Abstract

A rigorous solution is presented for plane-wave scattering by an asymmetric diffraction grating, which consists of a corrugated structure with three different groove depths per period. After establishing that asymmetry effects in the scattered amplitudes may be observed only if at least three spectral orders are present, we find that maximum asymmetry occurs when the angle of incidence closely satisfies a Bragg condition. For normal incidence, the peak asymmetry coincides with blazing, the degree of which may be quite significant. By employing simple phasing considerations, we derive a scheme for optimizing the grating design, and present examples illustrating that very high blazing efficiencies are easily attainable.

© 1973 Optical Society of America

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References

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  1. G. W. Stroke, in Handbuch der Physik, 29, edited by S. Flügge (Springer, Berlin, 1967), pp. 426–754.Refer also to the extensive bibliography therein.
    [Crossref]
  2. J. F. Carlson and A. E. Heins, Q. Appl. Math. 5, 82 (1947).
  3. A. Hessel and H. Hochstadt, Radio Sci. 3, 1019 (1968).
  4. D. Tseng, Ph.D. dissertation, Polytechnic Institute of Brooklyn, Brooklyn, N. Y. (1967) (University Microfilms, Ann Arbor, Mich., order No. 67-13425).
  5. R. J. Ikola, Ph.D. dissertation, Polytechnic Institute of Brooklyn, Brooklyn, N. Y. (1972) (University Microfilms, Ann Arbor, Mich., order No. 72-28243).
  6. R. E. Collin, Field Theory of Guided Waves (McGraw-Hill, New York, 1960), Sec. 5.7, pp. 209–214.
  7. Reference 6, Sec. 10.2, pp. 430–438.
  8. A. Wirgin and R. Deleuil, J. Opt. Soc. Am. 59, 1348 (1969).
    [Crossref]
  9. J. Shmoys and A. Hessel, J. Opt. Soc. Am. 62, 742 (1972).

1972 (1)

J. Shmoys and A. Hessel, J. Opt. Soc. Am. 62, 742 (1972).

1969 (1)

1968 (1)

A. Hessel and H. Hochstadt, Radio Sci. 3, 1019 (1968).

1947 (1)

J. F. Carlson and A. E. Heins, Q. Appl. Math. 5, 82 (1947).

Carlson, J. F.

J. F. Carlson and A. E. Heins, Q. Appl. Math. 5, 82 (1947).

Collin, R. E.

R. E. Collin, Field Theory of Guided Waves (McGraw-Hill, New York, 1960), Sec. 5.7, pp. 209–214.

Deleuil, R.

Heins, A. E.

J. F. Carlson and A. E. Heins, Q. Appl. Math. 5, 82 (1947).

Hessel, A.

J. Shmoys and A. Hessel, J. Opt. Soc. Am. 62, 742 (1972).

A. Hessel and H. Hochstadt, Radio Sci. 3, 1019 (1968).

Hochstadt, H.

A. Hessel and H. Hochstadt, Radio Sci. 3, 1019 (1968).

Ikola, R. J.

R. J. Ikola, Ph.D. dissertation, Polytechnic Institute of Brooklyn, Brooklyn, N. Y. (1972) (University Microfilms, Ann Arbor, Mich., order No. 72-28243).

Shmoys, J.

J. Shmoys and A. Hessel, J. Opt. Soc. Am. 62, 742 (1972).

Stroke, G. W.

G. W. Stroke, in Handbuch der Physik, 29, edited by S. Flügge (Springer, Berlin, 1967), pp. 426–754.Refer also to the extensive bibliography therein.
[Crossref]

Tseng, D.

D. Tseng, Ph.D. dissertation, Polytechnic Institute of Brooklyn, Brooklyn, N. Y. (1967) (University Microfilms, Ann Arbor, Mich., order No. 67-13425).

Wirgin, A.

J. Opt. Soc. Am. (2)

A. Wirgin and R. Deleuil, J. Opt. Soc. Am. 59, 1348 (1969).
[Crossref]

J. Shmoys and A. Hessel, J. Opt. Soc. Am. 62, 742 (1972).

Q. Appl. Math. (1)

J. F. Carlson and A. E. Heins, Q. Appl. Math. 5, 82 (1947).

Radio Sci. (1)

A. Hessel and H. Hochstadt, Radio Sci. 3, 1019 (1968).

Other (5)

D. Tseng, Ph.D. dissertation, Polytechnic Institute of Brooklyn, Brooklyn, N. Y. (1967) (University Microfilms, Ann Arbor, Mich., order No. 67-13425).

R. J. Ikola, Ph.D. dissertation, Polytechnic Institute of Brooklyn, Brooklyn, N. Y. (1972) (University Microfilms, Ann Arbor, Mich., order No. 72-28243).

R. E. Collin, Field Theory of Guided Waves (McGraw-Hill, New York, 1960), Sec. 5.7, pp. 209–214.

Reference 6, Sec. 10.2, pp. 430–438.

G. W. Stroke, in Handbuch der Physik, 29, edited by S. Flügge (Springer, Berlin, 1967), pp. 426–754.Refer also to the extensive bibliography therein.
[Crossref]

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Figures (14)

F. 1
F. 1

Geometry of the asymmetric grating.

F. 2
F. 2

Unit cell representing the electromagnetic problem of the grating in terms of a junction of four waveguides. Of these, region A indicates the air region, whereas regions 0, 1, and 2 denote three adjacent corrugations.

F. 3
F. 3

Equivalent representation of the unit cell in terms of transmission lines, each of which denotes a single propagating mode.

F. 4
F. 4

Intermediate steps for solving the scattering problem: (a) modified unit cell; (b) equivalent representation of the modified unit cell.

F. 5
F. 5

Relative powers P0, P−1, and P−2 as functions of the incidence angle θ>0. The curves shown were calculated for a grating with λ/d = 0.75, h1/a = 2.5, h2/a = 2.0, and h3/a = 1.5.

F. 6
F. 6

Relative powers P1 and P2 as functions of the incidence angle θ<0 for the same grating as in Fig. 5.

F. 7
F. 7

Relative powers P0, P−1, and P−2 as functions of the incidence angle. The curves shown are for a Symmetric grating with λ/d = 0.75, h1/a = h2/a = 2.0, and h2/a = 1.5.

F. 8
F. 8

Relative power P0, P−1, and P−2 as functions of the incidence angle. The curves shown are for a symmetric grating with λ/d = 0.75, h1/a = 2.5, and h2/a = h3/a = 2.0.

F. 9
F. 9

Relative power P0, P1, and P−1 as functions of normalized wavelength λ/a. The curves shown refer to normal incidence on a grating with h1/a = 2.5, h2/a = 2.0, and h3/a = 1.5.

F. 10
F. 10

Asymmetry ratio ξ1 as a function of λ/a for the same conditions as in Fig. 9.

F. 11
F. 11

Construction for deriving a constructive-interference argument for enhancing the nth spectral order. Note that L1 = 2ka sinϕn and L2 = ka sinϕn.

F. 12
F. 12

Relative powers P−1, P0, and P1 as functions of λ/a for a grating with corrugation depths optimized to produce a peak for P−1 at λ/a = 2.4: (a) choice of m = m1 = m2 = 0 and h3/a = 0.5 yields h1/a = 1.3 and h2/a = 0.9; (b) choice of m = 1 and h3/a = 0.5 yields h1/a = 2.5 and h2/a = 2.1.

F. 13
F. 13

Relative powers P0, P−1, and P−2 as functions of the incidence angle θ>0. The curves shown are for a grating with λ/d = 0.8, h1/a = 2.2035, h2/a = 1.4065, and h3/a = 0.6035.

F. 14
F. 14

Relative powers P0, P1, and P2 as functions of the incidence angle θ <0, for the same grating as in Fig. 13.

Equations (22)

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( 2 + k 2 ) H y = 0 ,
E z = 0 at z > 0 , x = n a ( n = 0 , ± 1 , ± 2 ) .
E z ( 3 a , z ) = E z ( 0 , z ) exp ( j 3 k s a ) ( < z < ) , E y ( 3 a , z ) = H y ( 0 , z ) exp ( j 3 k s a ) ( z < 0 ) ,
E x = 0 at z = h 1 , h 2 , or h 3 as appropriate .
H y ( x , z ) = H i exp ( j k s x Γ 0 z ) + n = I n exp [ j ( k s + 2 n π 3 a ) x + Γ n ( z ) ] .
Γ n = [ ( k s + 2 n π 3 a ) 2 k 2 ] 1 2 ,
Γ n > 0 for ( k s + 2 n π 3 a ) 2 > k 2 , Im ( Γ n ) > 0 for ( k s + 2 n π 3 a ) 2 < k 2 .
H y m ( x , z ) = n = 0 [ J m n exp ( γ n z ) + J m n exp ( γ n z ) cos n π ( x m a ) a ] for z > 0 ,
γ n = [ ( n π a ) 2 k 2 ] 1 2
γ n > 0 for ( n π a ) 2 > k 2 , Im ( γ n ) > 0 for ( n π a ) 2 < k 2 .
b = S a ,
[ b A b B ] = [ S A A S B A | S A B S B B ] [ a A a B ] ,
a B = R B b B ,
R B = [ exp ( 2 γ 0 h 1 ) 0 0 0 exp ( 2 γ 0 h 2 ) 0 0 0 exp ( 2 γ 0 h 3 ) ] .
b A = { S A A + S A B R B [ 1 S B B R B ] 1 S B A } a A ,
I 0 ( + θ ) = I 0 ( θ ) .
ξ n = 10 log P + n ( θ ) P n ( + θ ) dB ,
2 k h 1 + 2 k a sin ϕ n = 2 m 1 π + 2 k a sin θ + 2 k h 3
k a sin θ + 2 k h 2 + k a sin ϕ n = 2 m 2 π + 2 k a sin θ + 2 k h 3 ,
sin ϕ n = sin θ + n λ d , n = 0 , ± 1 , ± 2 ,
h 1 a = ( 3 m 1 2 n 6 ) λ a + h 3 a ,
h 2 a = ( 3 m 2 n 6 ) λ a + h 3 a .