiencies Calculated by Tremain and Meia and Zakib with Values Calculated in the Present Workc

Equations (30)

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

z = F ( x ) = F ( x + Λ ) ,
E 1 = exp ( j k 1 · r ) + i = R i exp ( j k 1 i · r ) ,
E 3 = i = T i exp [ j k 3 i · ( r d ) ] ,
n ( x , z n ) = I + ( III I ) h = + h , n exp ( j h K x ) ,
h , n = ( 1 / Λ ) 0 Λ f ( x , z n ) exp ( j h K x ) d x ,
E 2 , n = i = + S i , n ( z ) exp ( j σ i , n · r ) ,
σ i , n = k 2 , n i K ,
E 1 = exp { j [ k 1 ( sin θ x + cos θ z ) ] } + i = + R i exp ( j { ( k 1 sin θ i K ) x [ k 1 2 ( k 1 sin θ i K ) 2 ] 1 / 2 z } ) ,
E 2 , n = i = + S i , n ( z ) exp { j [ ( k 1 sin θ i K ) x + ( k 2 , n 2 k 1 2 sin 2 θ ) 1 / 2 z ] } ,
E 3 = i = T i exp ( j { ( k 1 sin θ i K ) x + [ k 3 2 ( k 1 sin θ i K ) 2 ] 1 / 2 ( z d ) } ) .
2 E 2 , n + k 2 n ( x , z n ) E 2 , n = 0 ,
d 2 S i , n ( z ) d z 2 j 2 ( k 2 , n 2 k 1 2 sin 2 θ ) 1 / 2 d S i , n ( z ) d z + K 2 i ( m i ) S i , n ( z ) + k 2 ( III I ) h = 1 [ h , n S i h , n ( z ) + h , n * S i + h , n ( z ) ] = 0 .
m = 2 Λ ( I ) 1 / 2 sin θ / λ .
S 1 , i , n ( z ) = S i , n ( z ) ,
S 2 , i , n ( z ) = d S i , n ( z ) / d z
d S 1 , i , n ( z ) d z = S 2 , i , n ( z ) ,
d S 2 , i , n ( z ) d z = k 2 ( III I ) h = 1 h , n S 1 , i h , n ( z ) k 2 i ( m i ) S 1 , i , n ( z ) k 2 ( III I ) h = 1 h , n * S 1 , i + h , n ( z ) + j 2 ( k 2 , n 2 k 1 2 sin 2 θ ) S 2 , i , n ( z ) .
[ S ˙ 1 , p , n S ˙ 2 , p , n ] = [ a p , q , n b p , q , n c p , q , n d p , q , n ] [ S 1 , q , n S 2 , q , n ] ,
S p , n ( z ) = q = 1 2 s C q , n w p , q , n exp ( λ q , n z ) ,
S i , n ( z ) = S p , n ( z ) ,
δ i 0 + R i = q = 1 2 s C q , 1 w p , q , 1 ,
j ( k 1 i · ) ( R i δ i 0 ) = q = 1 2 s C q , 1 w p , q , 1 [ λ q , 1 j ( σ i , 1 · ) ] ,
q = 1 2 s C q , n w p , q , n exp { [ λ q , n j ( σ i , n · ) ] n d / N } = q = 1 2 s C q , n + 1 w p , q , n + 1 exp { [ λ q , n + 1 j ( σ i , n + 1 · ) ] n d / N ,
q = 1 2 s C q , n w p , q , n [ λ q , n j ( σ i , n · ) ] × exp { [ λ q , n j ( σ i , n · ) ] n d / N } = q = 1 2 s C q , n + 1 w p , q , n + 1 [ λ q , n + 1 j ( σ i , n + 1 · ) ] × exp { [ λ q , n + 1 j ( σ i , n + 1 · ) ] n d / N } .
q = 1 2 s C q , N w p , q , N exp { [ λ q , N j ( σ i , N · ) ] d } = T i ,
q = 1 2 s C q , N w p , q , N [ λ q , N j ( σ i , N · ) ] × exp { [ λ q , N j ( σ i , N · ) ] d } = j ( k 3 i · ) T i .
DE 1 i = Re [ ( k 1 i · ) / ( k 10 · ) ] R i R i *
DE 3 i = Re [ ( k 3 i · ) / ( k 10 · ) ] T i T i * .
i ( DE 1 i + DE 3 i ) = 1 .
d Λ