Abstract

Many planar structures, including multilayered media and periodic configurations of the optical-grating type, are capable of supporting an electromagnetic field of the leaky-wave form. By exciting this field, an incident light beam transfers a portion of its energy into the leaky-wave structure; after being guided longitudinally for a certain distance along the structure, this energy is leaked back to form part of the reflected beam. Owing to the longitudinal energy flow, the complete reflected beam exhibits a lateral displacement that appears either as a forward beam shift, similar to the Goos-Hänchen effect along a single dielectric interface, or as a backward beam shift, which has not been identified before. By deriving a general expression for the field excited by a gaussian light beam incident upon a leaky-wave structure, we find that the reflected beam may undergo a lateral displacement of the order of the beam width; the magnitude of this beam shift may therefore be much larger than the maximum shift produced at a single dielectric boundary. In the case of periodic structures, all of the higher-order diffracted beams are shifted laterally whenever the specularly reflected beam is displaced. The dependence of the lateral displacement on the beam width, the angle of incidence, and the leakage distance is examined in detail and the relevance of the beam shift to optical-beam couplers is discussed.

© 1971 Optical Society of America

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References

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  1. H. K. V. Lotsch, Optik 32, 116, 189, 299, 553 (1970/1971).
  2. B. R. Horowitz and T. Tamir, J. Opt. Soc. Am. 61, 586 (1971).
    [Crossref]
  3. A. Schoch, Nuovo Cimento (Suppl.) 7, (9), 302 (1950).
    [Crossref]
  4. A. Schoch, Acustica 2, 1 (1952).
  5. Reference 1, Sec. 6, p. 554.
  6. J. E. Midwinter and F. Zernike, Appl. Phys. Letters 16, 198 (1970).
    [Crossref]
  7. H. K. V. Lotsch, J. Opt. Soc. Am. 58, 551 (1968).
    [Crossref]
  8. L. M. Brekhovskikh, Waves in Layered Media (Academic, New York, 1960), Ch. 22, pp. 281–292.
  9. T. Tamir and A. A. Oliner, J. Opt. Soc. Am. 59, 942 (1969).
  10. J. A. Stratton, Electromagnetic Theory (McGraw–Hill, New York, 1941), Sec. 9–13, pp. 516–520.
  11. R. E. Collin, Field Theory of Guided Waves (McGraw–Hill, New York, 1960), Ch. 11, pp. 453–506.
  12. T. Tamir and A. A. Oliner, Proc. IEE 110, 310 (1963).
  13. Antenna Theory, edited by R. E. Collin and F. J. Zucker (McGraw–Hill, New York, 1969), Chs. 19, 20, pp. 151–297.
  14. J. H. Harris, R. Shubert, and J. N. Polky, J. Opt. Soc. Am. 60, 1007 (1970).
    [Crossref]
  15. R. Ulrich, J. Opt. Soc. Am. 60, 1337 (1970).
    [Crossref]
  16. J. E. Midwinter, IEEE J. QE-6, 583 (1970).
    [Crossref]
  17. J. J. Burke, Appl. Opt. 9, 2444 (1970).
    [Crossref] [PubMed]
  18. T. Tamir and A. A. Oliner, Proc. IEEE 51, 317 (1963).
    [Crossref]
  19. M. L. Dakss, L. Kuhn, P. F. Heidrich, and B. A. Scott, Appl. Phys. Letters 16, 523 (1970).
    [Crossref]
  20. H. Kogelnik and T. P. Sosnowski, Bell System Tech. J. 49, 1602 (1970).
    [Crossref]
  21. A. Hessel and A. A. Oliner, Appl. Opt. 4, 1275 (1965).
    [Crossref]
  22. Reference 8, Sec. 8, pp. 100–117.
  23. Reference 8, Sec. 5, p. 44.
  24. Reference 8, Sec. 20, p. 261.
  25. Handbook of Mathematical Functions, edited by M. Abramowitz and I. A. Stegun (Dover, New York, 1965), Ch. 7, p. 297.
  26. T. Tamir and A. A. Oliner, Proc. IEE 110, 325 (1963).
  27. L. V. Iogansen, Sov. Phys.-Tech. Phys. 7, 295 (1962); Sov. Phys.-Tech. Phys. 8, 985 (1964); Sov. Phys.-Tech. Phys. 11, 1529 (1967).

1971 (1)

1970 (7)

J. E. Midwinter and F. Zernike, Appl. Phys. Letters 16, 198 (1970).
[Crossref]

J. H. Harris, R. Shubert, and J. N. Polky, J. Opt. Soc. Am. 60, 1007 (1970).
[Crossref]

R. Ulrich, J. Opt. Soc. Am. 60, 1337 (1970).
[Crossref]

J. E. Midwinter, IEEE J. QE-6, 583 (1970).
[Crossref]

J. J. Burke, Appl. Opt. 9, 2444 (1970).
[Crossref] [PubMed]

M. L. Dakss, L. Kuhn, P. F. Heidrich, and B. A. Scott, Appl. Phys. Letters 16, 523 (1970).
[Crossref]

H. Kogelnik and T. P. Sosnowski, Bell System Tech. J. 49, 1602 (1970).
[Crossref]

1969 (1)

1968 (1)

1965 (1)

1963 (3)

T. Tamir and A. A. Oliner, Proc. IEE 110, 325 (1963).

T. Tamir and A. A. Oliner, Proc. IEE 110, 310 (1963).

T. Tamir and A. A. Oliner, Proc. IEEE 51, 317 (1963).
[Crossref]

1962 (1)

L. V. Iogansen, Sov. Phys.-Tech. Phys. 7, 295 (1962); Sov. Phys.-Tech. Phys. 8, 985 (1964); Sov. Phys.-Tech. Phys. 11, 1529 (1967).

1952 (1)

A. Schoch, Acustica 2, 1 (1952).

1950 (1)

A. Schoch, Nuovo Cimento (Suppl.) 7, (9), 302 (1950).
[Crossref]

Brekhovskikh, L. M.

L. M. Brekhovskikh, Waves in Layered Media (Academic, New York, 1960), Ch. 22, pp. 281–292.

Burke, J. J.

Collin, R. E.

R. E. Collin, Field Theory of Guided Waves (McGraw–Hill, New York, 1960), Ch. 11, pp. 453–506.

Dakss, M. L.

M. L. Dakss, L. Kuhn, P. F. Heidrich, and B. A. Scott, Appl. Phys. Letters 16, 523 (1970).
[Crossref]

Harris, J. H.

Heidrich, P. F.

M. L. Dakss, L. Kuhn, P. F. Heidrich, and B. A. Scott, Appl. Phys. Letters 16, 523 (1970).
[Crossref]

Hessel, A.

Horowitz, B. R.

Iogansen, L. V.

L. V. Iogansen, Sov. Phys.-Tech. Phys. 7, 295 (1962); Sov. Phys.-Tech. Phys. 8, 985 (1964); Sov. Phys.-Tech. Phys. 11, 1529 (1967).

Kogelnik, H.

H. Kogelnik and T. P. Sosnowski, Bell System Tech. J. 49, 1602 (1970).
[Crossref]

Kuhn, L.

M. L. Dakss, L. Kuhn, P. F. Heidrich, and B. A. Scott, Appl. Phys. Letters 16, 523 (1970).
[Crossref]

Lotsch, H. K. V.

H. K. V. Lotsch, Optik 32, 116, 189, 299, 553 (1970/1971).

H. K. V. Lotsch, J. Opt. Soc. Am. 58, 551 (1968).
[Crossref]

Midwinter, J. E.

J. E. Midwinter, IEEE J. QE-6, 583 (1970).
[Crossref]

J. E. Midwinter and F. Zernike, Appl. Phys. Letters 16, 198 (1970).
[Crossref]

Oliner, A. A.

T. Tamir and A. A. Oliner, J. Opt. Soc. Am. 59, 942 (1969).

A. Hessel and A. A. Oliner, Appl. Opt. 4, 1275 (1965).
[Crossref]

T. Tamir and A. A. Oliner, Proc. IEE 110, 325 (1963).

T. Tamir and A. A. Oliner, Proc. IEEE 51, 317 (1963).
[Crossref]

T. Tamir and A. A. Oliner, Proc. IEE 110, 310 (1963).

Polky, J. N.

Schoch, A.

A. Schoch, Acustica 2, 1 (1952).

A. Schoch, Nuovo Cimento (Suppl.) 7, (9), 302 (1950).
[Crossref]

Scott, B. A.

M. L. Dakss, L. Kuhn, P. F. Heidrich, and B. A. Scott, Appl. Phys. Letters 16, 523 (1970).
[Crossref]

Shubert, R.

Sosnowski, T. P.

H. Kogelnik and T. P. Sosnowski, Bell System Tech. J. 49, 1602 (1970).
[Crossref]

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw–Hill, New York, 1941), Sec. 9–13, pp. 516–520.

Tamir, T.

B. R. Horowitz and T. Tamir, J. Opt. Soc. Am. 61, 586 (1971).
[Crossref]

T. Tamir and A. A. Oliner, J. Opt. Soc. Am. 59, 942 (1969).

T. Tamir and A. A. Oliner, Proc. IEE 110, 325 (1963).

T. Tamir and A. A. Oliner, Proc. IEE 110, 310 (1963).

T. Tamir and A. A. Oliner, Proc. IEEE 51, 317 (1963).
[Crossref]

Ulrich, R.

Zernike, F.

J. E. Midwinter and F. Zernike, Appl. Phys. Letters 16, 198 (1970).
[Crossref]

Acustica (1)

A. Schoch, Acustica 2, 1 (1952).

Appl. Opt. (2)

Appl. Phys. Letters (2)

M. L. Dakss, L. Kuhn, P. F. Heidrich, and B. A. Scott, Appl. Phys. Letters 16, 523 (1970).
[Crossref]

J. E. Midwinter and F. Zernike, Appl. Phys. Letters 16, 198 (1970).
[Crossref]

Bell System Tech. J. (1)

H. Kogelnik and T. P. Sosnowski, Bell System Tech. J. 49, 1602 (1970).
[Crossref]

IEEE J. (1)

J. E. Midwinter, IEEE J. QE-6, 583 (1970).
[Crossref]

J. Opt. Soc. Am. (5)

Nuovo Cimento (Suppl.) (1)

A. Schoch, Nuovo Cimento (Suppl.) 7, (9), 302 (1950).
[Crossref]

Optik (1)

H. K. V. Lotsch, Optik 32, 116, 189, 299, 553 (1970/1971).

Proc. IEE (2)

T. Tamir and A. A. Oliner, Proc. IEE 110, 310 (1963).

T. Tamir and A. A. Oliner, Proc. IEE 110, 325 (1963).

Proc. IEEE (1)

T. Tamir and A. A. Oliner, Proc. IEEE 51, 317 (1963).
[Crossref]

Sov. Phys.-Tech. Phys. (1)

L. V. Iogansen, Sov. Phys.-Tech. Phys. 7, 295 (1962); Sov. Phys.-Tech. Phys. 8, 985 (1964); Sov. Phys.-Tech. Phys. 11, 1529 (1967).

Other (9)

Reference 8, Sec. 8, pp. 100–117.

Reference 8, Sec. 5, p. 44.

Reference 8, Sec. 20, p. 261.

Handbook of Mathematical Functions, edited by M. Abramowitz and I. A. Stegun (Dover, New York, 1965), Ch. 7, p. 297.

Antenna Theory, edited by R. E. Collin and F. J. Zucker (McGraw–Hill, New York, 1969), Chs. 19, 20, pp. 151–297.

Reference 1, Sec. 6, p. 554.

J. A. Stratton, Electromagnetic Theory (McGraw–Hill, New York, 1941), Sec. 9–13, pp. 516–520.

R. E. Collin, Field Theory of Guided Waves (McGraw–Hill, New York, 1960), Ch. 11, pp. 453–506.

L. M. Brekhovskikh, Waves in Layered Media (Academic, New York, 1960), Ch. 22, pp. 281–292.

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Figures (13)

Fig. 1
Fig. 1

Lateral displacement D of a bounded beam at a dielectric interface. The dashed lines show the boundaries of the reflected beam as predicted by simple geometrical-optics considerations; the thick arrow indicates the direction of energy flow in the lower medium.

Fig. 2
Fig. 2

Leaky-wave fields: (a) forward variety; (b) backward variety. Equiphase contours are shown by dashed lines whereas equiamplitude contours are shown by solid lines; the density of the latter lines is a measure of the field amplitude. The shaded region between the arrows at the angle θ refers to the axis of an incident beam that can excite the leaky wave; the thick arrow in the region z>0 indicates the energy flux inside the leaky-wave structure.

Fig. 3
Fig. 3

Geometry and coordinate systems employed in deriving the field of the reflected beam. The coordinate set (xn,zn) shown dashed applies to periodic structures discussed in Sec. IV.

Fig. 4
Fig. 4

Intensity of normalized reflected field as a function of x/w0 for various values of αw0. The curves shown are for total-reflection and for phase-matching (β=k sinθ>0) conditions; the normalization quantity A is given by A=π(w0 cosθ)2=πw2.

Fig. 5
Fig. 5

Normalized values of the lateral displacement as a function of |α|w0=w0/L, for total-reflection and phase-matching (β=k sinθ>0) conditions. Portions of curves shown dashed denote considerable distortion in the gaussian amplitude profile of the beam.

Fig. 6
Fig. 6

Intensity of the normalized reflected field as a function of x/w0 for w0/L=|α|w0=0.68 and various values of |γ″| in the case of a totally reflecting structure. The normalization quantity A is given by A=π(w0 cosθ)2=πw2.

Fig. 7
Fig. 7

Normalized values of the lateral displacement as a function of |γ″| for |α|w0=w0/L=0.68 in the case of a totally reflecting structure.

Fig. 8
Fig. 8

Lateral beam shift due to reflection by a backward leaky-wave structure. The thick arrow indicates the direction of energy flow within the leaky-wave structure in the region z>0; the dashed lines show the reflected beam predicted by geometrical optics.

Fig. 9
Fig. 9

Schematic description for incidence upon a leaky-wave periodic structure: (a) The assumed form of a leaky wave that possesses only two (m=−1 and −2) radiating harmonics; (b) spectral orders of an incident plane wave that is phase matched to the m=−1 harmonic of the leaky wave shown in (a). The incident wave is indicated by a dashed line whereas the scattered waves are shown by solid lines.

Fig. 10
Fig. 10

Arrangement for the possible detection of a reverse lateral beam shift: (a) Desired distribution of the space harmonics for the leaky wave; (b) orientation of the incident beam (shown dashed) and its specular reflection (shown solid).

Fig. 11
Fig. 11

Variation of the cross-over distance xc and the corresponding maximum efficiency η(xc) with respect to |α|w0=w0/L, for θ=θp. The dashed curve refers to the approximation xc=D/2.

Fig. 12
Fig. 12

Variation of the cross-over distance xc and the corresponding maximum efficiency η(xc) with respect to γ″, for |α|w0=w0/L=0.68.

Fig. 13
Fig. 13

Basic configurations for thin-film optical beam couplers: (a) Air-gap coupler; (b) grating coupler.

Equations (76)

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sin θ c = ( k / k ) < 1.
( 2 + k 2 ) ψ = 0 ,
ψ = exp [ i ( k x x - k z z - ω t ) ] ,
k z = ( k 2 - k x 2 ) 1 2 .
k p = β + i α .
sin θ = β / k = sin θ p ,
x i r = x cos θ - ( h ± z ) sin θ ,
z i r = x sin θ + ( h ± z ) cos θ ,
E w ( x i , 0 ) = { exp [ - ( x i / w ) 2 ] } / π 1 2 w ,
k w = 2 π ( w / λ ) 1 ,
E ap ( x , - h ) = E w ( x i , 0 ) exp ( i k z i ) z = - h = exp { - [ ( x cos θ ) / w ] 2 + i k x sin θ } π 1 2 w ,
E inc ( x , z ) = 1 2 π - Φ ( k x ) exp [ i k x x + i k z ( z + h ) ] d k x ,
Φ ( k x ) = - E ap ( x , - h ) exp ( - i k x x ) d x ,
Φ ( k x ) = exp { - [ w ( k x - k sin θ ) / ( 2 cos θ ) ] 2 } cos θ .
k z = k cos θ - ( k x - k sin θ ) tan θ - [ ( k x - k sin θ ) 2 / ( 2 k cos 3 θ ) ] .
E inc = { exp [ - ( x i / w i ) 2 + i k z i ] } / π 1 2 w i ,
w i = w { 1 + i [ 2 ( z + h ) / ( k w 2 cos θ ) ] } 1 2 .
E refl = 1 2 π - ρ ( k x ) Φ ( k x ) exp [ i k x x - i k z ( z - h ) ] d k x .
ρ ( k x ) = ( k x - k 1 ) ( k x - k 2 ) ( k x - k 3 ) ( k x - k 1 ) ( k x - k 2 ) ( k x - k 3 ) .
ρ ( k x ) = r ( k x ) [ ( k x - k 0 ) / ( k x - k p ) ] .
[ ( k s - k sin θ ) / ( 2 cos θ ) ] w r 1 ,
r ( k x ) r ( k sin θ ) = R .
ρ t ( k x ) = [ ( k x - k p * ) / ( k x - k p ) ] e i Δ ,
ρ ( k x ) = ρ ( 0 ) + ρ ( 1 ) ( k x ) ,
ρ ( 0 ) = ρ ( k sin θ ) = R [ ( k 0 - k sin θ ) / ( k p - k sin θ ) ] ,
ρ ( 1 ) ( k x ) = R { [ ( k x - k sin θ ) ( k p - k 0 ) ] / [ ( k p - k sin θ ) ( k x - k p ) ] } .
E refl = E ( 0 ) + E ( 1 ) ,
E ( 0 ) = ρ ( k sin θ ) { exp [ - ( x r / w r ) 2 + i k z r ] } / π 1 2 w r ,
w r = w { 1 - i [ 2 ( z - h ) / ( k w 2 cos θ ) ] } 1 2 .
E ( 1 ) = E ( 0 ) k p - k 0 k 0 - k sin θ × [ 1 ± i π 1 2 w r 2 cos θ ( k p - k sin θ ) exp ( γ 2 ) erfc ( ± γ ) ] ,             as             α 0.
γ = γ + i γ = α w r 2 cos θ - x r w r + i ( k sin θ - β ) w r 2 cos θ ,
( z r / k w 2 ) 1.
w r w i w 0 cos θ ,
E ( 0 ) = - e i Δ π 1 2 w r exp [ - ( x r / w r ) 2 + i k z r ] ,
E ( 1 ) = - E ( 0 ) { 2 π 1 2 α w 0 exp [ ( γ ) 2 ] erfc ( ± γ ) } ,             as             α 0.
exp [ ( γ ) 2 ] erfc ( γ ) { 2 exp [ ( γ ) 2 ] , for γ - 1 ( π 1 2 γ ) - 1 , for γ 1.
E ( 1 ) 2 π 1 2 α w 0 E ( 0 ) exp [ ( γ ) 2 ] ,
E ( 1 ) - E ( 0 ) ( 2 - α w 0 γ ) = - E ( 0 ) 1 - [ α w r 2 / ( 2 x r cos θ ) ] ,
E ( 1 ) - 2 α sec θ exp [ ( α w 0 2 ) 2 + i ( β cot θ - i α tan θ ) ( h - z ) ] · e i ( β + i α ) x .
γ = ( k w 0 / 2 ) ( sin θ - sin θ p ) ( k w / 2 ) ( θ - θ p ) = ( π w / λ ) ( θ - θ p ) ,
E ( 1 ) ( x ; α ) = E ( 1 ) ( - x ; - α ) ,
ψ = m = - ψ m = m = - a m exp [ i k p m x - i ( k 2 - k p m 2 ) 1 2 z ] ,
k p m = k p + 2 m π / d .
k p m = β m + i α ,
β m = β + 2 m π / d ,
k sin θ = β m = k sin θ p m ,
k sin θ + 2 n π / d = k sin θ n             ( n = 0 , ± 1 , ± 2 ) ,
σ n ( k x ) = S m n [ ( k z - k 0 n ) / ( k x - k p m ) ] ,
E n = 1 2 π - σ n ( k x ) Φ ( k x ) × exp [ i ( k z h + k x n x - k z n z ) ] d k x ,
k x n = k x + 2 n π / d ,
k z n = ( k 2 - k x n 2 ) 1 2 .
x n = x cos θ n + z sin θ n - h tan θ cos θ n ,
z n = x sin θ n - z cos θ n + h cos θ ,
σ n ( k x ) = σ n ( 0 ) + σ n ( 1 ) ( k x ) ,
σ n ( 0 ) = σ n ( k sin θ ) = S m n k 0 n - k sin θ k p m - k sin θ ,
σ n ( 1 ) ( k x ) = S m n ( k x - k sin θ ) ( k p m - k 0 n ) ( k p m - k sin θ ) ( k x - k p m ) .
E n = E n ( 0 ) + E n ( 1 ) .
E n ( 0 ) = σ n ( k sin θ ) π 1 2 w n exp [ - ( x n cos θ w n cos θ n ) 2 + i k z n ] ,
E n ( 1 ) = E n ( 0 ) k p m - k 0 n k 0 n - k sin θ [ 1 ± i π 1 2 w n 2 cos θ ( k p m - k sin θ ) × exp ( γ m n 2 ) erfc ( ± γ m n ) ] ,             as             α 0.
γ m n = α w n 2 cos θ - x n cos θ w n cos θ n + i ( k sin θ - β m ) w n 2 cos θ ,
w n = w [ 1 + 2 i k w 2 ( h cos θ - z cos 2 θ cos 3 θ n ) ] 1 2 .
η ( x 0 ) = - x 0 [ E inc ( x ) 2 - E refl ( x ) 2 ] d x / - E inc ( x ) 2 d x .
η ( x 0 ) / x 0 = 0.
E inc ( x c ) = E refl ( x c ) .
x c D / 2.
k z n = k cos θ n - ( k x - k sin θ ) tan θ n - ( k x - k sin θ ) 2 2 k cos 3 θ n .
E n ( 1 ) = S m n 2 π k p m - k 0 n k p m - k sin θ exp ( i k z r ) cos θ - ( 1 + k p m - k sin θ k x - k p m ) × exp [ - ( k x - k sin θ 2 cos θ w n ) 2 + i ( k x - k sin θ ) x n cos θ n ] d k x .
E n ( 1 ) = E n ( 0 ) k p m - k 0 n k 0 n - k sin θ [ 1 + ( k p m - k sin θ ) w n 2 π 1 2 cos θ I ] ,
κ = ( k x - k sin θ ) / ( 2 k cos θ ) ,
I = exp [ ( x n cos θ w n cos θ n ) 2 ] - d κ κ - κ p × exp [ - ( k w n κ ) 2 + 2 i k x n cos θ cos θ n κ ] .
I = exp [ ( x n cos θ w n cos θ n ) 2 ] - F 1 ( u ) F 2 ( u n - u ) d u ,
u n = 2 k x n cos θ cos θ n ,
F 1 ( u ) = - exp [ - ( k w n κ ) 2 + i u κ ] d κ = π 1 2 k w n exp [ - ( u 2 k w n ) 2 ] ,
F 2 ( u ) = 1 2 π - e i u κ κ - κ p d κ .
F 2 ( u ) = { 0 , if u α < 0 ; i exp ( i u κ p ) , if u , α > 0 ; - i exp ( i u κ p ) , if u , α < 0.
I = i π 1 2 k w n exp [ ( x n cos θ w n cos θ n ) 2 ] × - u n exp [ - ( u 2 k w n ) 2 + i ( u n - u ) κ p ] d u = i π exp [ ( u n 2 k w n + i k w n κ p ) 2 ] × erfc ( - u n 2 k w n - i k w n κ p ) .