Abstract

An exact solution to the four-order acousto-optic (AO) Bragg diffraction problem with arbitrary initial conditions compatible with exact Bragg angle incident light is developed. The solution, obtained by solving a 4th-order differential equation, is formalized into a transition matrix operator predicting diffracted light orders at the exit of the AO cell in terms of the same diffracted light orders at the entrance. It is shown that the transition matrix is unitary and that this unitary matrix condition is sufficient to guarantee energy conservation. A comparison of analytical solutions with numerical predictions validates the formalism. Although not directly related to the approach used to obtain the solution, it was discovered that all four generated eigenvalues from the four-order AO differential matrix operator are expressed simply in terms of Euclid’s Divine Proportion.

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References

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  1. P. Debye and F. W. Sears, “On the scattering of light by supersonic waves,” Proc. Natl. Acad. Sci. USA 18, 409-414(1932).
    [CrossRef] [PubMed]
  2. A. Korpel, Acousto-Optics (Marcel Dekker, 1988).
  3. R. J. Pieper, A. Korpel, and W. Hereman, “Extension of the acoustic-optic Bragg regime through Hamming apodization of the sound field,” J. Opt. Soc. Am. A 3, 1608-1619 (1986).
    [CrossRef]
  4. R. Pieper and A. Korpel, “A comparison of phased-array Bragg cells operating in the second order,” Appl. Opt. 23, 2921-2934(1984).
    [CrossRef] [PubMed]
  5. T.-C. Poon and A. Korpel, “Feynman diagram approach to acousto-optic scattering in the near-Bragg region,” J. Opt. Soc. Am. 71, 1202-1208 (1981).
    [CrossRef]
  6. T.-C. Poon, “A Feynman diagram approach to multiple plane wave scattering in acousto-optic interactions,” Ph.D. dissertation (University of Iowa, 1982), pp. 64-68.
  7. E. Blomme and O. Leroy, “Diffraction of light by ultrasound at oblique incidence; An exact 4-order solution,” Acustica 59, 182-192 (1986).
  8. R. J. Pieper and A. Korpel, “A matrix formalism for the analysis of acousto-optic beam steering,” Appl. Opt. 22, 4073-4081 (1983).
    [CrossRef] [PubMed]
  9. A. Korpel, “Two-dimensional plane wave theory of strong acousto-optic interaction in isotropic media,” J. Opt. Soc. Am. 69, 678-683 (1979).
    [CrossRef]
  10. E. D. Nering, Elementary Linear Algebra (Saunders, 1974), p. 297.
  11. P. Hemenway, Divine Proportion Φ (Phi) in Art, Nature and Science (Stering, 2005), p. 14.
  12. W. R. Klein and B. D. Cook, “Unified approach to ultrasonic light diffraction,” IEEE Trans. Sonics Ultrason. SU-14, 123-133 (1967).
  13. P. M. DeRusso, R. J. Roy, C. M. Close, and A. A. Desrochers, State Variables for Engineers, 2nd ed. (Wiley, 1966), p. 356.
  14. S.-T. Chen and M. R. Chatterjee, “Dual-input acousto-optic set reset flip-flop and its nonlinear dynamics,” Appl. Opt. 36, 3147-3154 (1997).
    [CrossRef] [PubMed]
  15. P. M. De Russo, R. J. Roy, C. M. Close, and A. A. Desrochers, State Variables for Engineers, 2nd ed. (Wiley, 1966), pp. 276-281.

1997 (1)

1986 (2)

R. J. Pieper, A. Korpel, and W. Hereman, “Extension of the acoustic-optic Bragg regime through Hamming apodization of the sound field,” J. Opt. Soc. Am. A 3, 1608-1619 (1986).
[CrossRef]

E. Blomme and O. Leroy, “Diffraction of light by ultrasound at oblique incidence; An exact 4-order solution,” Acustica 59, 182-192 (1986).

1984 (1)

1983 (1)

1981 (1)

1979 (1)

1967 (1)

W. R. Klein and B. D. Cook, “Unified approach to ultrasonic light diffraction,” IEEE Trans. Sonics Ultrason. SU-14, 123-133 (1967).

1932 (1)

P. Debye and F. W. Sears, “On the scattering of light by supersonic waves,” Proc. Natl. Acad. Sci. USA 18, 409-414(1932).
[CrossRef] [PubMed]

Blomme, E.

E. Blomme and O. Leroy, “Diffraction of light by ultrasound at oblique incidence; An exact 4-order solution,” Acustica 59, 182-192 (1986).

Chatterjee, M. R.

Chen, S.-T.

Close, C. M.

P. M. DeRusso, R. J. Roy, C. M. Close, and A. A. Desrochers, State Variables for Engineers, 2nd ed. (Wiley, 1966), p. 356.

P. M. De Russo, R. J. Roy, C. M. Close, and A. A. Desrochers, State Variables for Engineers, 2nd ed. (Wiley, 1966), pp. 276-281.

Cook, B. D.

W. R. Klein and B. D. Cook, “Unified approach to ultrasonic light diffraction,” IEEE Trans. Sonics Ultrason. SU-14, 123-133 (1967).

De Russo, P. M.

P. M. De Russo, R. J. Roy, C. M. Close, and A. A. Desrochers, State Variables for Engineers, 2nd ed. (Wiley, 1966), pp. 276-281.

Debye, P.

P. Debye and F. W. Sears, “On the scattering of light by supersonic waves,” Proc. Natl. Acad. Sci. USA 18, 409-414(1932).
[CrossRef] [PubMed]

DeRusso, P. M.

P. M. DeRusso, R. J. Roy, C. M. Close, and A. A. Desrochers, State Variables for Engineers, 2nd ed. (Wiley, 1966), p. 356.

Desrochers, A. A.

P. M. DeRusso, R. J. Roy, C. M. Close, and A. A. Desrochers, State Variables for Engineers, 2nd ed. (Wiley, 1966), p. 356.

P. M. De Russo, R. J. Roy, C. M. Close, and A. A. Desrochers, State Variables for Engineers, 2nd ed. (Wiley, 1966), pp. 276-281.

Hemenway, P.

P. Hemenway, Divine Proportion Φ (Phi) in Art, Nature and Science (Stering, 2005), p. 14.

Hereman, W.

Klein, W. R.

W. R. Klein and B. D. Cook, “Unified approach to ultrasonic light diffraction,” IEEE Trans. Sonics Ultrason. SU-14, 123-133 (1967).

Korpel, A.

Leroy, O.

E. Blomme and O. Leroy, “Diffraction of light by ultrasound at oblique incidence; An exact 4-order solution,” Acustica 59, 182-192 (1986).

Nering, E. D.

E. D. Nering, Elementary Linear Algebra (Saunders, 1974), p. 297.

Pieper, R.

Pieper, R. J.

Poon, T.-C.

T.-C. Poon and A. Korpel, “Feynman diagram approach to acousto-optic scattering in the near-Bragg region,” J. Opt. Soc. Am. 71, 1202-1208 (1981).
[CrossRef]

T.-C. Poon, “A Feynman diagram approach to multiple plane wave scattering in acousto-optic interactions,” Ph.D. dissertation (University of Iowa, 1982), pp. 64-68.

Roy, R. J.

P. M. De Russo, R. J. Roy, C. M. Close, and A. A. Desrochers, State Variables for Engineers, 2nd ed. (Wiley, 1966), pp. 276-281.

P. M. DeRusso, R. J. Roy, C. M. Close, and A. A. Desrochers, State Variables for Engineers, 2nd ed. (Wiley, 1966), p. 356.

Sears, F. W.

P. Debye and F. W. Sears, “On the scattering of light by supersonic waves,” Proc. Natl. Acad. Sci. USA 18, 409-414(1932).
[CrossRef] [PubMed]

Acustica (1)

E. Blomme and O. Leroy, “Diffraction of light by ultrasound at oblique incidence; An exact 4-order solution,” Acustica 59, 182-192 (1986).

Appl. Opt. (3)

IEEE Trans. Sonics Ultrason. (1)

W. R. Klein and B. D. Cook, “Unified approach to ultrasonic light diffraction,” IEEE Trans. Sonics Ultrason. SU-14, 123-133 (1967).

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (1)

Proc. Natl. Acad. Sci. USA (1)

P. Debye and F. W. Sears, “On the scattering of light by supersonic waves,” Proc. Natl. Acad. Sci. USA 18, 409-414(1932).
[CrossRef] [PubMed]

Other (6)

A. Korpel, Acousto-Optics (Marcel Dekker, 1988).

T.-C. Poon, “A Feynman diagram approach to multiple plane wave scattering in acousto-optic interactions,” Ph.D. dissertation (University of Iowa, 1982), pp. 64-68.

E. D. Nering, Elementary Linear Algebra (Saunders, 1974), p. 297.

P. Hemenway, Divine Proportion Φ (Phi) in Art, Nature and Science (Stering, 2005), p. 14.

P. M. DeRusso, R. J. Roy, C. M. Close, and A. A. Desrochers, State Variables for Engineers, 2nd ed. (Wiley, 1966), p. 356.

P. M. De Russo, R. J. Roy, C. M. Close, and A. A. Desrochers, State Variables for Engineers, 2nd ed. (Wiley, 1966), pp. 276-281.

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Figures (7)

Fig. 1
Fig. 1

Selected physical parameters and possible incident light rays identified on incident light side. L is the length of the acoustic transducer along the z direction.

Fig. 2
Fig. 2

General AO interaction diagram showing (dashed) Bragg lines (after Pieper et al. [3]).

Fig. 3
Fig. 3

Comparison for four-order numerical and analytical solutions; all power is initially in the 0 order.

Fig. 4
Fig. 4

Comparison for four-order numerical and analytical solutions; all power is initially in the 1 order.

Fig. 5
Fig. 5

Comparison for four-order numerical and analytical solutions; all power is initially in the + 1 order.

Fig. 6
Fig. 6

Comparison for 10-order numerical and analytical solutions; all power is initially in the 2 order. The arrow designates the approximate location of the maximum discrepancy between numerical and analytical curves, estimated to be less than 10% of incident power.

Fig. 7
Fig. 7

Comparison for four-order numerical and analytical solutions; power is initially 50% in the 0th order and 50% in the 1 order.

Tables (2)

Tables Icon

Table 1 Fundamental Parameters (After Pieper et al. [3])

Tables Icon

Table 2 Normalized Parameters a

Equations (85)

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K = 2 π Λ = Ω V s ,
k = 2 π λ = ω c .
ϕ B = sin 1 ( K 2 k ) K / ( 2 k ) = Ω / ( 2 k V s ) .
ϕ n = ϕ 0 + 2 n ϕ B ,
Δ ϕ ϕ 0 ϕ B 0 ( Bragg regime condition ) ,
Q 2 π L λ Λ 2 > 2 π ( Bragg regime condition ) ,
Δ ψ n ( z ) = j a ( S n 1 + ( z ) ψ n 1 ( z ) + S n + 1 ( z ) ψ n + 1 ( z ) ) Δ z .
a = k C / 4 ,
x n ± z tan ( ϕ n ± ϕ B ) z ( ϕ n ± ϕ B ) .
S n 1 + ( z ) S ˜ ( z , x n 1 + ) S ˜ ( z , z ( ϕ n 1 + ϕ B ) ) ,
S n + 1 ( z ) S ˜ * ( z , x n + 1 ) S ˜ * ( z , z ( ϕ n + 1 ϕ B ) ) ,
S ˜ ( z , x ) = { | S ˜ ( z ) | exp ( j K x + j θ ( z ) ) z   in   [ 0 , L ] 0 otherwise ,
S n 1 + ( z ) = S ˜ ( z ) exp ( j K z ( ϕ 0 + ( 2 n 1 ) ϕ B ) + j θ ( z ) ) ,
S n + 1 ( z ) = S ˜ * ( z ) exp ( + j K z ( ϕ 0 + ( 2 n + 1 ) ϕ B ) j θ ( z ) ) .
S n 1 + ( ξ ) = S ˜ ( ξ ) exp ( j 1 2 Q c Ω ¯ 2 ξ ( ϕ 0 ¯ Ω ¯ + ( 2 n 1 ) ) + j θ ( ξ ) ) ,
S n + 1 ( ξ ) = S ˜ * ( ξ ) exp ( + j 1 2 Q c Ω ¯ 2 ξ ( ϕ 0 ¯ Ω ¯ + ( 2 n + 1 ) ) j θ ( ξ ) ) ,
Δ ψ n ( ξ ) = j a L ( S n 1 + ( ξ ) ψ n 1 ( ξ ) + S n + 1 ( ξ ) ψ n + 1 ( ξ ) ) Δ ξ .
θ ( ξ ) = 0 ( no design phase shift ) ,
S ( ξ ) = | S ˜ | ( uniform sound field ) ,
Ω ¯ = 1 ( operating at the center frequency ) ,
ϕ 0 ¯ = 1 ( exact Bragg condition   Δ ϕ = 0 ) .
S n 1 + ( ξ ) / | S ˜ | = exp ( j n Q c ξ ) ,
S n + 1 ( ξ ) / | S ˜ | = exp ( + j ( n + 1 ) Q c ξ ) .
Δ ψ n ( ξ ) = j α ( e j n Q c ξ ψ n 1 ( ξ ) + e j ( n + 1 ) Q c ξ ψ n + 1 ( ξ ) ) Δ ξ ,
α a | S ˜ | L
d Ψ ( ξ ) d ξ = j α A ( ξ ) Ψ ( ξ ) ,
A = ( 0 exp ( j Q c ξ ) 0 0 exp ( + j Q c ξ ) 0 1 0 0 1 0 exp ( + j Q c ξ ) 0 0 exp ( j Q c ξ ) 0 ) .
d 4 ψ 2 d ξ 4 + 2 j Q c d 3 ψ 2 d ξ 3 + ( 3 α 2 Q c 2 ) d 2 ψ 2 d ξ 2 + 2 j Q c α 2 d ψ 2 d ξ + α 4 ψ 2 = 0 .
r 4 + 2 Q c r 3 + ( Q c 2 3 α 2 ) r 2 2 Q c α 2 r + α 4 = 0 ,
r 1 = 1 2 ( Q c + α + ( Q c 2 + 2 Q c α + 5 α 2 ) 1 / 2 ) ,
r 2 = 1 2 ( Q c + α ( Q c 2 + 2 Q c α + 5 α 2 ) 1 / 2 ) ,
r 3 = 1 2 ( Q c α + ( Q c 2 2 Q c α + 5 α 2 ) 1 / 2 ) ,
r 4 = 1 2 ( Q c α ( Q c 2 2 Q c α + 5 α 2 ) 1 / 2 ) ,
Q c 2 ± 2 Q c α + 5 α 2 = ( Q c ± α ) 2 + 4 α 2 .
ψ 2 ( ξ ) = A exp ( j r 1 ξ ) + B exp ( j r 2 ξ ) + C exp ( j r 3 ξ ) + D exp ( j r 4 ξ ) ,
ψ + 1 ( ξ ) = A exp ( j r 1 ξ ) + B exp ( j r 2 ξ ) C exp ( j r 3 ξ ) D exp ( j r 4 ξ ) ,
ψ 0 ( ξ ) = 1 α { r 1 A exp [ j ( r 1 + Q c ) ξ ] + r 1 B exp [ j ( r 2 + Q c ) ξ ] r 1 C exp [ j ( r 3 + Q c ) ξ ] r 1 D exp [ j ( r 4 + Q c ) ξ ] } ,
ψ 1 ( ξ ) = 1 α { r 1 A exp [ j ( r 1 + Q c ) ξ ] + r 1 B exp [ j ( r 2 + Q c ) ξ ] + r 1 C exp [ j ( r 3 + Q c ) ξ ] + r 1 D exp [ j ( r 4 + Q c ) ξ ] } ,
ψ 2 ( ξ ) = A exp ( j r 1 ξ ) + B exp ( j r 2 ξ ) + C exp ( j r 3 ξ ) + D exp ( j r 4 ξ ) .
A = 1 2 ( r 2 r 1 ) ( r 2 ψ + 1 ( 0 ) + α ψ 0 ( 0 ) + α ψ 1 ( 0 ) + r 2 ψ 2 ( 0 ) ) ,
B = 1 2 ( r 2 r 1 ) ( r 1 ψ + 1 ( 0 ) α ψ 0 ( 0 ) α ψ 1 ( 0 ) r 1 ψ 2 ( 0 ) ) ,
C = 1 2 ( r 3 r 4 ) ( r 4 ψ + 1 ( 0 ) + α ψ 0 ( 0 ) α ψ 1 ( 0 ) r 4 ψ 2 ( 0 ) ) ,
D = 1 2 ( r 3 r 4 ) ( r 3 ψ + 1 ( 0 ) α ψ 0 ( 0 ) + α ψ 1 ( 0 ) + r 3 ψ 2 ( 0 ) ) ,
ψ + 1 ( ξ ) = 1 2 [ ( σ + τ ) ψ + 1 ( 0 ) + α ( μ ρ ) ψ 0 ( 0 ) + α ( μ + ρ ) ψ 1 ( 0 ) + ( σ τ ) ψ 2 ( 0 ) ] ,
ψ 0 ( ξ ) = 1 2 exp ( j Q c ξ ) [ α ( μ ρ ) ψ + 1 ( 0 ) + ( θ + γ ) ψ 0 ( 0 ) + ( θ γ ) ψ 1 ( 0 ) + α ( μ + ρ ) ψ 2 ( 0 ) ] ,
ψ 1 ( ξ ) = 1 2 exp ( j Q c ξ ) [ α ( μ + ρ ) ψ + 1 ( 0 ) + ( θ γ ) ψ 0 ( 0 ) + ( θ + γ ) ψ 1 ( 0 ) + α ( μ ρ ) ψ 2 ( 0 ) ] ,
ψ 2 ( ξ ) = 1 2 [ ( σ τ ) ψ + 1 ( 0 ) + α ( μ + ρ ) ψ 0 ( 0 ) + α ( μ ρ ) ψ 1 ( 0 ) + ( σ + τ ) ψ 2 ( 0 ) ] ,
μ exp ( j r 1 ξ ) exp ( j r 2 ξ ) r 2 r 1 ,
ρ exp ( j r 3 ξ ) exp ( j r 4 ξ ) r 3 r 4 ,
σ r 2 exp ( j r 1 ξ ) r 1 exp ( j r 2 ξ ) r 2 r 1 ,
τ r 4 exp ( j r 3 ξ ) + r 3 exp ( j r 4 ξ ) r 3 r 4 ,
θ r 1 exp ( j r 1 ξ ) + r 2 exp ( j r 2 ξ ) r 2 r 1 ,
γ r 3 exp ( j r 3 ξ ) r 4 exp ( j r 4 ξ ) r 3 r 4 .
Ψ ( ξ ) = G ( ξ ) Ψ ( 0 ) ,
Ψ ( ξ ) ( ψ + 1 ( ξ ) ψ 0 ( ξ ) ψ 1 ( ξ ) ψ 2 ( ξ ) ) ,
G ( ξ ) 1 2 [ σ + τ α ( μ ρ ) α ( μ + ρ ) σ τ α ( μ ρ ) exp ( j Q c ξ ) ( θ + γ ) exp ( j Q c ξ ) ( θ γ ) exp ( j Q c ξ ) α ( μ + ρ ) exp ( j Q c ξ ) α ( μ + ρ ) exp ( j Q c ξ ) ( θ γ ) e x o ( j Q c ξ ) ( θ + γ ) exp ( j Q c ξ ) α ( μ ρ ) exp ( j Q c ξ ) σ τ α ( μ + ρ ) α ( μ ρ ) σ + τ ] .
G ( G T ) * = G G = G G = I ,
Ψ T ( 0 ) ( 0 , 1 , 0 , 0 ) Ψ T ( ξ ) = 1 2 ( α ( μ ρ ) , ( θ + γ ) exp ( j Q c ξ ) , ( θ γ ) exp ( j Q c ξ ) , α ( μ + ρ ) ) ,
Ψ T ( 0 ) ( 0 , 0 , 1 , 0 ) Ψ T ( ξ ) = 1 2 ( α ( μ + ρ ) , ( θ γ ) exp ( j Q c ξ ) , ( θ + γ ) exp ( j Q c ξ ) , α ( μ ρ ) ) ,
Ψ T ( 0 ) ( 1 , 0 , 0 , 0 ) Ψ T ( ξ ) = 1 2 ( σ + τ , α ( μ ρ ) exp ( j Q c ξ ) , α ( μ + ρ ) exp ( j Q c ξ ) , σ τ ) ,
Ψ T ( 0 ) ( 0 , 0 , 0 , 1 ) Ψ T ( ξ ) = 1 2 ( σ τ , α ( μ + ρ ) exp ( j Q c ξ ) , α ( μ ρ ) exp ( j Q c ξ ) , σ + τ ) .
Ψ T ( ξ ) ( ψ + 1 ( ξ ) , ψ 0 ( ξ ) , ψ 1 ( ξ ) , ψ 2 ( ξ ) ) ,
Ψ ( ξ ) · Ψ ( ξ ) = i = + 1 2 | ψ i ( ξ ) | 2 .
Ψ ( ξ ) = ( G ( ξ ) Ψ ( 0 ) ) = Ψ ( 0 ) G ( ξ ) .
v I v = v v
Ψ ( ξ ) · Ψ ( ξ ) = Ψ ( 0 ) G ( ξ ) G ( ξ ) Ψ ( 0 ) = Ψ ( 0 ) Ψ ( 0 ) .
i = + 1 2 | ψ i ( ξ ) | 2 = i = + 1 2 | ψ i ( 0 ) | 2 ,
I T = I + 1 + I 0 + I 1 + I 2 = i = + 1 2 | ψ i ( ξ ) | 2 ,
I 0 I 1 , I + 1 I 2 .
G ( ξ , ξ o ) = exp ( A ( ξ ξ o ) ) ,
Ψ ( ξ ) = G ( ξ , ξ o ) · Ψ ( ξ o ) ,
Φ = 1 + 5 2 1.61.
| A ( ξ ) λ I | = | λ exp ( j Q c ξ ) 0 0 exp ( + j Q c ξ ) λ 1 0 0 1 λ exp ( + j Q c ξ ) 0 0 exp ( j Q c ξ ) λ | ,
λ 4 3 λ 2 + 1 = 0 ,
3 + 5 2 = ( 1 + 5 2 ) 2 , 3 5 2 = 2 3 + 5 ,
± 3 + 5 2 = ± Φ , ± 3 5 2 = ± 1 Φ .
d ψ + 1 d ξ = j α e j Q c ξ ψ 0 ,
d ψ 0 d ξ = j α e j Q c ξ ψ + 1 j α ψ 1 ,
d ψ 1 d ξ = j α e j Q c ξ ψ 2 j α ψ 0 ,
d ψ 2 d ξ = j α e j Q c ξ ψ 1 .
ψ 0 = j e j Q c ξ α d ψ + 1 d ξ ,
ψ 1 = j e j Q c ξ α d ψ 2 d ξ .
d 2 ψ + 1 d ξ 2 + j Q c d ψ + 1 d ξ + α 2 ψ + 1 = j α d ψ 2 d ξ .
d 2 ψ 2 d ξ 2 + j Q c d ψ 2 d ξ + α 2 ψ 2 = j α d ψ + 1 d ξ .
d 3 ψ + 1 d ξ 3 + j Q c d 2 ψ + 1 d ξ 2 + α 2 d ψ + 1 d ξ = j α d 2 ψ 2 d ξ 2 .

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