Abstract

By using an angular spectral representation, we show that the fields of Gaussian beams scattered by reflection gratings differ markedly from those predicted by geometrical considerations. We find that, in general, each diffracted beam exhibits a lateral displacement, a focal shift, and an angular deflection; in addition, the size of the beam width is enlarged or reduced. The beam changes are largest if the incidence angle is phase matched to a leaky wave that may be supported by the grating. This phase condition is identical to that for which Wood’s anomalies of the resonant variety occur if plane waves, instead of bounded beams, are incident. By evaluating the spatial modifications of beams diffracted at a canonic grating structure consisting of a sinusoidal reactance plane, we show that the magnitudes of the beam effects can be considerably large. We also examine the special case of blazed diffracted orders and find that their corresponding beams are not extinguished completely but appear with reduced intensity and strong profile distortion.

© 1989 Optical Society of America

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References

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  1. H. K. V. Lotsch, “Beam displacement at total reflection: the Goos–Hänchen effect,” Optik 32, 116–137, 189–204 (1970); Optik 32, 299–319, 553–569 (1971).
  2. T. Tamir, H. L. Bertoni, “Lateral displacement of optical beams at multilayered and periodic structures,”J. Opt. Soc. Am. 61, 1397–1413 (1971).
    [CrossRef]
  3. V. Shah, T. Tamir, “Absorption and lateral shift of beams incident upon lossy multilayered media,”J. Opt. Soc. Am. 73, 37–44 (1983).
    [CrossRef]
  4. C. W. Hsue, T. Tamir, “Lateral beam displacement and distortion of beams incident upon a transmitting layer configuration,” J. Opt. Soc. Am. A 2, 978–988 (1985).
    [CrossRef]
  5. R. P. Riesz, R. Simon, “Reflection of a Gaussian beam from a dielectric slab,” J. Opt. Soc. Am. A 2, 1809–1817 (1985).
    [CrossRef]
  6. S. L. Chuang, “Lateral shift of an optical beam due to leaky surface-plasmon excitations,” J. Opt. Soc. Am. A 3, 593–599 (1986).
    [CrossRef]
  7. T. Tamir, “Nonspecular phenomena in beam fields reflected by multilayered media,” J. Opt. Soc. Am. A 3, 558–565 (1986).
    [CrossRef]
  8. M. A. Breazeale, M. A. Torbett, “Backward displacement of waves reflected from an interface having superimposed periodicity,” Appl. Phys. Lett. 29, 456–458 (1976).
    [CrossRef]
  9. A. Hessel, A. A. Oliner, “A new theory of Wood’s anomalies on optical gratings,” Appl. Opt. 4, 1275–1297 (1965).
    [CrossRef]
  10. N. W. MacLachlan, Theory and Applications of Mathieu Functions (Oxford U. Press, Oxford, 1951).
  11. T. Tamir, Integrated Optics (Springer-Verlag, Berlin, 1982), Sec. 3.1.4, p. 98.
  12. See Ref. 11, Sec. 3.1.6, p. 110.
  13. C. C. Chan, T. Tamir, “Angular shift of a Gaussian beam reflected near the Brewster angle,” Opt. Lett. 10, 378–380 (1985).
    [CrossRef] [PubMed]
  14. H. L. Bertoni, C. W. Hsue, T. Tamir, “Non-specular reflection of convergent beams from liquid–solid interface,” Trait. Signal 2, 201–205 (1985).
  15. P. B. Nagy, K. Cho, L. Adler, D. E. Chimenti, “Focal shift of convergent ultrasonic beams reflected from a liquid–solid interface,”J. Acoust. Soc. Am. 81, 835–839 (1987).
    [CrossRef]
  16. R. Simon, T. Tamir, “Nonspecular phenomena in partly coherent beams reflected by multilayered structures,” J. Opt. Soc. Am. A 6, 18–22 (1989).
    [CrossRef]

1989 (1)

1987 (1)

P. B. Nagy, K. Cho, L. Adler, D. E. Chimenti, “Focal shift of convergent ultrasonic beams reflected from a liquid–solid interface,”J. Acoust. Soc. Am. 81, 835–839 (1987).
[CrossRef]

1986 (2)

1985 (4)

1983 (1)

1976 (1)

M. A. Breazeale, M. A. Torbett, “Backward displacement of waves reflected from an interface having superimposed periodicity,” Appl. Phys. Lett. 29, 456–458 (1976).
[CrossRef]

1971 (1)

1970 (1)

H. K. V. Lotsch, “Beam displacement at total reflection: the Goos–Hänchen effect,” Optik 32, 116–137, 189–204 (1970); Optik 32, 299–319, 553–569 (1971).

1965 (1)

Adler, L.

P. B. Nagy, K. Cho, L. Adler, D. E. Chimenti, “Focal shift of convergent ultrasonic beams reflected from a liquid–solid interface,”J. Acoust. Soc. Am. 81, 835–839 (1987).
[CrossRef]

Bertoni, H. L.

H. L. Bertoni, C. W. Hsue, T. Tamir, “Non-specular reflection of convergent beams from liquid–solid interface,” Trait. Signal 2, 201–205 (1985).

T. Tamir, H. L. Bertoni, “Lateral displacement of optical beams at multilayered and periodic structures,”J. Opt. Soc. Am. 61, 1397–1413 (1971).
[CrossRef]

Breazeale, M. A.

M. A. Breazeale, M. A. Torbett, “Backward displacement of waves reflected from an interface having superimposed periodicity,” Appl. Phys. Lett. 29, 456–458 (1976).
[CrossRef]

Chan, C. C.

Chimenti, D. E.

P. B. Nagy, K. Cho, L. Adler, D. E. Chimenti, “Focal shift of convergent ultrasonic beams reflected from a liquid–solid interface,”J. Acoust. Soc. Am. 81, 835–839 (1987).
[CrossRef]

Cho, K.

P. B. Nagy, K. Cho, L. Adler, D. E. Chimenti, “Focal shift of convergent ultrasonic beams reflected from a liquid–solid interface,”J. Acoust. Soc. Am. 81, 835–839 (1987).
[CrossRef]

Chuang, S. L.

Hessel, A.

Hsue, C. W.

C. W. Hsue, T. Tamir, “Lateral beam displacement and distortion of beams incident upon a transmitting layer configuration,” J. Opt. Soc. Am. A 2, 978–988 (1985).
[CrossRef]

H. L. Bertoni, C. W. Hsue, T. Tamir, “Non-specular reflection of convergent beams from liquid–solid interface,” Trait. Signal 2, 201–205 (1985).

Lotsch, H. K. V.

H. K. V. Lotsch, “Beam displacement at total reflection: the Goos–Hänchen effect,” Optik 32, 116–137, 189–204 (1970); Optik 32, 299–319, 553–569 (1971).

MacLachlan, N. W.

N. W. MacLachlan, Theory and Applications of Mathieu Functions (Oxford U. Press, Oxford, 1951).

Nagy, P. B.

P. B. Nagy, K. Cho, L. Adler, D. E. Chimenti, “Focal shift of convergent ultrasonic beams reflected from a liquid–solid interface,”J. Acoust. Soc. Am. 81, 835–839 (1987).
[CrossRef]

Oliner, A. A.

Riesz, R. P.

Shah, V.

Simon, R.

Tamir, T.

Torbett, M. A.

M. A. Breazeale, M. A. Torbett, “Backward displacement of waves reflected from an interface having superimposed periodicity,” Appl. Phys. Lett. 29, 456–458 (1976).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

M. A. Breazeale, M. A. Torbett, “Backward displacement of waves reflected from an interface having superimposed periodicity,” Appl. Phys. Lett. 29, 456–458 (1976).
[CrossRef]

J. Acoust. Soc. Am. (1)

P. B. Nagy, K. Cho, L. Adler, D. E. Chimenti, “Focal shift of convergent ultrasonic beams reflected from a liquid–solid interface,”J. Acoust. Soc. Am. 81, 835–839 (1987).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (5)

Opt. Lett. (1)

Optik (1)

H. K. V. Lotsch, “Beam displacement at total reflection: the Goos–Hänchen effect,” Optik 32, 116–137, 189–204 (1970); Optik 32, 299–319, 553–569 (1971).

Trait. Signal (1)

H. L. Bertoni, C. W. Hsue, T. Tamir, “Non-specular reflection of convergent beams from liquid–solid interface,” Trait. Signal 2, 201–205 (1985).

Other (3)

N. W. MacLachlan, Theory and Applications of Mathieu Functions (Oxford U. Press, Oxford, 1951).

T. Tamir, Integrated Optics (Springer-Verlag, Berlin, 1982), Sec. 3.1.4, p. 98.

See Ref. 11, Sec. 3.1.6, p. 110.

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Figures (10)

Fig. 1
Fig. 1

Geometry of the reflection grating. See the text for details.

Fig. 2
Fig. 2

Interaction of an incident beam with a leaky wave supported by the grating: (a) Incidence conditions for coupling to the q = −1 forward harmonic of a leaky wave, whose fundamental (q = 0) harmonic travels in the +x direction. (b) Incidence conditions for coupling to the q = 2 backward harmonic of the same leaky wave, whose q = 0 harmonic travels in the −x direction.

Fig. 3
Fig. 3

Variation of the amplitude Rn and the phase ψn of rn versus κi for n = 0 and n = −1 at a grating with λ/d = 0.93, X s ( / μ ) 1 / 2 = ( 11 ) / 5 = 0.6633, and M = 0.4. The exact result given by Eqs. (36) and (37) is shown by solid curves, whereas the approximate values obtained from Eq. (40) are indicated by thick dashed curves. The approximate values were calculated by using the quantities in Table 1.

Fig. 4
Fig. 4

Variation of the normalized lateral displacement Lnq′/λ versus κi at the same grating as in Fig. 3 but with M = 0.2 and M = 0.4. Note that the vertical scales for L0q′ are markedly different.

Fig. 5
Fig. 5

Variation of the normalized focal shift Fnq′/λ versus κi for the same situation as in Fig. 4. Note that most vertical scales are logarithmic.

Fig. 6
Fig. 6

Variation of the maximum values of Ln,−1′/λ and Fn,−1′/λ versus M at the same grating as in Fig. 3.

Fig. 7
Fig. 7

Variation of the normalized angular deflection Ln″/λ versus κi for the same situation as in Fig. 4. The dotted curves refer to the angular shifts of the beam lobes discussed in Section 6.

Fig. 8
Fig. 8

Variation of the normalized beam-waist change parameter Fn″/λ versus κi for the same situation as in Fig. 4. The dotted curves refer to the waist changes of the beam lobes discussed in Section 6.

Fig. 9
Fig. 9

Variation of the intensity profile |H−1| versus Δ−1 = x−1/z−1 in the far field of a beam with kw = 2,000 incident at or near the blazing angle given by sin θe = κ−1,−1+ = 0.278 for the same grating as in Fig. 3. The vertical scales have been normalized to a convenient arbitrary constant.

Fig. 10
Fig. 10

Characteristic values of the functions rn(κ) in the complex κ plane: (a) branch points (shown as squares) and branch cuts (shown by wavy lines); the horizontal portions of the latter actually fall on the real κ′ axis, but they are shown away from that axis for clarity; (b) poles κpq of rn (shown as crosses); (c) zeros κ0q of r0; (d) zeros κ ^ 0 , q of r ^ 0; (e) zeros κ−1,q of r−1; (f) zeros κ−2,q of r−2.

Tables (1)

Tables Icon

Table 1 Locations of Poles and Zeros for a Grating with λ/d = 0.93 and X s ( I ^ / μ ) 1 / 2 = ( 11 ) / 5 = 0 . 6633

Equations (75)

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sin θ n = sin θ i + ( n λ / d )             for n = 0 , ± 1 , ± 2 , ,
H i = ( w / w i ) exp [ - ( x i / w i ) 2 + i k z i ] ,
w i 2 = w 2 + i ( 2 z i / k ) .
H i = k w 2 π - exp [ - ( k w s / 2 ) 2 - i k ( s x i - c z i ) ] d s ,
s = sin ( ϕ - θ i ) ,
c = cos ( ϕ - θ i ) ,
c = 1 - s 2 / 2 ,
H s = n = - H n ,
H n = - k w 2 π r n ( κ ) exp [ - ( k w s / 2 ) 2 + i k ( s n x n + c n z n ) ] d s ,
s n = s n + i s n = sin ( ϕ n - θ n ) ,
c n = c n + i c n = cos ( ϕ n - θ n ) .
sin ϕ n = sin ϕ 0 + ( n λ / d ) ,
s 2 = s 0 2 ( p n s n ) 2 ,
c n 1 - s n 2 / 2 ,
p n = cos θ n / cos θ 0 .
r n ( κ ) d s d s n = ρ n ( s n ) exp ( ln ρ n ) ,
ρ n ( s n ) = ρ n ( 0 ) exp ( - i k L n s n ) exp ( i k F n s n 2 ) ,
ρ n ( 0 ) = p n r n ( κ i ) ,             κ i = sin θ i ,
L n = L n + i L n = i k ρ n d ρ n d s n | s n = 0 ,
F n = F n + i F n = - i k d d s n ( 1 ρ n d ρ n d s n ) s n = 0 .
H n = k w n 2 π r n ( κ i ) exp ( i k z n ) × P exp [ - ( k w f n s n / 2 ) 2 + i k ( x n - L n ) s n ] d s n ,
w f n 2 = w n 2 + i 2 ( z n - F n ) / k ,
w n = p n w = p n w 0 .
H n = ( w n / w f n ) r n ( κ i ) exp [ - ( x n - L n ) 2 / w f n 2 + i k z n ] .
w m n 2 = w n 2 ( 1 + μ n ) ,
μ n = 2 F n / k w m n 2 ,
a n tan α n = 2 L n / k w n 2 .
E x ( x , 0 ) + Z s ( x ) H y ( x , 0 ) = 0 ,
Z s ( x ) = - i X s [ 1 + M cos ( 2 π x / d ) ] = - i X s { 1 + ( M / 2 ) [ exp ( i 2 π x / d ) + exp ( - i 2 π x / d ) ] } ,
H y ( ) = exp [ i k ( κ x - τ 0 z ) ] + n = - r n exp [ i k ( κ n x + τ n z ) ] ,
κ n = κ n + i κ n = sin ϕ n ,
τ n = τ n + i τ n = cos ϕ n = ( 1 - κ n 2 ) 1 / 2 ,
r n + 1 + ( 2 / M ) ( 1 + y n ) ( r n + δ n 0 ) + r n - 1 = ( 4 y n / M ) δ n 0 ,
y n = i ( 1 / X s ) ( μ / ) 1 / 2 τ n
C ± n = 2 M r n r n 1 + δ n 1 , 0 = - 1 1 + y n - ( M / 2 ) 2 1 + y n ± 1 - ( M / 2 ) 2 1 + y n ± 2 - - ,
r 0 = y 0 - 1 - ( M / 2 ) 2 ( C 1 + C - 1 ) y 0 + 1 + ( M / 2 ) 2 ( C 1 + C - 1 ) .
r ± n = 2 y 0 ( M / 2 ) n y 0 + 1 + ( M / 2 ) 2 ( C 1 + C - 1 ) ν = ± 1 ± n C ν .
κ p q = κ p 0 + ( q λ / d ) ,             q = 0 , ± 1 , ± 2 , ,
κ p 0 M = 0 = κ s = [ 1 + ( / μ ) X s 2 ] 1 / 2 > 1.
r n q ( κ ) = R n q exp ( i ψ n q ) = A n q κ - κ n q κ - κ p q ,
L n i k r n d r n d s n | s n = 0 = i p n k r n d r n d θ i ,
F n - i p n 2 k d d θ i ( 1 r n d r n d θ i ) = - p n d L n d θ i ,
L n q = - p n q k d ψ n q d θ i = ( κ p q κ p q - κ i 2 - κ n q κ n q - κ i 2 ) cos θ n k ,
F n q = - p n q d L n q d θ i - [ ( κ p q - κ i ) κ p q κ p q - κ i 4 - ( κ n q - κ i ) κ n q κ n q - κ i 4 ] 2 cos 2 θ n k .
L n q = p n q k R n q d R n q d θ i = ( κ p q - κ i κ p q - κ i 2 - κ n q - κ i κ n q - κ i 2 ) cos θ n k .
F n q = - p n q d L n q d θ i - [ ( κ p q - κ i ) 2 - ( κ p q ) 2 κ p q - κ i 4 - ( κ n q - κ i ) 2 - ( κ n q ) 2 κ n q - κ i 4 ] cos 2 θ n k .
ρ n ( s n ) = p n ( b n 0 + b n 1 s n + b n 2 s n 2 + ) ,
b n ν 1 ν ! ( p n d d θ i ) ν r n .
H n = w n w r n { b n 0 + i 2 b n 1 x n k w r n 2 + 2 b n 2 ( k w r n ) 2 [ 1 - 2 ( x n w r n ) 2 ] } × exp [ - ( x n w r n ) 2 + i k z n ] ,
w r n 2 = w n 2 + i 2 z n / k .
Δ n = x n / z n ,
1 / w r n 2 - i k / 2 z n + ( k w n / 2 z n ) 2 .
H n = ( w n / w r n ) ( b n 0 + b n 1 Δ n + b n 2 Δ n 2 ) × exp [ - ( k w n Δ n / 2 ) 2 + i k z n ( 1 + Δ n 2 / 2 ) ] .
H n 2 = w n / w r n 2 b n 0 + b n 1 Δ n + b n 2 Δ n 2 2 × exp [ - ( k w n Δ n ) 2 / 2 ] .
y ± j ( κ n ) = y ± j ( κ + n λ / d ) = y n ± j ( κ )             with j = 0 , 1 , 2 , ,
C ± j ( κ n ) = C ± j ( κ + n λ / d ) = C n ± j ( k )             as n ± j 0 ,
C ± j = - 1 1 + y ± j + ( M / 2 ) 2 C ± ( j + 1 )             for all j 0 ,
r ^ 0 = r 0 + 1 = 2 y 0 1 + y 0 + ( M / 2 ) 2 ( C 1 + C - 1 ) ,
r n = r ^ 0 ( M / 2 ) n ν = ± 1 n C ν .
κ b n = ± 1 + n λ / d .
τ n > 0 if - 1 + n λ / d < κ i < 1 + n λ / d , τ n > 0 otherwise .
P 0 = 1 + y 0 + ( M / 2 ) 2 ( C 1 + C - 1 ) = 0.
P 1 = 1 + y 1 + ( M / 2 ) 2 [ C 2 - 1 1 + y 0 + ( M / 2 ) 2 C - 1 ] = 0 ,
P 0 ( κ r 1 ) = P 1 ( κ r 0 ) = 0.
P ± j = 1 + y ± j + ( M / 2 ) 2 [ C ± j ± 1 - 1 1 + y ± j 1 - ( M / 2 ) 2 1 + y ± j 2 - - ( M / 2 ) 2 1 + y 0 + ( M / 2 ) 2 C 1 ] = 0.
κ r , ± j = κ s j λ / d = κ p , j = κ p , - q .
y 0 ( N + 1 ) = - 1 - ( M / 2 ) 2 [ C 1 ( N ) + C - 1 ( N ) ] ,
Z ± 1 , 0 = 1 + y ± 1 + ( M / 2 ) 2 C ± 2 = 0.
Z ± 1 , ± j = 1 + y ± 1 ± j + ( M / 2 ) 2 × [ C ± j ± 2 - 1 1 + y ± j - ( M / 2 ) 2 1 + y ± j 1 - - ( M / 2 ) 2 1 + y ± 1 ] = 0.
y - q ( N + 1 ) = - 1 - ( M / 2 ) 2 [ C - q 1 ( N ) - 1 1 + y - q ± 1 ( N ) - ( M / 2 ) 2 1 + y - q ± 2 ( N ) - - ( M / 2 ) 2 1 + y 1 ( N ) ]             as q 0 ,
Z - 2 , - j = 1 + y - j - 2 + ( M / 2 ) 2 × [ C - j - 3 - 1 1 + y - j + 1 - ( M / 2 ) 2 1 + y - j + 2 - - ( M / 2 ) 2 1 + y - 2 ] = 0.
Z - 1 , 1 - q ( κ ^ 0 , q ) = Z - 2 , 1 - q ( κ ^ 0 , q + λ / d ) = Z - 2 , 1 - q ( κ - 1 , q + 1 ) = 0             for all q > 1 ,
κ - 1 , q = κ ^ 0 , q - 1 + λ / d ;
κ n q = { κ ^ 0 , q for q 0 κ ^ 0 , q - n λ / d for q + n 0 ,
Z 0 , ± j = 1 + y ± j + ( M / 2 ) 2 [ C ± j ± 1 - 1 1 + y ± j 1 - ( M / 2 ) 2 1 + y ± j 2 - - ( M / 2 ) 2 1 - y 0 + ( M / 2 ) 2 C 1 ] = 0.

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