Abstract

A unified approach is developed to three important areas of signal and image processing: pulse compression and real-time spectrum analysis, fiber-cable communications and dispersion, and Fresnel diffraction and optical filtering. The results are based on the properties of quadratic phase filters and Fresnel transforms, and they lead to a variety of analogies among the time responses of fiber cables, the spatial variations of diffracted fields, and the role of frequency modulation in narrow-band systems. The analysis includes deterministic and stochastic excitations.

© 1994 Optical Society of America

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References

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  1. B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).
    [CrossRef]
  2. T. Jannson, J. Jannson, “Temporal self-imaging effect in single-mode fibers,”J. Opt. Soc. Am. 71, 1373–1376 (1981).
  3. A. Papoulis, Signal Analysis (McGraw-Hill, New York, 1977), Chap. 8, p. 263.
  4. A. Papoulis, The Fourier Integral and Its Applications (McGraw-Hill, New York, 1962), Chap. 7, pp. 120–134.
  5. A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd ed. (McGraw-Hill, New York, 1991), Chap. 10, pp. 285–332.
  6. H. A. Haus, Waves and Fields in Optoelectronics, Series in Solid State Physical Electronics (Prentice-Hall, Englewood Cliffs, N.J., 1984).
  7. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968), Chap. 3.
  8. A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968; reprinted by Kreiger, Malabar, Fla., 1981), Chap. 9.
  9. E. E. Kriezis, D. P. Chryssoulides, A. G. Papayannakis, Electromagnetics and Optics (World Scientific, River Edge, N.J., 1992).
  10. A. Papoulis, “Fourier optics,” Electromagnetics 9, 1–16 (1989).
    [CrossRef]
  11. F. Zernike, Z. Tech. Phys. 16, 454 (1935).

1989

A. Papoulis, “Fourier optics,” Electromagnetics 9, 1–16 (1989).
[CrossRef]

1981

1935

F. Zernike, Z. Tech. Phys. 16, 454 (1935).

Chryssoulides, D. P.

E. E. Kriezis, D. P. Chryssoulides, A. G. Papayannakis, Electromagnetics and Optics (World Scientific, River Edge, N.J., 1992).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968), Chap. 3.

Haus, H. A.

H. A. Haus, Waves and Fields in Optoelectronics, Series in Solid State Physical Electronics (Prentice-Hall, Englewood Cliffs, N.J., 1984).

Jannson, J.

Jannson, T.

Kriezis, E. E.

E. E. Kriezis, D. P. Chryssoulides, A. G. Papayannakis, Electromagnetics and Optics (World Scientific, River Edge, N.J., 1992).

Papayannakis, A. G.

E. E. Kriezis, D. P. Chryssoulides, A. G. Papayannakis, Electromagnetics and Optics (World Scientific, River Edge, N.J., 1992).

Papoulis, A.

A. Papoulis, “Fourier optics,” Electromagnetics 9, 1–16 (1989).
[CrossRef]

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968; reprinted by Kreiger, Malabar, Fla., 1981), Chap. 9.

A. Papoulis, Signal Analysis (McGraw-Hill, New York, 1977), Chap. 8, p. 263.

A. Papoulis, The Fourier Integral and Its Applications (McGraw-Hill, New York, 1962), Chap. 7, pp. 120–134.

A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd ed. (McGraw-Hill, New York, 1991), Chap. 10, pp. 285–332.

Saleh, B. E. A.

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).
[CrossRef]

Teich, M. C.

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).
[CrossRef]

Zernike, F.

F. Zernike, Z. Tech. Phys. 16, 454 (1935).

Electromagnetics

A. Papoulis, “Fourier optics,” Electromagnetics 9, 1–16 (1989).
[CrossRef]

J. Opt. Soc. Am.

Z. Tech. Phys.

F. Zernike, Z. Tech. Phys. 16, 454 (1935).

Other

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).
[CrossRef]

A. Papoulis, Signal Analysis (McGraw-Hill, New York, 1977), Chap. 8, p. 263.

A. Papoulis, The Fourier Integral and Its Applications (McGraw-Hill, New York, 1962), Chap. 7, pp. 120–134.

A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd ed. (McGraw-Hill, New York, 1991), Chap. 10, pp. 285–332.

H. A. Haus, Waves and Fields in Optoelectronics, Series in Solid State Physical Electronics (Prentice-Hall, Englewood Cliffs, N.J., 1984).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968), Chap. 3.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968; reprinted by Kreiger, Malabar, Fla., 1981), Chap. 9.

E. E. Kriezis, D. P. Chryssoulides, A. G. Papayannakis, Electromagnetics and Optics (World Scientific, River Edge, N.J., 1992).

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Figures (14)

Fig. 1
Fig. 1

Real part exp(−αt2)cos βt2 and imaginary part exp(−αt2)sin βt2 of a complex Gaussian beam.

Fig. 2
Fig. 2

Fresnel cosine integral C(t) and Fresnel sine integral S(t); Cornu spiral Fc(t) = C(t) + jS(t).

Fig. 3
Fig. 3

Duration D ¯ of the output f ¯(t) of a QPF as a function of the duration D of its input f(t).

Fig. 4
Fig. 4

Real-time spectrum analyzer: (a) phase modulated, (b) unmodulated.

Fig. 5
Fig. 5

Narrow-band systems and lowpass bandpass transformations.

Fig. 6
Fig. 6

Response of a system to a slowly varying input.

Fig. 7
Fig. 7

Fiber cable with modulated input.

Fig. 8
Fig. 8

Aperture S illuminated by a harmonic field gi(x, y, z).

Fig. 9
Fig. 9

Diffracted field g(x, z) as response to a QPF with input the aperture function f(x).

Fig. 10
Fig. 10

Diffracted field: z1, near zone; z2, intermediate zone; z3, distant zone.

Fig. 11
Fig. 11

(a) General transparency q(x), (b) thin lens as a transparency.

Fig. 12
Fig. 12

(a) Field w(x, z) diffracted from an aperture S; (b) field w(x, z) diffracted from a lens.

Fig. 13
Fig. 13

P1, P2: Fourier planes.

Fig. 14
Fig. 14

Optical filtering.

Equations (128)

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exp ( - s t 2 ) = exp ( - α t 2 ) exp ( j β t 2 ) ,             s = α - j β ,             α > 0.
exp ( - s t 2 ) π s exp ( - ω 2 / 4 s ) .
α π exp ( - α t 2 ) exp ( - ω 2 / 4 α ) ,
β j π exp ( i β t 2 ) exp ( - j ω 2 / 4 β )
β j π - exp ( j β t 2 ) d t = 1.
β j π - exp ( j β t 2 ) f ( t ) d t β f ( 0 ) ,
β j π exp ( j β t 2 ) β δ ( t ) .
F c ( t ) = 2 π 0 t exp ( j τ 2 ) d τ = C ( t ) + j S ( t ) .
F c ( ) = 1 + j 2 ,             C ( ) = S ( ) = 1 2 .
s 1 π exp ( - s 1 t 2 ) * s 2 π exp ( - s 2 t 2 ) exp { - ω 2 4 ( 1 s 1 + 1 s 2 ) } .
s π exp ( - s t 2 ) ,
1 s = 1 s 1 + 1 s 2 ,             s = α - j β .
- j α 1 β 2 π exp ( - α 1 t 2 ) * exp ( j β 2 t 2 ) = α - j β π exp ( - α t 2 ) exp ( j β t 2 ) ,
α = α 1 1 + ( α 1 / β 2 ) 2 ,             β = β 2 1 + ( β 2 / α 1 ) 2 .
q β ( t ) = β j π exp ( j β t 2 ) Q β ( ω ) = exp ( - j ω 2 / 4 β ) .
f ¯ ( t ) = β j π - f ( τ ) exp [ - j β ( t - τ ) 2 ] d τ = f ( t ) * q β ( t ) .
F ¯ ( ω ) = F ( ω ) exp ( - j ω 2 / 4 β ) = F ( ω ) Q β ( ω ) .
f ¯ ( t ) = exp ( - c t 2 ) * β j π exp ( j β t 2 ) = s c exp ( - s t 2 ) ,             1 s = 1 c - 1 j β ,
s = a - j b ,             1 a = 1 c + c β 2 .
D = 1 2 c ,             D ¯ = 1 2 a
1 a = 1 c + c β 2 ,             D ¯ = ( D 2 + 1 16 β 2 D 2 ) 1 / 2 .
D ¯ = D ¯ m = 1 2 β             for             c = c m = β .
f 1 ( t ) = U ( t ) = { 1 t > 0 0 t < 0 .
f ¯ 1 ( t ) = β j π - t exp ( j β τ 2 ) d τ = 1 2 j [ F c ( β t ) + 1 ] .
f 2 ( t ) = U ( t + T ) - U ( t - T ) = { 1 t < T 0 t > T .
f ¯ 2 ( t ) = f ¯ 1 ( t + T ) - f ¯ 1 ( t - T ) = 1 2 j { F c [ β ( t + T ) ] - F c [ β ( t - T ) ] } .
f ¯ ( t ) = f ( t ) * q B ( t ) β f ( t ) * δ ( t ) = f ( t ) .
f ( t ) = 0             for             t > T ,             β T 2 1 ,
f ¯ ( t ) β j π exp ( j β t 2 ) F ( 2 β t ) = q β ( t ) F ( 2 β t ) ,
f ¯ ( t ) = β j π - T T f ( τ ) exp ( j β t 2 ) exp ( - j 2 β t τ ) exp ( j β τ 2 ) d τ β j π exp ( j β t 2 ) - T T f ( τ ) exp ( - j 2 β t τ ) d τ ,
f ¯ ( t ) β π F ( 2 β t ) ,
w ( t ) = f ( t ) exp ( - j β t 2 ) .
g ( t ) = β j π exp ( j β t 2 ) F ( 2 β t ) = q β ( t ) F ( 2 β t ) .
w ¯ ( t ) = β j π - f ( τ ) exp ( - j β τ 2 ) exp [ j β ( t - τ ) 2 ] d τ = β j π exp ( j β t 2 ) - f ( τ ) exp ( - j 2 β t τ ) d τ ,
g ( t ) = w ¯ ( t ) ,             G ( ω ) = W ( ω ) exp ( - j ω 2 / 4 β ) .
g ( t ) = β j π P ( 2 β t ) .
F ( ω ) = P ( ω ) exp ( - j ω 2 / 4 β ) .
g ( t ) = q β ( t ) F ( 2 β t ) = β j π exp ( j β t 2 ) P ( 2 β t ) exp ( - 4 β 2 t 2 / 4 β ) ,
D f 2 = 1 E f - t 2 f ( t ) 2 d t ,             E f = - f ( t ) 2 d t .
D g = D F 2 β .
r = D f D g = 2 β D f D F .
f ( t ) = exp ( - c t 2 ) F ( ω ) = π c exp ( - j ω 2 / 4 c ) .
E f = π 2 c ,             D f 2 = 2 c π - t 2 exp ( - 2 c t 2 ) d t = 1 4 c 2 , D f = 1 2 c ,             D F = c ,             r = β c .
f ( t ) = { 1 t < T 0 t > T F ( ω ) = sin T ω ω .
D f = 2 T ,             D F = 2 π T ,             r = 2 β T 2 π .
D w D W 1 2 ,
w ( t ) = f ( t ) exp ( - j β t 2 ) ,
D w D W β d f D F .
D = D f ,             D W = D G ,             D g D G 1 2 , 1 D g 2 D G = 2 D W , 2 β D f D F = D f D g = D w D g 2 D w D W ,
x ( t ) = f ( t ) n ( t ) ,             y ( t ) = x ( t ) * q β ( t ) ,
R n ( τ ) = E { n ( t + τ ) n ( t ) } , S n ( ω ) = - R n ( τ ) exp ( - j ω τ ) d τ .
R n ( τ ) 0 for τ > τ c f ( t + Δ ) f ( t ) for Δ < τ c .
E { y ( t ) 2 } = f 2 ( t ) * [ R ¯ n ( t ) q β * ( t ) ] , q β ( t ) = β j π exp ( j β t 2 ) ,
E { y ( t ) 2 } = | - x ( t - γ ) q β ( γ ) d γ | 2 = - - f ( t - γ 1 ) n ( t - γ 1 ) q β ( γ 1 ) × f ( t - γ 2 ) n ( t - γ 2 ) q β * ( γ 2 ) d γ 1 d γ 2 .
E { y ( t ) 2 } = - f 2 ( t - γ 2 ) q β * ( γ 2 ) × - R n ( γ 2 - γ 1 ) q β ( γ 1 ) d γ 1 d γ 2 ,
β τ c 2 1 ,
E { y ( t ) 2 } f 2 ( t ) * β π S n ( 2 β t ) ,
R ¯ n ( t ) = q β ( t ) S n ( 2 β t ) . R ¯ n ( t ) q β * ( t ) = S n ( 2 β t ) q β ( t ) 2 = β π S n ( 2 β t ) ,
f ( t ) = exp ( - c t 2 ) ,             R n ( τ ) = exp ( - b τ 2 ) , S n ( ω ) = π b exp ( - ω 2 / 4 b ) .
R ¯ n ( t ) ~ exp ( - b t 2 ) * exp ( j β t ) ~ exp ( - s t 2 ) ,             1 s - 1 b - 1 j β .
E { y ( t ) 2 } ~ exp ( - 2 c t 2 ) * [ exp ( - s t 2 ) exp ( - j β t 2 ) ] ~ exp ( - s 1 t 2 ) ,             1 s 1 = 1 2 c + 1 s + j β .
E { y ( t ) 2 } ~ exp ( - 2 c t 2 ) * exp ( - β 2 t 2 / b ) ~ exp ( s 2 t 2 ) ,             1 s 2 = 1 2 c + b β 2 .
H 1 ( ω ) = A 1 ( ω ) exp [ - j φ 1 ( ω ) ] ,             H 1 ( ω ) = 0             for             ω > ω c ,
f ( t ) F ( ω ) ,             F ( ω ) = 0             for             ω > ω 1 < ω c ,
H ( ω ) = H 1 ( ω - ω o ) exp ( - j φ o ) ,             ω > 0 ,             H ( - ω ) = H * ( ω ) .
g ( t ) = exp [ j ( ω o t - φ o ) ] g 1 ( t ) .
A 1 ( ω ) A 1 ( 0 ) ,             φ 1 ( ω ) φ 1 ( 0 ) ω + φ 1 ( 0 ) 2 ω 2             for             ω < ω c .
G 1 ( ω ) = F ( ω ) H 1 ( ω ) A 1 ( 0 ) F ( ω ) exp { - j [ φ 1 ( 0 ) ω + φ 1 ( 0 ) 2 ω 2 ] } .
g 1 ( t ) = A 1 ( 0 ) f ¯ ( t - t 1 ) ,             t 1 = φ 1 ( 0 ) ,             β = 1 / 2 φ 1 ( 0 ) .
g ( t ) = A ( ω o ) f ¯ ( t - t 1 ) exp [ j ω o ( t - t o ) ] ,             t o = φ o ω o = φ ( ω o ) ω o .
L ( ω , z ) = exp [ - k ( ω , z ) ] ,             k ( ω , z ) = α ( ω , z ) + j θ ( ω , z ) .
A ( ω ) = exp [ α ( ω , z ) ] ,             ϕ ( ω ) = θ ( ω , z ) .
L ( ω , z 1 + z 2 ) = L ( ω , z 1 ) L ( ω , z 2 ) .
k ( ω , z ) = k o ( ω ) z = α o ( ω ) a + j θ o ( ω ) z .
β = 1 2 2 θ ( ω o , z ) / ω 2 ,             t 1 = θ ( ω o , z ) ω ,             t o = θ ( ω o , z ) ω o .
g ( t ) ~ exp [ j ω o ( t - t o ) ] exp [ - s ( t - t 1 ) 2 ] ,             s = a - j b .
D g 2 = 1 4 a = 1 4 c + c θ o ( ω o ) z .
x ( t ) = f ( t ) n ( t ) exp ( j ω o t ) .
R x ( t 1 , t 2 ) = f ( t 1 ) f ( t 2 ) R n ( t 1 - t 2 ) ,
E { y ( t ) 2 } = A 2 ( ω o ) f 2 ( t - t 1 ) * [ R ¯ n ( t ) q β * ( t ) ] .
l β τ c 2 = 1 2 θ o ( ω o ) z ,
E { y ( t ) 2 } = A 2 ( ω o ) f 2 ( t - t 1 ) * β π S n ( 2 β t ) ,
E { y 2 ( t ) } = f 2 ( t ) * h 2 ( t ) .
β π A 2 ( ω o ) S n [ 2 β ( t - t 1 ) ] ,
g i ( P ) exp ( - j ω t ) ,             P : ( x , y , z ) ,
2 g x 2 + 2 g y 2 + 2 g z 2 + k 2 g = 0 ,             k = ω c = 2 π λ .
g ( x , y , 0 + ) = { g i ( x , y , 0 ) = f ( x , y ) ( x , y ) S 0 ( x , y ) S .
g ( x , y , z ) = S f ( ξ , η ) exp ( j k r ) j λ r d ξ d η , r = [ ( x - ξ ) 2 + ( y - η ) 2 + z 2 ] 1 / 2 ,
[ ( x - ξ ) 2 + ( y - η ) 2 ] 1 / 2 z ,             ( ξ , η ) S ,
r = z [ 1 + ( x - ξ ) 2 + ( y - η ) 2 z ] 1 / 2 z + ( x - ξ ) 2 + ( y - η ) 2 2 z
g ( x , y , z ) exp ( j k z ) j λ z S f ( ξ , η ) × exp [ j k [ ( x - ξ ) 2 + ( y - η ) 2 ] / 2 z ] d ξ d η .
w ( x , y , z ) = g ( x , y , z ) exp ( - j k z ) .
2 w x 2 + 2 w y 2 + 2 j k w z = 0 ,             k = 2 π λ .
2 w x 2 + 2 w y 2 + 2 w z 2 + 2 j k w z - k 2 w + k 2 w = 0.
1 j λ z exp [ j k ( x 2 + y 2 ) / 2 z ] × 1 j λ z - exp [ j k ( x 2 + y 2 ) / 2 z ] d x d y = 1
w ( x , z ) = g ( x , z ) exp ( - j k z ) .
2 w x 2 + 2 j k w z = 0 ,             k = 2 π λ .
1 j λ z exp ( j k x 2 / 2 z ) ,             1 j λ z - exp ( j k x 2 / 2 z ) d x = 1
w ( x , y ) = 1 j λ z - d d f ( ξ ) exp [ j k ( x - ξ ) 2 / 2 z ] d ξ ,
q β ( x ) = 1 j λ z exp ( j β x 2 ) ,             β = k 2 z = π λ z .
w ( x , z ) = f ¯ ( x ) ,
w ( x , z ) = f ¯ ( x ) = a - j b c exp ( - a x 2 ) exp ( j b x 2 ) ,             1 a = 1 c + 4 c k 2 z 2 .
w ( x , z ) = A 2 j { F c [ k 2 z ( x + d ) ] - F c [ k 2 z ( x - d ) ] } ,
w ( x , z ) = 1 j λ z F ( 2 π λ z x ) exp ( j k x 2 / 2 z ) .
g ( x , z ) = w ( x , z ) exp ( j k z ) = 1 j λ z F ( 2 π λ θ ) exp ( j ρ o x ) ,
ρ o = ( x 2 + z 2 ) 1 / 2 z + x 2 2 z ,             θ = x z .
g ( x , 0 + ) = g i ( x , 0 - ) t ( x ) .
t ( x ) = a ( x ) exp [ j φ ( x ) ] .
t ( x ) = exp [ - j φ ( x ) ] ,             φ ( x ) = k ( n - 1 ) q ( x ) ,             k = ω c .
φ ( x ) = ω Δ = ω c q ( x ) ( n - 1 ) ,
w ( x , z ) = [ f ( x ) t ( x ) ] * q β ( x ) ,             β = k 2 z .
t ( x ) = exp ( - j γ x 2 ) ,             γ = k 2 l .
q ( x ) = q 1 ( x ) + q 2 ( x ) x 2 2 ( 1 r 1 + 1 r 2 ) .
1 l = ( n - 1 ) ( 1 r 1 + 1 r 2 ) ,
w ( x , z ) = [ f ( x ) exp ( - j γ x 2 ) ] * β j π exp ( j β x 2 ) ,             β = k 2 z .
w ( x , z ) = [ f ( x ) exp ( - j γ x 2 ) ] * β j π exp ( j β x 2 )
w ( x , l ) = q γ ( x ) F ( 2 γ x ) ,             w ( x , l ) = 1 λ l F ( 2 γ x ) e .
w ( x , l ) = 1 j λ l F ( 2 γ x ) .
W ( ω , l ) = - j λ l f ( - ω 2 γ ) .
G ( ω ) = - g ( x ) exp ( - j ω x ) d x
G ( ω ) = - F ( - ω ) T ( - ω ) ,
p ( x , 0 - ) = 1 j λ l F ( 2 γ x ) ,             p ( x , 0 + ) = p ( x , 0 - ) T ( 2 γ x ) .
G ( ω ) = - j λ l p ( - ω 2 γ , 0 + ) = - j λ l 1 j λ l F ( - 2 γ ω 2 γ ) T ( - 2 γ ω 2 γ ) ,
f ( x ) = exp [ j α s ( x ) ] = 1 + j α s ( x ) .
F ( ω ) = 2 π δ ( ω ) + j α S ( ω ) .
T ( u ) = { j u < Δ 1 u > Δ .
G ( ω ) = [ 2 π δ ( ω ) + j α S ( ω ) ] T ( ω ) 2 π j δ ( ω ) + j α S ( ω ) ,
g ( x ) j [ 1 + α s ( x ) ] ,             g ( x ) 2 1 + 2 α s ( x ) .

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