Abstract

Several theorems are formulated, regarding symmetry relations between two monochromatic fields that propagate either into the same half-space (z > 0) or into two complementary half-spaces (z > 0 and z < 0) and that satisfy one of two simple phase-conjugacy conditions in a cross sectional plane z = constant. The theorems are rigorously valid for fields whose two-dimensional spatial-frequency spectrum in the cross sectional plane is bandlimited to a circle of radius equal to the wave number of the field. One of the theorems elucidates some recently predicted symmetry properties of focused fields.

© 1980 Optical Society of America

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References

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  1. For reviews of this subject see, for example, A. Yariv, “Phase Conjugate Optics and Real-Time Holography,” IEEE J. Quantum Electron. QE-14, 650–660 (1978)or J. AuYeung and A. Yariv, “Phase-Conjugate Optics,” Opt. News,  5, 13–17 (1979).
    [CrossRef]
  2. E. Collett and E. Wolf, “Symmetry Properties of Focused Fields,” Opt. Lett.,  5, 264–266 (1980).
    [CrossRef] [PubMed]
  3. Sufficiency conditions are discussed in papers by (a)W. D. Montgomery, “Algebraic Formulation of Diffraction Applied to Self Imaging,” J. Opt. Soc. Am. 58, 1112–1124 (1968);(b)E. Lalor, “Conditions for the Validity of the Angular Spectrum of Plane Waves,” J. Opt. Soc. Am. 58, 1235–1237 (1968);(c)G. C. Sherman, “Diffracted Wavefields Expressible by Plane-Wave Expansions Containing Only Homogeneous Waves,” J. Opt. Soc. Am. 59, 697–711 (1969).
  4. Although not explicity stated, this theorem is implicit in the analysis of J. R. Shewell and E. Wolf, “Inverse Diffraction and a New Reciprocity Theorem,” J. Opt. Soc. Am. 58, 1596–1603 (1968), Sec. III.
    [CrossRef]
  5. Ref. 3(c) above. See also G. C. Sherman, “Diffracted Wavefields Expressible by Plane-Wave Expansions Containing Only Homogeneous Waves,” Phys. Rev. Lett.,  21, 761–764 (1968);ibid. 21, 1220(E) (1968).
    [CrossRef]
  6. An example is provided by a Gaussian laser beam. If the spot size of the beam is w0, the angular spread of the beam is well known to be given by Θ = λ/πw0, where λ is the wavelength. With w0 ≫ λ, as is always the case in practice, the effective values of p and q are then restricted to the domainp2+q2⩽sin2Θ≈Θ2=(λ/πw0)2≪1,as may readily be established with the help of the asymptotic formula (1.15a). Now, as is clear from (1.8), there is a one to one correspondence between the directional parameters (p,q) and the spatial frequencies (u,υ), namely, p = u/k, q = υ/k. Hence the above inequality implies that in any cross section perpendicular to the beam axis, the beam field is effectively bandlimited to the spatial frequency domainu2+υ2=k2(λ/πw0)2≪k2.Typically, the ratio λ/πw0 may be of the order of 10−4.
  7. Theorem II, p. 701 of Ref. 3(c) above.
  8. It is not difficult to show that if the bandlimited wavefield U(x,y,z) is defined throughout the whole space, it is bandlimited to the spatial-frequency domain u2+υ2⩽k2 in any planar cross section, whether or not the cross section is perpendicular to the z direction.
  9. K. Miyamoto and E. Wolf, “Generalization of the Maggi-Rubinowicz Theory of the Boundary Diffraction Wave—Part I,” J. Opt. Soc. Am,  52, 615–625 (1962), Appendix.
    [CrossRef]
  10. M. Born and E. Wolf, Principles of Optics, 5th Ed. (Pergamon, Oxford, 1975), Sec. 8.8.4.
  11. A strictly analogous result to that stated in Footnote 8 then applies: The“extended” wavefield V(x,y,z) may be shown to be bandlimited to the spatial-frequency domain u2+υ2⩽k2 in any planar cross section.
  12. W. Lukosz, “Equivalent-Lens Theory of Holographic Imaging,” J. Opt. Soc. Am. 58, 1084–1091 (1968), Sec. III.Our method of proof is similar to that given by Lukosz but it is free of an inaccuracy contained in his derivation. Lukosz did not impose any restriction regarding bandlimitation; however, it is not difficult to see that the theorem then no longer holds. In fact, the angular spectrum integral then diverges in the half-space z < 0.
    [CrossRef]
  13. R. Mittra and P. L. Ransom, “Imaging with Coherent Fields,” in Modern Optics, edited by J. Fox, (Polytechnic, Brooklyn, 1967;Wiley, New York, distr.),pp. 619–647.
  14. P. Debye, “Das Verhalten von Lichtwellen in der Nähe eines Brennpunktes oder einer Brennlinie,” Ann. Phys. 30, 755–776 (1909).
    [CrossRef]

1980 (1)

1978 (1)

For reviews of this subject see, for example, A. Yariv, “Phase Conjugate Optics and Real-Time Holography,” IEEE J. Quantum Electron. QE-14, 650–660 (1978)or J. AuYeung and A. Yariv, “Phase-Conjugate Optics,” Opt. News,  5, 13–17 (1979).
[CrossRef]

1968 (4)

1962 (1)

K. Miyamoto and E. Wolf, “Generalization of the Maggi-Rubinowicz Theory of the Boundary Diffraction Wave—Part I,” J. Opt. Soc. Am,  52, 615–625 (1962), Appendix.
[CrossRef]

1909 (1)

P. Debye, “Das Verhalten von Lichtwellen in der Nähe eines Brennpunktes oder einer Brennlinie,” Ann. Phys. 30, 755–776 (1909).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 5th Ed. (Pergamon, Oxford, 1975), Sec. 8.8.4.

Collett, E.

Debye, P.

P. Debye, “Das Verhalten von Lichtwellen in der Nähe eines Brennpunktes oder einer Brennlinie,” Ann. Phys. 30, 755–776 (1909).
[CrossRef]

Lukosz, W.

Mittra, R.

R. Mittra and P. L. Ransom, “Imaging with Coherent Fields,” in Modern Optics, edited by J. Fox, (Polytechnic, Brooklyn, 1967;Wiley, New York, distr.),pp. 619–647.

Miyamoto, K.

K. Miyamoto and E. Wolf, “Generalization of the Maggi-Rubinowicz Theory of the Boundary Diffraction Wave—Part I,” J. Opt. Soc. Am,  52, 615–625 (1962), Appendix.
[CrossRef]

Montgomery, W. D.

Ransom, P. L.

R. Mittra and P. L. Ransom, “Imaging with Coherent Fields,” in Modern Optics, edited by J. Fox, (Polytechnic, Brooklyn, 1967;Wiley, New York, distr.),pp. 619–647.

Sherman, G. C.

Ref. 3(c) above. See also G. C. Sherman, “Diffracted Wavefields Expressible by Plane-Wave Expansions Containing Only Homogeneous Waves,” Phys. Rev. Lett.,  21, 761–764 (1968);ibid. 21, 1220(E) (1968).
[CrossRef]

Shewell, J. R.

Wolf, E.

E. Collett and E. Wolf, “Symmetry Properties of Focused Fields,” Opt. Lett.,  5, 264–266 (1980).
[CrossRef] [PubMed]

Although not explicity stated, this theorem is implicit in the analysis of J. R. Shewell and E. Wolf, “Inverse Diffraction and a New Reciprocity Theorem,” J. Opt. Soc. Am. 58, 1596–1603 (1968), Sec. III.
[CrossRef]

K. Miyamoto and E. Wolf, “Generalization of the Maggi-Rubinowicz Theory of the Boundary Diffraction Wave—Part I,” J. Opt. Soc. Am,  52, 615–625 (1962), Appendix.
[CrossRef]

M. Born and E. Wolf, Principles of Optics, 5th Ed. (Pergamon, Oxford, 1975), Sec. 8.8.4.

Yariv, A.

For reviews of this subject see, for example, A. Yariv, “Phase Conjugate Optics and Real-Time Holography,” IEEE J. Quantum Electron. QE-14, 650–660 (1978)or J. AuYeung and A. Yariv, “Phase-Conjugate Optics,” Opt. News,  5, 13–17 (1979).
[CrossRef]

Ann. Phys. (1)

P. Debye, “Das Verhalten von Lichtwellen in der Nähe eines Brennpunktes oder einer Brennlinie,” Ann. Phys. 30, 755–776 (1909).
[CrossRef]

IEEE J. Quantum Electron. (1)

For reviews of this subject see, for example, A. Yariv, “Phase Conjugate Optics and Real-Time Holography,” IEEE J. Quantum Electron. QE-14, 650–660 (1978)or J. AuYeung and A. Yariv, “Phase-Conjugate Optics,” Opt. News,  5, 13–17 (1979).
[CrossRef]

J. Opt. Soc. Am (1)

K. Miyamoto and E. Wolf, “Generalization of the Maggi-Rubinowicz Theory of the Boundary Diffraction Wave—Part I,” J. Opt. Soc. Am,  52, 615–625 (1962), Appendix.
[CrossRef]

J. Opt. Soc. Am. (3)

Opt. Lett. (1)

Phys. Rev. Lett. (1)

Ref. 3(c) above. See also G. C. Sherman, “Diffracted Wavefields Expressible by Plane-Wave Expansions Containing Only Homogeneous Waves,” Phys. Rev. Lett.,  21, 761–764 (1968);ibid. 21, 1220(E) (1968).
[CrossRef]

Other (6)

An example is provided by a Gaussian laser beam. If the spot size of the beam is w0, the angular spread of the beam is well known to be given by Θ = λ/πw0, where λ is the wavelength. With w0 ≫ λ, as is always the case in practice, the effective values of p and q are then restricted to the domainp2+q2⩽sin2Θ≈Θ2=(λ/πw0)2≪1,as may readily be established with the help of the asymptotic formula (1.15a). Now, as is clear from (1.8), there is a one to one correspondence between the directional parameters (p,q) and the spatial frequencies (u,υ), namely, p = u/k, q = υ/k. Hence the above inequality implies that in any cross section perpendicular to the beam axis, the beam field is effectively bandlimited to the spatial frequency domainu2+υ2=k2(λ/πw0)2≪k2.Typically, the ratio λ/πw0 may be of the order of 10−4.

Theorem II, p. 701 of Ref. 3(c) above.

It is not difficult to show that if the bandlimited wavefield U(x,y,z) is defined throughout the whole space, it is bandlimited to the spatial-frequency domain u2+υ2⩽k2 in any planar cross section, whether or not the cross section is perpendicular to the z direction.

M. Born and E. Wolf, Principles of Optics, 5th Ed. (Pergamon, Oxford, 1975), Sec. 8.8.4.

A strictly analogous result to that stated in Footnote 8 then applies: The“extended” wavefield V(x,y,z) may be shown to be bandlimited to the spatial-frequency domain u2+υ2⩽k2 in any planar cross section.

R. Mittra and P. L. Ransom, “Imaging with Coherent Fields,” in Modern Optics, edited by J. Fox, (Polytechnic, Brooklyn, 1967;Wiley, New York, distr.),pp. 619–647.

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Figures (9)

FIG. 1
FIG. 1

Illustrating the asymptotic behavior of the spatially bandlimited wavefields U(x,y,z) [Eq. (1.14)] and V(x,y,z) [Eq. (1.17)]. (a) As kR → ∞ in fixed directions, U(x,y,z) behaves as a diverging wave in the half-space z > 0 and as a converging wave in the half-space z < 0. The values of the field at points P and P+ that are diametrically opposite to each other with respect to the origin are given by Eqs. (1.15a) and (1.15b), respectively. (b) As kR → ∞ in fixed directions, V(x,y,z) behaves as a diverging wave in the half-space z < 0 and as a converging wave in the half-space z > 0. The values of the fields at points Q and Q+ that are diametrically opposite to each other with respect to the origin are given by Eqs. (1.18a) and (1.18b), respectively.

FIG. 2
FIG. 2

Illustrating Theorems III and IV: (a) If U(1) and U(2) are complex conjugates of each other at every point P in a plane z = z0, then they are also complex conjugates of each other at all pairs of points C and C+ that possess reflection symmetry with respect to the plane z = z0 [Eqs. (2.2) and (2.3)]. (b) If U(1) and U(2) are complex conjugates of each other at all pairs of points P1 and P2—in the plane z = z0—that possess inversion symmetry with respect to the axial point x = y = 0, z = z0 they are complex conjugates of each other at all pairs of points D and D+ that possess inversion symmetry with respect to that point [Eqs. (2.10) and (2.11)].

FIG. 3
FIG. 3

Illustrating Theorem V: If U(1) and U(2) are both rotationally symmetric about the axial point in some plane z = z0 and are, at each point P in that plane, complex conjugates of each other, then the two fields are necessarily rotationally symmetric about the z axis; and moreover, they are also complex conjugates of each other at all pairs of points E and E+ that possess reflection symmetry with respect to the plane z = z0 [Eqs. (2.19) and (2.20)].

FIG. 4
FIG. 4

Illustrating Theorems VI and VII: (a) If U and V are complex conjugates of each other at every point P in the plane z = z0, they are complex conjugates of each other at all points F [Eqs. (3.2) and (3.3)]. (b) If U and V are complex conjugates of each other at all pairs of points P1 and P2—located in a place z = z0—that possess inversion symmetry with respect to the axial point x = y = 0, z = z0, then they are also complex conjugates of each other at all pairs of points G+ and G—in every plane z = ζ—that possess inversion symmetry with respect to the axial point x = y = 0, z = ζ [Eqs. (3.10) and (3.11)].

FIG. 5
FIG. 5

Illustrating Theorem VIII: If U and V are both rotationally symmetric about the axial point in some plane z = z0 and are at each point P in that plane complex conjugates of each other, then the two fields are necessarily rotationally symmetric about the z axis; and moreover, they are also complex conjugates of each other at all points H [Eqs. (3.17) and (3.18)].

FIG. 6
FIG. 6

Notation used in Eqs. (4.1) and (4.2), relating to the diffraction of a uniform converging monochromatic spherical wave at an aperture.

FIG. 7
FIG. 7

Illustrating the relation (4.5), (first derived in Ref. 2), for focused fields. The values of the field at points K and K+ that have inversion symmetry with respect to the geometrical focus O are related by U(K) = −[U(K+)], [Eq.(4.5)].

FIG. 8
FIG. 8

Isophotes [contours of the intensity] in a meridional plane in the neighborhood of the focus of a uniform, converging, monochromatic spherical wave diffracted at a circular aperture. The intensity is normalized to unity at the focus. The dotted lines represent the boundary of the geometrical shadow, making an angle θ with the z axis (normal to the aperture plane). The coordinates u and v are u = kzsin2 θ, v = k x 2 + y 2 sin Θ (k = wave number). [Adapted from E. H. Linfoot and E. Wolf, “Phase distribution near focus in an aberration-free diffraction image,” Proc. Phys. Soc. B 69, 823–832 (1956)].

FIG. 9
FIG. 9

Profiles of the surfaces of constant phase in a meridional plane in the neighborhood of the geometrical focal plane of a uniform, converging, monochromatic spherical wave diffracted at a circular aperture. The angle of convergence is that appropriate to an f/3.5 pencil of rays. [Adapted from E. H. Linfoot and E. Wolf, “Phase distribution near focus in an aberration-free diffraction image,” Proc. Phys. Soc. B 69, 823–832 (1956)].

Equations (85)

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u ( x , y , z , t ) = U ( x , y , z ) e i ω t ,
2 U + k 2 U = 0 ,
k = ω / c ,
U ( s x R , s y R , s z R ) ~ a ( s x , s y ) e i k R / R
U ( x , y , z ) = A ( p , q ) e i k ( p x + q y + m z ) d p d q ,
m = + 1 p 2 q 2 when p 2 + q 2 1
= + i p 2 + q 2 1 when p 2 + q 2 > 1
Ũ ( u , υ ; z ) = 1 ( 2 π ) 2 U ( x , y , z ) e i ( u x + υ y ) d x d y
A ( u / k , υ / k ) = ( 1 / k 2 ) Ũ ( u , υ ; z ) e i w z ,
w = + k 2 u 2 υ 2 when u 2 + υ 2 k 2
+ i u 2 + υ 2 k 2 when u 2 + υ 2 > k 2 .
u 2 + υ 2 k 2 ,
Ũ ( u , υ ; z 0 ) = 0 when u 2 + υ 2 > k 2 .
A ( p , q ) = 0 when p 2 + q 2 > 1
Ũ ( u , υ ; z ) = 0 when u 2 + υ 2 > k 2 .
U ( x , y , z ) = p 2 + q 2 1 A ( p , q ) e i k ( p x + q y + m z ) d p d q .
U ( s x R , s y R , s z R ) ~ 2 π i k s z A ( s x , s y ) e i k R R as k R ,
U ( s x R , s y R , s z R ) ~ + 2 π i k s z A ( s x , s y ) e i k R R as k R .
V ( x , y , z ; t ) = V ( x , y , z ) e i ω t .
V ( x , y , z ) = p 2 + q 2 1 B ( p , q ) e i k ( p x + q y m z ) d p d q .
V ( s x R , s y R , s z R ) ~ 2 π i k s z B ( s x , s y ) e i k R R as k R ,
V ( s x R , s y R , s z R ) ~ 2 π i k s z B ( s x , s y ) e i k R R as k R .
U ( 1 ) ( x , y , z ) e i ω t , U ( 2 ) ( x , y , z ) e i ω t ,
U ( 2 ) ( x , y , z 0 ) = [ U ( 1 ) ( x , y , z 0 ) ] *
U ( 2 ) ( x , y , z 0 + d ) = [ U ( 1 ) ( x , y , z 0 d ) ] * .
U ( j ) ( x , y , z ) = p 2 + q 2 1 A ( j ) ( p , q ) e i k ( p x + q y + m z ) d p d q , ( j = 1 , 2 ) ,
m = + 1 p 2 q 2 .
p 2 + q 2 1 A ( 2 ) ( p , q ) e i k ( p x + q y + m z 0 ) d p d q = p 2 + q 2 1 [ A ( 1 ) ( p , q ) ] * e i k ( p x + q y + m z 0 ) d p d q .
A ( 2 ) ( p , q ) e i k m z 0 = [ A ( 1 ) ( p , q ) ] * e i k m z 0 .
U ( 2 ) ( x , y , z 0 + d ) = p 2 + q 2 1 [ A ( 1 ) ( p , q ) ] * e 2 i k m z 0 e i k [ p z + q y + m ( z 0 + d ) ] d p d q ,
U ( 2 ) ( x , y , z 0 + d ) = p 2 + q 2 1 [ A ( 1 ) ( p , q ) ] * e i k [ p z + q y + m ( z 0 d ) ] d p d q .
U ( 2 ) ( x , y , z 0 + d ) = [ U ( 1 ) ( x , y , z 0 d ) ] * ,
U ( 2 ) ( x , y , z 0 ) = [ U ( 1 ) ( x , y , z 0 ) ] *
U ( 2 ) ( x , y , z 0 + d ) = [ U ( 1 ) ( x , y , z 0 d ) ] * .
A ( 2 ) ( p , q ) e i k m z 0 = [ A ( 1 ) ( p , q ) ] * e i k m z 0 .
U ( j ) ( x , y , z 0 ) = U ( j ) ( r , z 0 ) , ( j = 1 , 2 ) ,
r = x 2 + y 2 .
U ( j ) ( x , y , z ) = U ( j ) ( r , z ) , ( j = 1 , 2 ) ,
A ( j ) ( p , q ) = A ( j ) ( ρ ) , ( j = 1 , 2 ) ,
ρ = p 2 + q 2 1 .
U ( 1 ) ( x , y , z ) e i ω t , U ( 2 ) ( x , y , z ) e i ω t
U ( 2 ) ( r , z 0 ) = [ U ( 1 ) ( r , z 0 ) ] * ,
U ( 2 ) ( r , z 0 + d ) = [ U ( 1 ) ( r , z 0 d ) ] * .
A ( 2 ) ( ρ ) e i k m z 0 = [ A ( 1 ) ( ρ ) ] * e i k m z 0 ,
U ( x , y , z ) e i ω t , V ( x , y , z ) e i ω t
V ( x , y , z 0 ) = [ U ( x , y , z 0 ) ] *
V ( x , y , z ) = [ U ( x , y , z ) ] * .
U ( x , y , z ) = p 2 + q 2 1 A ( p , q ) e i k ( p x + q y + m z ) d p d q ,
V ( x , y , z ) = p 2 + q 2 1 B ( p , q ) e i k ( p x + q y m z ) d p d q ,
m = + 1 p 2 q 2 .
p 2 + q 2 1 B ( p , q ) e i k ( p x + q y m z 0 ) d p d q = p 2 + q 2 1 [ A ( p , q ) ] * e i k ( p x + q y + m z 0 ) d p d q .
B ( p , q ) = [ A ( p , q ) ] * .
V ( x , y , z ) = [ A ( p , q ) ] * e i k ( p x + q y m z ) d p d q
V ( x , y , z ) = [ A ( p , q ) ] * e i k ( p x + q y + m z ) d p d q .
V ( x , y , z ) = [ U ( x , y , z ) ] * ,
V ( x , y , z 0 ) = [ U ( x , y , z 0 ) ] *
V ( x , y , z ) = [ U ( x , y , z ) ] * .
B ( p , q ) = [ A ( p , q ) ] * .
U ( x , y , z 0 ) = U ( r , z 0 ) , V ( x , y , z 0 ) = V ( r , z 0 ) ,
U ( x , y , z ) = U ( r , z ) , V ( x , y , z ) = V ( r , z )
A ( p , q ) = A ( ρ ) , B ( p , q ) = B ( ρ ) ,
U ( x , y , z ) e i ω t , V ( x , y , z ) e i ω t
V ( r , z 0 ) = [ U ( r , z 0 ) ] * ,
V ( r , z ) = [ U ( r , z ) ] * .
B ( ρ ) = [ A ( ρ ) ] * .
U ( R ŝ ) = a ( ŝ ) e i k R R ,
U ( x , y , z ) = i k 2 π Ω a ( s x , s y ) s z e i k ( s x x + s y y + s z z ) d s x d s y ,
U ( x , y , 0 ) = i k 2 π Ω a ( s z , s y ) s z e i k ( s x x + s y y ) d s x d s y .
U ( x , y , 0 ) = [ U ( x , y , 0 ) ] * .
U ( x , y , z ) = [ U ( x , y , z ) ] * .
| U ( x , y , z ) | = | U ( x , y , z ) | ,
ϕ ( x , y , z ) = ϕ ( x , y , z ) π , ( mod 2 π ) .
r = x 2 + y 2 .
| U ( r , z ) | = | U ( r , z ) | ,
ϕ ( r , z ) = ϕ ( r , z ) π , ( mod 2 π ) .
U ( x , y , z ) = U ( r , z ) ,
r = x 2 + y 2 .
A ( p , q ) = A ( ρ ) ,
ρ = p 2 + q 2 1 .
U ( x , y , z 0 ) = U ( r , z 0 ) ,
V ( x , y , z ) = V ( r , z ) ,
B ( p , q ) = B ( ρ ) ,
V ( x , y , z 0 ) = V ( r , z 0 ) ,
p2+q2sin2ΘΘ2=(λ/πw0)21,
u2+υ2=k2(λ/πw0)2k2.