Abstract

A beam incident upon a slightly lossy layered dielectric structure at special (phase-matched) angles displays abnormal absorption and lateral beam shift on reflection. The amplitude of the reflected beam can be calculated either by a conventional transform technique or by a quasi-particle method. Different wave-number (k) expansions of the reflection coefficient may be employed for calculation of reflected beam properties. The convergence of these expansions is limited by the presence of poles and zeros and by the rapidity of dispersion in k of the reflection coefficient. The series expansions used in the quasi-particle method are not limited by the zeros and converge better for far fields than for fields on the reflecting surface; moreover, they permit faster numerical calculation of beam properties.

© 1983 Optical Society of America

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References

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  1. T. Tamir and H. L. Bertoni, “Lateral displacement of optical beams at multilayered and periodic structures,” J. Opt. Soc. Am. 61, 1397–1413 (1971).
    [Crossref]
  2. V. Shah and T. Tamir, “Absorption and lateral shift of beams incident upon lossy multilayered media,” J. Opt. Soc. Am. 73, 37–44 (1983).
    [Crossref]
  3. N. Marcuvitz, “Quasi-particle view of wave propagation,” Proc. IEEE 68, 1380–1395 (1980).
    [Crossref]

1983 (1)

1980 (1)

N. Marcuvitz, “Quasi-particle view of wave propagation,” Proc. IEEE 68, 1380–1395 (1980).
[Crossref]

1971 (1)

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Figures (5)

Fig. 1
Fig. 1

Single dielectric slab layer with metal substrate. d = 5λ0 and η 1 = 3 ( 1 + i ν 1 ).

Fig. 2
Fig. 2

Reflected beam amplitude squared, calculated at z = 0 and k0 = 200 by FFT (solid line), by quasi-particle methods with second-order (dotted line) and third-order (dashed line) reflection coefficient expansion of Eq. (3a), and by incident beam amplitude squared (dotted line). a(x, 0) = exp[−x2 + nx]; ν1 = 0, βp = 154.99.

Fig. 3
Fig. 3

Reflected beam amplitude squared, calculated at z = 0 and k0= 200 by FFT (solid line) and by quasi-particle methods with second-order (dotted line) and third-order (dashed line) reflection coefficient expansion of Eq. (3a). a(x, 0) = exp[−x2 + px]; ν1 = 0.007 (critical absorption); βp = 154.56.

Fig. 4
Fig. 4

Reflected beam amplitude squared versus x + z tan θ calculated by FFT (solid line) and by quasi-particle methods (dotted line) with second-order reflection coefficient expansion of Eq. (3a): a(x, 0) = exp(−x2 + px); ν1 = 0.

Fig. 5
Fig. 5

Reflected beam amplitude squared versus x + z tan θ calculated by FFT (solid line) and by quasi-particle methods (dotted line) with second-order reflection coefficient expansion of Eq. (3a): a(x, 0) = exp(−x2 + px); ν1 =0.007.

Equations (26)

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b ( x , 0 ) = a k ( 0 ) Γ ( k ) exp ( i k x ) d k / 2 π = a ( x , 0 ) Γ ( k ) exp [ i k ( x - x ) ] d k d x / 2 π ,
B ( x , 0 ) = [ Γ ( k x i ) - i d Γ ( k x i ) d k x - 1 2 d 2 Γ ( k x i ) d k 2 2 x 2 + ] A ( x , 0 )
B ( x , 0 ) = Γ ( k x i ) [ 1 - i [ d d k ln Γ ( k x i ) ] r x - 1 2 ( d 2 d k 2 ln Γ ( k x i ) + { [ d d k ln Γ ( k x i ) ] r } 2 ) 2 x 2 + ] A { x + [ d d k ln Γ ( k x i ) ] i , 0 } ,
a ( x , 0 ) = A ( x , 0 ) exp ( i k x i x ) , b ( x , 0 ) = B ( x , 0 ) exp ( i k x i x ) ,
F a z + k ( k 0 2 - k 2 ) 1 / 2 F a x = 0 , F b z - k ( k 0 2 - k 2 ) 1 / 2 F b x = 0 ,
F a ( k , x , 0 ) = a ( x + ξ / 2 , 0 ) a * ( x - ξ / 2 , 0 ) exp ( - i k ξ ) d ξ
F b ( k , x , 0 ) = b ( x + ξ / 2 , 0 ) b * ( x - ξ / 2 , 0 ) exp ( - i k ξ ) d ξ ,
F b ( k , x , 0 ) = Γ ( k - i 1 2 x ) Γ * ( k + i 1 2 x ) F a ( k , x , 0 ) ,
F b ( k , x , 0 ) = [ Γ ( k ) 2 + ( Γ * d Γ d k ) i x - 1 4 ( Γ * d 2 Γ d k 2 - | d Γ d k | 2 ) r 2 x 2 - ] F a ( k , x , 0 )
F b ( k , x , 0 ) = Γ 2 { 1 - 1 4 [ Γ Γ - ( Γ Γ ) 2 ] r 2 x 2 + } F a [ k , x + ( Γ / Γ ) i , 0 ] ,
a ( x , 0 ) = exp ( - x 2 / 2 h 2 + i k x i x )
F a ( k , x , 0 ) = 4 π h exp [ - x 2 / h 2 - ( k - k x i ) 2 h 2 ] ,
F b ( k , x , 0 ) 4 π h 2 H Γ 2 × exp { - [ x + ( Γ / Γ ) i ] 2 H 2 - ( k - k x i ) 2 h 2 } ,
H = { h 2 - [ Γ / Γ - ( Γ / Γ ) 2 ] r } 1 / 2
Γ ( k ) Γ 0 N - exp { 2 i k 0 d [ - ( k / k 0 ) 2 ] 1 / 2 } P - exp { 2 i k 0 d [ - ( k / k 0 ) 2 ] 1 / 2 } ,
P = exp { 2 i k 0 d [ - ( k p / k 0 ) 2 ] 1 / 2 } , N = exp { 2 i k 0 d [ - ( k n / k 0 ) 2 ] 1 / 2 } ,
d m Γ / d k m | α p - α n α n Γ α p m | | α p 1 - m d Γ d k | ,
Γ ( k ) Γ 0 [ 1 + i ( M n - M p ) k 0 η 2 β d × 1 ( 1 + 2 M p ) 1 / 2 ( 1 k - β + i C - 1 k - β + i D ) ] ,
η = [ - ( β / k 0 ) 2 ] 1 / 2 , C = k 0 η 2 d β [ 1 + ( 1 + 2 M p ) 1 / 2 ] , D = k 0 η 2 d β [ 1 - ( 1 + 2 M p ) 1 / 2 ] , M p = exp ( 2 i k 0 d { [ - ( k p / k 0 ) 2 ] 1 / 2 - η } ) - 1 , M n = exp ( 2 i k 0 d { [ - ( k n / k 0 ) 2 ] 1 / 2 - η } ) - 1 ,
Γ 0 = M p M n Γ ( β )             if Γ ( β ) 0 , = - i k 0 η 2 d β d Γ ( β ) d k M p             if Γ ( β ) = 0 ,
B ( x , z ) = Γ ( k ) a k ( 0 ) exp { i [ ( k - k x i ) x - k z z ] } d k / 2 π = n = 0 R n ( k x i ) ( - i x ) n A ( x , - z )
F b ( k , x , z ) = Γ 1 Γ 2 * a k 1 ( 0 ) a k 2 * ( 0 ) × exp { - i [ k z ( k 1 ) - k z ( k 2 ) ] z } exp [ i κ x ] d κ / 2 π = n = 0 S n ( k ) ( - i x ) n F a ( k , x , - z ) ,
F b ( k , x , - z ) = F b ( k , x - k k z z , 0 ) .
b ( x , z ) 2 n = 0 h H ( z ) S n ( k x i ) ( - i x ) n exp [ - ( x - k z z ) 2 H 2 ( z ) ] ,
k z = ( k 0 2 - k 2 ) 1 / 2 k z = d k z / d k , k z = d 2 k z / d k 2 ,
H ( z ) = [ h 2 + ( k z z / h ) 2 ] 1 / 2 .