Abstract

The theory of electromagnetic Bloch waves in periodic stratified media is applied to the problems of birefringence and group velocity in these media. The relevance of periodic media to phase matching in nonlinear mixing experiments and to laser action in the x-ray region is discussed.

© 1977 Optical Society of America

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References

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  1. P. Yen, A. Yariv, and C. S. Hong, “Electromagnetic propagation in periodic stratified media. I. General theory,” J. Opt. Soc. Am. 67, 423–438 (1977) (preceding paper).
    [Crossref]
  2. See, for example, A. Yariv, Quantum Electronics, 2nd. ed. (Wiley, New York, 1975), p. 85.
  3. S. M. Rytov, Sov. Phys. -JETP 2, 446 (1956).
  4. A. Ashkin and A. Yariv, Bell Labs. Tech. Memo No. MM-61-124-46 (13November1961) (unpublished).
  5. N. Bloembergen and A. J. Sievers, Appl. Phys. Lett. 17, 483 (1970).
    [Crossref]
  6. C. L. Tang and P. P. Bey, IEEE J. Quant. Electron. QE-9, 9 (1973).
    [Crossref]
  7. J. P. van der Ziel and M. Ilegems, Appl. Phys. Lett. 28, 437 (1976).
    [Crossref]
  8. A. Yariv, Appl. Phys. Lett. 25, 105 (1974).
    [Crossref]
  9. B. W. Batterman, Rev. Mod. Phys. 36, 681 (1964).
    [Crossref]

1977 (1)

1976 (1)

J. P. van der Ziel and M. Ilegems, Appl. Phys. Lett. 28, 437 (1976).
[Crossref]

1974 (1)

A. Yariv, Appl. Phys. Lett. 25, 105 (1974).
[Crossref]

1973 (1)

C. L. Tang and P. P. Bey, IEEE J. Quant. Electron. QE-9, 9 (1973).
[Crossref]

1970 (1)

N. Bloembergen and A. J. Sievers, Appl. Phys. Lett. 17, 483 (1970).
[Crossref]

1964 (1)

B. W. Batterman, Rev. Mod. Phys. 36, 681 (1964).
[Crossref]

1956 (1)

S. M. Rytov, Sov. Phys. -JETP 2, 446 (1956).

Ashkin, A.

A. Ashkin and A. Yariv, Bell Labs. Tech. Memo No. MM-61-124-46 (13November1961) (unpublished).

Batterman, B. W.

B. W. Batterman, Rev. Mod. Phys. 36, 681 (1964).
[Crossref]

Bey, P. P.

C. L. Tang and P. P. Bey, IEEE J. Quant. Electron. QE-9, 9 (1973).
[Crossref]

Bloembergen, N.

N. Bloembergen and A. J. Sievers, Appl. Phys. Lett. 17, 483 (1970).
[Crossref]

Hong, C. S.

Ilegems, M.

J. P. van der Ziel and M. Ilegems, Appl. Phys. Lett. 28, 437 (1976).
[Crossref]

Rytov, S. M.

S. M. Rytov, Sov. Phys. -JETP 2, 446 (1956).

Sievers, A. J.

N. Bloembergen and A. J. Sievers, Appl. Phys. Lett. 17, 483 (1970).
[Crossref]

Tang, C. L.

C. L. Tang and P. P. Bey, IEEE J. Quant. Electron. QE-9, 9 (1973).
[Crossref]

van der Ziel, J. P.

J. P. van der Ziel and M. Ilegems, Appl. Phys. Lett. 28, 437 (1976).
[Crossref]

Yariv, A.

P. Yen, A. Yariv, and C. S. Hong, “Electromagnetic propagation in periodic stratified media. I. General theory,” J. Opt. Soc. Am. 67, 423–438 (1977) (preceding paper).
[Crossref]

A. Yariv, Appl. Phys. Lett. 25, 105 (1974).
[Crossref]

See, for example, A. Yariv, Quantum Electronics, 2nd. ed. (Wiley, New York, 1975), p. 85.

A. Ashkin and A. Yariv, Bell Labs. Tech. Memo No. MM-61-124-46 (13November1961) (unpublished).

Yen, P.

Appl. Phys. Lett. (3)

N. Bloembergen and A. J. Sievers, Appl. Phys. Lett. 17, 483 (1970).
[Crossref]

J. P. van der Ziel and M. Ilegems, Appl. Phys. Lett. 28, 437 (1976).
[Crossref]

A. Yariv, Appl. Phys. Lett. 25, 105 (1974).
[Crossref]

IEEE J. Quant. Electron. (1)

C. L. Tang and P. P. Bey, IEEE J. Quant. Electron. QE-9, 9 (1973).
[Crossref]

J. Opt. Soc. Am. (1)

Rev. Mod. Phys. (1)

B. W. Batterman, Rev. Mod. Phys. 36, 681 (1964).
[Crossref]

Sov. Phys. -JETP (1)

S. M. Rytov, Sov. Phys. -JETP 2, 446 (1956).

Other (2)

A. Ashkin and A. Yariv, Bell Labs. Tech. Memo No. MM-61-124-46 (13November1961) (unpublished).

See, for example, A. Yariv, Quantum Electronics, 2nd. ed. (Wiley, New York, 1975), p. 85.

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Figures (9)

FIG. 1
FIG. 1

Contours of constant frequency in β-K plane.

FIG. 2
FIG. 2

Section of normal surface in kxkz plane.

FIG. 3
FIG. 3

Double refraction at the boundary of a periodic stratified medium.

FIG. 4
FIG. 4

Typical periodicity dispersion neff vs ω.

FIG. 5
FIG. 5

Typical overall dispersion neff vs ω.

FIG. 6
FIG. 6

Contours of equal reflectivity in ωα2, plane.

FIG. 7
FIG. 7

Field distribution near oscillation. The dashed arrows indicate incident and reflected waves, respectively. The solid arrow at the right-hand side indicates the transmitted wave. The inset in the upper part is the gain-loss profile.

FIG. 8
FIG. 8

Periodicity dispersion when n1a = n2b (or ν = 0).

FIG 9
FIG 9

Graphic method of finding band edges and gap sizes.

Equations (119)

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E ( x , z , t ) = E K ( x ) e i K x e i β z e i ω t ,
cos K Λ = 1 2 ( A + D ) = cos k 1 x a cos k 2 x b Δ sin k 1 x a sin k 2 x b ,
Δ = { 1 2 ( k 2 x / k 1 x + k 1 x / k 2 x ) , TE waves , 1 2 ( n 2 2 k 1 x / n 1 2 k 2 x + n 1 2 k 2 x / n 2 2 k 1 x ) , TM waves ;
k 1 x = { [ ( ω / c ) n 1 ] 2 β 2 } 1 / 2 ,
k 2 x = { [ ( ω / c ) n 2 ] 2 β 2 } 1 / 2 .
E K ( x ) = n e K ( n ) e i n ( 2 π / Λ ) x ,
E ( x , z , t ) = n e K ( n ) e i [ K + n ( 2 π / Λ ) ] x e i β z e i ω t ,
cos K Λ = cos [ k x ( a + b ) ] = cos k x Λ ,
| e K ( 0 ) | | e K ( n ) |
1 Λ 0 Λ E K ( x ) d x E K
V p = ω / ( K 2 + β 2 ) 1 / 2 .
E ( x , z , t ) = E K e i K x e i β z e i ω t .
V g = ( ω K ) β â x + ( ω β ) K â z .
V e = ( 1 Λ 0 Λ ( Poynting vector ) d x ) × ( 1 Λ 0 Λ ( energy density ) d x ) 1
E ( r , t ) = E e i [ ( ω / c ) n s · r ω t ]
S x 2 n 2 x / 0 + S y 2 n 2 y / 0 + S z 2 n 2 z / 0 = 1 n 2 ,
E i = n 2 S i ( S · E ) n 2 i / 0 , i = x , y , z ,
k x 2 k 2 ( ω 2 / c 2 ) x / 0 + k y 2 k 2 ( ω 2 / c 2 ) y / 0 + k z 2 k 2 ( ω 2 / c 2 ) z / 0 = 1 ,
k 2 = k x 2 + k y 2 + k z 2 .
× ( × E ) + μ ¯ ¯ 2 t 2 E = 0 .
k × ( k × E ) = μ ¯ ¯ 2 t 2 E ,
( ω 2 μ x k y 2 k z 2 k x k y k x k z k y k x ω 2 μ y k x 2 k z 2 k y k z k z k x k z k y ω 2 μ z k x 2 k y 2 ) × ( E x E y E z ) = 0 .
det | ω 2 μ x k y 2 k z 2 k x k y k x k z k y k x ω 2 μ y k z 2 k x 2 k y k z k z k x k z k y ω 2 μ z k x 2 k y 2 | = 0 .
k x 2 n o 2 + k y 2 + k z 2 n e 2 = ω 2 c 2 ,
k x 2 n o 2 + k y 2 + k z 2 n o 2 = ω 2 c 2 ,
n e 2 = x / 0 , n o 2 = y / 0 .
S = E × H
V g = k ω ( k ) .
k × ( k × E ) + ω 2 μ ¯ ¯ E = 0 .
δ k ( k · E ) + k ( δ k · E ) + k ( k · δ E ) δ E ( k · k ) 2 E ( k · δ k ) + ω 2 μ ¯ ¯ δ E = 0
2 δ k · [ k ( E · E ) E ( k · E ) ] + δ E · [ k ( E · k ) E ( k · k ) + ω 2 μ ¯ ¯ E ] = 0 .
δ k · ( E × H ) = 0 ,
κ = â x K + â z β .
K 2 / n o 2 + β 2 / n o 2 = ω 2 / c 2 , TE ,
K 2 / n o 2 + β 2 / n e 2 = ω 2 / c 2 , TM ,
n o 2 = ( a / Λ ) n 1 2 + ( b / Λ ) n 2 2 ,
1 / n e 2 = ( a / Λ ) · ( 1 / n 1 2 ) + ( 1 / Λ ) · ( 1 / n 2 2 ) .
K = m ( π / Λ ) , m = integer ,
β = k 0 sin θ .
tan θ TE = β / K TE ,
tan θ TM = β / K TM .
E ( ω l ) ( x , z , t ) = E K l ( x ) e i K l x e i β l z e i ω l t , l = 1 , 2 , 3 ,
ω 3 = ω 1 + ω 2 .
P i ( ω 3 ) ( x , z ) = d i j k E j ( ω 1 ) ( x , z ) E k ( ω 2 ) ( x , z ) .
d i j k ω 1 + ω 2 ω 3 ( x + Λ ) = d i j k ω 1 + ω 2 ω 3 ( x ) .
K 3 | d | K 1 K 2 = d i j k ( x ) E i K 1 ( x ) E j K 2 ( x ) ( E k K 3 ( x ) ) * × e i ( K 1 + K 2 K 3 ) x e i ( β 1 + β 2 β 3 ) d x d z ,
d i j k ( x ) = m D i j k m e i m ( 2 π / Λ ) x ,
E i K 1 ( x ) = n A i n e i n ( 2 π / Λ ) x ,
E j K 2 ( x ) = l B j e i l ( 2 π / Λ ) x ,
E k K 3 ( x ) = p C k p e i p ( 2 π / Λ ) x .
K 3 | d | K 1 K 2 = ( 2 π ) 2 m , n , l , p D i j k m A i n B j l ( C k p ) * × δ ( K 1 + K 2 K 3 + ( m + n + l p ) 2 π Λ ) δ ( β 1 + β 2 β 3 ) .
β 3 = β 1 + β 2 ,
K 3 = K 1 + K 2 + s ( 2 π / Λ ) , s = m + n + l p .
n eff ( ω ) = c K ( ω ) / ω .
n ¯ = ( n 1 a + n 2 b ) / Λ .
ω u , l ( c / n ¯ Λ ) { ( 2 l + 1 ) π ± 2 [ ( Δ 1 ) / ( Δ + 1 ) ] 1 / 2 } , Δ 1 , ( ν / n ¯ ) ( l + 1 2 ) π 1 .
Δ n 1 / 2 = n eff ( ω l ) n ¯ = c K ( ω l ) / ω l n ¯ .
K ( ω l ) Λ = ( 2 l + 1 ) π ,
Δ n 1 / 2 = c ( 2 l + 1 ) π / ω l Λ n ¯ .
Δ n 1 / 2 = [ 2 n ¯ / ( 2 l + 1 ) π ] [ ( Δ 1 ) ( Δ + 1 ) ] 1 / 2 .
Δ ω 1 / 2 1 2 ( ω u ω l ) ( 2 c / n ¯ Λ ) [ ( Δ 1 ) / ( Δ + 1 ) ] 1 / 2 .
n o n e = ( 4 a b / Λ 2 ) [ n e 2 / ( n o + n e ) ] ( Δ 2 1 ) .
r N = C U N 1 / ( A U N 1 U N 2 ) ,
t N = ( A U N 1 U N 2 ) 1 .
n ̂ 1 = n 1 + i κ 1 ,
n ̂ 2 = n 2 + i κ 2 .
α 1 , 2 = 2 κ 1 , 2 ω / c .
J = α ( x ) E 2 ( x ) d x ,
α ( x ) = { α 1 > 0 , layer 1 , α 2 < 0 , layer 2 .
α ( x ) d x > 0 .
α L < 0 .
S = 1 2 Re [ E × H * ] .
U = 1 4 ( | E | 2 + μ | H | 2 ) .
U = 1 Λ 0 Λ U ( x ) d x ,
S = 1 Λ 0 Λ S ( x ) d x .
V e = S / U ,
V g = ( ω K ) β â x + ( ω β ) K â z .
V g = V e .
E y ( x , z ) = E K ( x ) e i K x e i β z
= { ( a 0 e i k 1 x x + b 0 e i k 1 x x ) e i β z e i ω t , a < x < 0 , ( c 0 e i k 2 x x + d 0 e i k 2 x x ) e i β z e i ω t , Λ < x < a .
H x ( x , z ) = ( β / ω μ ) E y ( x , z ) ,
H z ( x , z ) = { ( k 1 x / ω μ ) ( a 0 e i k 1 x x b 0 e i k 1 x x ) × e i β z , a < x < 0 , ( k 2 x / ω μ ) ( c 0 e i k 2 x x d 0 e i k 2 x x ) × e i β z , Λ < x < a ,
( a 0 b 0 ) = ( B e i K Λ A ) ,
( c 0 e i k 2 x a d 0 e i k 2 x a ) = ( 1 2 ( 1 + k 1 x / k 2 x ) 1 2 ( 1 k 1 x / k 2 x ) 1 2 ( 1 k 1 x / k 2 x ) 1 2 ( 1 + k 1 x / k 2 x ) ) × ( a 0 e i k 1 x a b 0 e i k 1 x a ) .
U = 1 2 [ a Λ 1 ( | a 0 | 2 + | b 0 | 2 ) + b Λ 2 ( | c 0 | 2 + | d 0 | 2 ) + B + C Λ · β 2 ω 2 μ k 1 x ( 1 k 1 x 2 1 k 2 x 2 ) ( sin K Λ i A D 2 ) ] .
S x = S x = 1 2 k 1 x ω μ ( | a 0 | 2 | b 0 | 2 ) = 1 2 k 2 x ω μ ( | c 0 | 2 | d 0 | 2 ) ,
S z = 1 2 β ω μ [ a Λ ( | a 0 | 2 + | b 0 | 2 ) + b Λ ( | c 0 | 2 + | d 0 | 2 ) + B + C Λ · k 1 x ( 1 k 1 x 2 1 k 2 x 2 ) ( sin K Λ i A D 2 ) ] .
V g = [ ( F K ) ω , β / ( F ω ) K , β ] â x [ ( F β ) ω , K / ( F ω ) K , β ] â z ,
F ( ω , β , K ) = cos K Λ 1 2 ( A + D ) .
V g x = Λ sin K Λ / [ i ( A D 2 ) a n 1 2 ω k 1 x c 2 + i ( Ã D 2 ) b n 2 2 ω k 2 x c 2 B + C 2 ( n 1 2 ω k 1 x 2 c 2 n 2 2 ω k 2 x 2 c 2 ) ] ,
V g z = [ i ( A D 2 ) a β k 1 x + i ( Ã D 2 ) b β k 2 x B + C 2 β ( 1 k 1 x 2 1 k 2 x 2 ) ] × [ i ( A D 2 ) a n 1 2 ω k 1 x c 2 + i ( Ã D 2 ) b n 2 2 ω k 2 x c 2 B + C 2 ( n 1 2 ω k 1 x 2 c 2 n 2 2 ω k 2 x 2 c 2 ) ] 1 ,
à = e i k 2 x b [ cos k 1 x a 1 2 i ( k 2 x k 1 x + k 1 x k 2 x ) sin k 1 x a ] ,
D = e i k 2 x b [ cos k 1 x a + 1 2 i ( k 2 x k 1 x + k 1 x k 2 x ) sin k 1 x a ] .
V e x = [ a Λ ω μ 1 k 1 x | a 0 | 2 + | b 0 | 2 | a 0 | 2 | b 0 | 2 + b Λ ω μ 2 k 2 x | c 0 | 2 + | d 0 | 2 | c 0 | 2 | d 0 | 2 + B + C Λ β 2 ω ( 1 k 1 x 2 1 k 2 x 2 ) sin K Λ i ( A D ) / 2 | a 0 | 2 | b 0 | 2 ] 1 ,
V e z = β ω μ [ a Λ ( | a 0 | 2 + | b 0 | 2 ) + b Λ ( | c 0 | 2 + | d 0 | 2 ) + B + C Λ k 1 x ( 1 k 1 x 2 1 k 2 x 2 ) ( sin K Λ i A D 2 ) ] × [ a 1 Λ ( | a 0 | 2 + | b 0 | 2 ) + b 2 Λ ( | c 0 | 2 + | d 0 | 2 ) + B + C Λ · β 2 ω 2 μ k 1 x ( 1 k 1 x 2 1 k 2 x 2 ) ( sin K Λ i A D 2 ) ] 1 .
X ( | a 0 | 2 + | b 0 | 2 ) / ( | a 0 | 2 | b 0 | 2 ) ,
Y ( | c 0 | 2 + | d 0 | 2 ) / ( | c 0 | 2 | d 0 | 2 ) ,
Z [ sin K Λ i ( A D ) / 2 ] / ( | a 0 | 2 | b 0 | 2 ) ,
V e x = [ a Λ · ω μ 1 k 1 x X + b Λ ω μ 2 k 2 x Y + B + C Λ · β 2 ω ( 1 k 1 x 2 1 k 2 x 2 ) Z ] 1 ,
V e z = [ a Λ · β k 1 x X + b Λ · β k 2 x Y + B + C Λ ( 1 k 1 x 2 1 k 2 x 2 ) Z ] × [ a Λ · ω μ 1 k 1 x X + b Λ · ω μ 2 k 2 x Y + B + C Λ · β 2 ω ( 1 k 1 x 2 1 k 2 x 2 ) Z ] 1 .
X = i [ ( A D ) / 2 ] sin K Λ ,
Y = i [ ( Ã D ) / 2 ] sin K Λ ,
Z = ( 2 sin K Λ ) 1 .
Δ = 1 2 ( n 2 / n 1 + n 1 / n 2 ) ,
n ¯ = ( n 1 a + n 2 b ) / Λ ,
ν ( n 1 a n 2 b ) / Λ .
cos K Λ = ( Δ + 1 2 ) cos n ¯ ω c Λ ( Δ 1 2 ) cos ν ω c Λ .
cos x = 1 2 sin 2 x / 2 ,
sin 2 K Λ 2 = ( Δ + 1 2 ) sin 2 n ¯ ω 2 c Λ ( Δ 1 2 ) sin 2 ν ω 2 c Λ .
sin 2 ( n ¯ ω 2 c Λ ) = { ( Δ 1 Δ + 1 ) sin 2 ( ν ω 2 c Λ ) , K Λ = 2 l π , 1 ( Δ 1 Δ + 1 ) cos 2 ( ν ω 2 c Λ ) , K Λ = ( 2 l + 1 ) π ,
ω u , l { c n ¯ Λ { 2 l π ± 2 sin 1 [ ( Δ 1 Δ + 1 ) 1 / 2 sin ( ν n ¯ l π ) ] } , K Λ = 2 l π , c n ¯ Λ { ( 2 l + 1 ) π ± 2 sin 1 [ ( Δ 1 Δ + 1 ) 1 / 2 cos ( ν n ¯ ( l + 1 2 ) π ) ] } , K Λ = ( 2 l + 1 ) π .
Δ ω max = 4 c n ¯ Λ sin 1 ( Δ 1 Δ + 1 ) 1 / 2 ,
ω = ( c / n ¯ ) K .
Δ ω gap = { 4 c n ¯ Λ sin 1 ( Δ 1 Δ + 1 ) 1 / 2 sin ( ν n ¯ l π ) , K Λ = 2 l π , 4 c n ¯ Λ sin 1 ( Δ 1 Δ + 1 ) 1 / 2 cos ( ν n ¯ ( l + 1 2 ) π ) , K Λ = ( 2 l + 1 ) π .
Δ = { 1 2 ( n 2 cos θ 2 n 1 cos θ 1 + n 1 cos θ 1 n 2 cos θ 2 ) , TE waves , 1 2 ( n 1 cos θ 2 n 2 cos θ 1 + n 2 cos θ 1 n 1 cos θ 2 ) , TM waves ,
n ¯ = ( n 1 a cos θ 1 + n 2 b cos θ 2 ) / Λ ,
ν = ( n 1 a cos θ 1 n 2 b cos θ 2 ) / Λ ,
cos θ 1 = c k 1 x / n 1 ω ,
cos θ 2 = c k 2 x / n 2 ω .