Abstract

In 1955 Wolf noticed that the mutual coherence function Γ obeys two wave equations [Proc. R. Soc. London 230, 246 (1955)]. The physical optics of this finding is thoroughly presented in Mandel and Wolf’s recent monograph [Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK1995)]. We discuss those coherence waves in the spirit of optical signal processing. The term coherence waves is justified if one accepts the following paradigm: A wave is whatever obeys the wave equation.

© 1999 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995), Chap. 4, pp. 147–228.
  2. E. Wolf, “A macroscopic theory of interference and diffraction of light from finite sources. II. Fields with a spectral range of arbitrary width,” Proc. R. Soc. London 230, 246–265 (1955).
    [CrossRef]
  3. L. Mandel, E. Wolf, “Coherence properties of optical fields,” Rev. Mod. Phys. 37, 231–287 (1965).
    [CrossRef]
  4. L. Mandel, E. Wolf, “Spectral coherence and the concept of cross-spectral purity,” J. Opt. Soc. Am. 66, 529–535 (1976).
    [CrossRef]
  5. E. Wolf, “Coherence and radiometry,” J. Opt. Soc. Am. 68, 6–17 (1978).
    [CrossRef]
  6. M. Born, E. Wolf, Principles of Optics, Electromagnetic Theory of Propagation, Interference and Diffraction (Pergamon, New York, 1980), Chap. 10, pp. 491–555.
  7. M. F. Erden, H. M. Ozaktas, D. Mendlovic, “Synthesis of mutual intensity distribution using the fractional Fourier transform,” Opt. Commun. 125, 288–301 (1996).
    [CrossRef]
  8. M. F. Erden, H. M. Ozaktas, D. Mendlovic, “Propagation of mutual intensity expressed in terms of the fractional Fourier transform,” J. Opt. Soc. Am. A 13, 1068–1071 (1996).
    [CrossRef]
  9. M. J. Beran, G. R. Parrent, Theory of Partial Coherence (Prentice-Hall, Englewood Cliffs, N.J., 1964), Chaps. 3 and 4, pp. 36–64.
  10. J. J. McCoy, M. J. Beran, “Propagation of beamed signals through inhomogeneous media: a diffraction theory,” J. Acoust. Soc. Am. 59, 1142–1149 (1976).
    [CrossRef]
  11. M. J. Beran, “Propagation of a finite beam in a random medium,” J. Opt. Soc. Am. 60, 518–521 (1970).
    [CrossRef]
  12. A. W. Lohmann, D. Mendlovic, Z. Zalevsky, G. Shabtay, “The use of Ewald’s surfaces in triple correlation optics,” Opt. Commun. 144, 170–172 (1997).
    [CrossRef]
  13. A. S. Marathay, Elements of Optical Coherence Theory (Wiley, New York, 1982), Chaps. 4 and 5, pp. 60–181.
  14. A. S. Marathay, “Noncoherent-object hologram: its reconstruction and optical processing,” J. Opt. Soc. Am. A 4, 1861–1868 (1987).
    [CrossRef]
  15. C. Iaconis, I. A. Walmsley, “Direct measurement of the two-point field correlation function,” Opt. Lett. 21, 1783–1785 (1996).
    [CrossRef] [PubMed]
  16. J. Tu, S. Tamura, “Analytic relation for recovering the mutual intensity by means of intensity information,” J. Opt. Soc. Am. A 15, 202–206 (1998).
    [CrossRef]
  17. D. Mendlovic, G. Shabtay, A. W. Lohmann, N. Konforti, “Display of spatial coherence,” Opt. Lett. 23, 1084–1086 (1998).
    [CrossRef]

1998

1997

A. W. Lohmann, D. Mendlovic, Z. Zalevsky, G. Shabtay, “The use of Ewald’s surfaces in triple correlation optics,” Opt. Commun. 144, 170–172 (1997).
[CrossRef]

1996

1987

1978

1976

L. Mandel, E. Wolf, “Spectral coherence and the concept of cross-spectral purity,” J. Opt. Soc. Am. 66, 529–535 (1976).
[CrossRef]

J. J. McCoy, M. J. Beran, “Propagation of beamed signals through inhomogeneous media: a diffraction theory,” J. Acoust. Soc. Am. 59, 1142–1149 (1976).
[CrossRef]

1970

1965

L. Mandel, E. Wolf, “Coherence properties of optical fields,” Rev. Mod. Phys. 37, 231–287 (1965).
[CrossRef]

1955

E. Wolf, “A macroscopic theory of interference and diffraction of light from finite sources. II. Fields with a spectral range of arbitrary width,” Proc. R. Soc. London 230, 246–265 (1955).
[CrossRef]

Beran, M. J.

J. J. McCoy, M. J. Beran, “Propagation of beamed signals through inhomogeneous media: a diffraction theory,” J. Acoust. Soc. Am. 59, 1142–1149 (1976).
[CrossRef]

M. J. Beran, “Propagation of a finite beam in a random medium,” J. Opt. Soc. Am. 60, 518–521 (1970).
[CrossRef]

M. J. Beran, G. R. Parrent, Theory of Partial Coherence (Prentice-Hall, Englewood Cliffs, N.J., 1964), Chaps. 3 and 4, pp. 36–64.

Born, M.

M. Born, E. Wolf, Principles of Optics, Electromagnetic Theory of Propagation, Interference and Diffraction (Pergamon, New York, 1980), Chap. 10, pp. 491–555.

Erden, M. F.

M. F. Erden, H. M. Ozaktas, D. Mendlovic, “Synthesis of mutual intensity distribution using the fractional Fourier transform,” Opt. Commun. 125, 288–301 (1996).
[CrossRef]

M. F. Erden, H. M. Ozaktas, D. Mendlovic, “Propagation of mutual intensity expressed in terms of the fractional Fourier transform,” J. Opt. Soc. Am. A 13, 1068–1071 (1996).
[CrossRef]

Iaconis, C.

Konforti, N.

Lohmann, A. W.

D. Mendlovic, G. Shabtay, A. W. Lohmann, N. Konforti, “Display of spatial coherence,” Opt. Lett. 23, 1084–1086 (1998).
[CrossRef]

A. W. Lohmann, D. Mendlovic, Z. Zalevsky, G. Shabtay, “The use of Ewald’s surfaces in triple correlation optics,” Opt. Commun. 144, 170–172 (1997).
[CrossRef]

Mandel, L.

L. Mandel, E. Wolf, “Spectral coherence and the concept of cross-spectral purity,” J. Opt. Soc. Am. 66, 529–535 (1976).
[CrossRef]

L. Mandel, E. Wolf, “Coherence properties of optical fields,” Rev. Mod. Phys. 37, 231–287 (1965).
[CrossRef]

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995), Chap. 4, pp. 147–228.

Marathay, A. S.

A. S. Marathay, “Noncoherent-object hologram: its reconstruction and optical processing,” J. Opt. Soc. Am. A 4, 1861–1868 (1987).
[CrossRef]

A. S. Marathay, Elements of Optical Coherence Theory (Wiley, New York, 1982), Chaps. 4 and 5, pp. 60–181.

McCoy, J. J.

J. J. McCoy, M. J. Beran, “Propagation of beamed signals through inhomogeneous media: a diffraction theory,” J. Acoust. Soc. Am. 59, 1142–1149 (1976).
[CrossRef]

Mendlovic, D.

D. Mendlovic, G. Shabtay, A. W. Lohmann, N. Konforti, “Display of spatial coherence,” Opt. Lett. 23, 1084–1086 (1998).
[CrossRef]

A. W. Lohmann, D. Mendlovic, Z. Zalevsky, G. Shabtay, “The use of Ewald’s surfaces in triple correlation optics,” Opt. Commun. 144, 170–172 (1997).
[CrossRef]

M. F. Erden, H. M. Ozaktas, D. Mendlovic, “Synthesis of mutual intensity distribution using the fractional Fourier transform,” Opt. Commun. 125, 288–301 (1996).
[CrossRef]

M. F. Erden, H. M. Ozaktas, D. Mendlovic, “Propagation of mutual intensity expressed in terms of the fractional Fourier transform,” J. Opt. Soc. Am. A 13, 1068–1071 (1996).
[CrossRef]

Ozaktas, H. M.

M. F. Erden, H. M. Ozaktas, D. Mendlovic, “Propagation of mutual intensity expressed in terms of the fractional Fourier transform,” J. Opt. Soc. Am. A 13, 1068–1071 (1996).
[CrossRef]

M. F. Erden, H. M. Ozaktas, D. Mendlovic, “Synthesis of mutual intensity distribution using the fractional Fourier transform,” Opt. Commun. 125, 288–301 (1996).
[CrossRef]

Parrent, G. R.

M. J. Beran, G. R. Parrent, Theory of Partial Coherence (Prentice-Hall, Englewood Cliffs, N.J., 1964), Chaps. 3 and 4, pp. 36–64.

Shabtay, G.

D. Mendlovic, G. Shabtay, A. W. Lohmann, N. Konforti, “Display of spatial coherence,” Opt. Lett. 23, 1084–1086 (1998).
[CrossRef]

A. W. Lohmann, D. Mendlovic, Z. Zalevsky, G. Shabtay, “The use of Ewald’s surfaces in triple correlation optics,” Opt. Commun. 144, 170–172 (1997).
[CrossRef]

Tamura, S.

Tu, J.

Walmsley, I. A.

Wolf, E.

E. Wolf, “Coherence and radiometry,” J. Opt. Soc. Am. 68, 6–17 (1978).
[CrossRef]

L. Mandel, E. Wolf, “Spectral coherence and the concept of cross-spectral purity,” J. Opt. Soc. Am. 66, 529–535 (1976).
[CrossRef]

L. Mandel, E. Wolf, “Coherence properties of optical fields,” Rev. Mod. Phys. 37, 231–287 (1965).
[CrossRef]

E. Wolf, “A macroscopic theory of interference and diffraction of light from finite sources. II. Fields with a spectral range of arbitrary width,” Proc. R. Soc. London 230, 246–265 (1955).
[CrossRef]

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995), Chap. 4, pp. 147–228.

M. Born, E. Wolf, Principles of Optics, Electromagnetic Theory of Propagation, Interference and Diffraction (Pergamon, New York, 1980), Chap. 10, pp. 491–555.

Zalevsky, Z.

A. W. Lohmann, D. Mendlovic, Z. Zalevsky, G. Shabtay, “The use of Ewald’s surfaces in triple correlation optics,” Opt. Commun. 144, 170–172 (1997).
[CrossRef]

J. Acoust. Soc. Am.

J. J. McCoy, M. J. Beran, “Propagation of beamed signals through inhomogeneous media: a diffraction theory,” J. Acoust. Soc. Am. 59, 1142–1149 (1976).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

A. W. Lohmann, D. Mendlovic, Z. Zalevsky, G. Shabtay, “The use of Ewald’s surfaces in triple correlation optics,” Opt. Commun. 144, 170–172 (1997).
[CrossRef]

M. F. Erden, H. M. Ozaktas, D. Mendlovic, “Synthesis of mutual intensity distribution using the fractional Fourier transform,” Opt. Commun. 125, 288–301 (1996).
[CrossRef]

Opt. Lett.

Proc. R. Soc. London

E. Wolf, “A macroscopic theory of interference and diffraction of light from finite sources. II. Fields with a spectral range of arbitrary width,” Proc. R. Soc. London 230, 246–265 (1955).
[CrossRef]

Rev. Mod. Phys.

L. Mandel, E. Wolf, “Coherence properties of optical fields,” Rev. Mod. Phys. 37, 231–287 (1965).
[CrossRef]

Other

M. Born, E. Wolf, Principles of Optics, Electromagnetic Theory of Propagation, Interference and Diffraction (Pergamon, New York, 1980), Chap. 10, pp. 491–555.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995), Chap. 4, pp. 147–228.

M. J. Beran, G. R. Parrent, Theory of Partial Coherence (Prentice-Hall, Englewood Cliffs, N.J., 1964), Chaps. 3 and 4, pp. 36–64.

A. S. Marathay, Elements of Optical Coherence Theory (Wiley, New York, 1982), Chaps. 4 and 5, pp. 60–181.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1
Fig. 1

Gabor’s puzzle: Why doesn’t the photo capture the information traveling from the object to the camera?

Fig. 2
Fig. 2

Ewald’s sphere corresponding to the Helmholtz equation.

Fig. 3
Fig. 3

Ewald’s sphere related to the rigorous paraxial wave equation.

Fig. 4
Fig. 4

Ewald’s paraboloid for the approximated paraxial wave equation.

Equations (76)

Equations on this page are rendered with MathJax. Learn more.

Ex(R, t)=Re[V(R, t)],R=(x, y, z),
Γ(R1, R2, t1, t2)=V(R1, t1)V*(R2, t2).
Δ1Γ-(1/c2) 2Γt12=0,
Δ2Γ-(1/c2) 2Γt22=0.
Δ1=2x12+2y12+2z12.
V(R, t)=U(R, t)exp(-iωt),
Γ(R1, R2, t1, t2)=U(R1, t1)U*(R2, t2)×exp[-iω(t1-t2)],
Γ(R1, R2, t, t)=U(R1, t)U*(R2, t).
Δ1V(R1, t1)-(1/c2) 2V(R1, t1)t12=0,
Δ2V(R2, t2)-(1/c2) 2V(R2, t2)t22=0.
Δ1[V(R1, t1)V*(R2, t2)]
-(1/c2) 2[V(R1, t1)V*(R2, t2)]t12=0.
Δ1Γ+k2Γ=0,
Δ2Γ+k2Γ=0,
V(R, t)=v(R)exp[i(kz-ωt)].
Δv(R)+2ik v(R)z=0.
2vz2|Δv|,Δ̲=2x2+2y2;
Δ̲v(R)+2ik v(R)z=0.
v(R)=(1/z)expiπ x2+y2λz,
v(R)=exp[i2π(xνx+yνy)-iπλz(νx2+νy2)].
Δ̲1Γ+2ik Γz1=0,
Δ̲2Γ+2ik Γz2=0,
Γ(R1, R2)=v(R1)v*(R2).
(Δ1+Δ2)Γ-(1/c2)2t12+2t22Γ=0,
Δ1Δ2Γ-(1/c4) 4t12t22Γ=0.
exp[i(k1·R1-k2·R2-ω1t1+ω2t2)].
(Δ1+Δ2)Γ+(k12+k22)Γ=0,
Δ1Δ2Γ-k12k22Γ=0.
(Δ̲1+Δ̲2)Γ+2ik1 z1+k2 z2Γ=0,
Δ̲1Δ̲2Γ+4k1k2 2z1z2Γ=0.
k12+k22=2k2,
k12k22=k4.
k12=k22=k2.
Δu(R)+k2u(R)=0.
u(x, y, z)= u¯(ν)exp(2πiν·R)d3ν,
ν=(νx, νy, νz)
u˜(ν)0onlyifν·ν=1/λ2.
u(R)=v(R)exp(ikz),
v˜(ν)0onlyifνx2+νy2+(νz+1/λ)2=1/λ2.
νx2+νy2+2νz/λ=0,
2vz20,νz21/λ2
ν·ν=(ω/2πc)2.
ν1·ν1-(ω1/2πc)2=0,
ν2·ν2-(ω2/2πc)2=0.
ν1·ν1+ν2·ν2-(ω1/2πc)2-(ω2/2πc)2=0,
(ν1·ν1)(ν2·ν2)-(ω1ω2/2πc)4=0.
V(R, t)V(R+R¯, t+t¯),
Γ(R1, R2, t1, t2)Γ(R1+R¯, R2+R¯, t1+t¯, t2+t¯).
V(R, t)V(R, t)exp(2πiν¯·R-iω¯t),
V˜(ν, ω)V˜(ν-ν¯, ω-ω¯),
Γ˜(ν, ω)Γ˜(ν1-ν¯, ν2+ν¯, ω1-ω¯, ω2+ω¯),
V˜(ν, ω)= V(R, t)exp(-2πiνR+iωt)dRdt.
V(R, t)V(R, t)P(R, t),
Γ(R1, R2, t1, t2)Γ(R1, R2, t1, t2)×P(R1, t1)P*(R2, t2).
V(R, t)VA(R, t)VB(R, t),
Γ(R1, R2, t1, t2)ΓA(R1, R2, t1, t2)×ΓB(R1, R2, t1, t2).
V(R, t)V˜(ν, ω),
Γ(R1, R2, t1, t2)Γ˜(ν1, ν2, ω1, ω2).
V0V˜0V˜0P˜=V˜BVB,
Γ0Γ˜0Γ˜0Γ˜P=Γ˜BΓB,
Γ˜P(ν1, ν2, ω1, ω2)=P˜(ν1, ω1)P˜*(-ν2,-ω2).
Γ0Γ0ΓP=ΓB,
ΓP(x1, x2, t1, t2)=P(x1, t1)P*(x2, t2).
t1=t2=t,z1=z2=z,
u(x, z)=u˜0(ν)exp{2πi[xν+(z/λ)1-λ2ν2]}dν,
u˜0(ν)=u0(x)exp(-i2πνx)dx,
u0(x)=u(x, z=0).
Γ0(x1, x2)=u0(x1)u0*(x2)Γ(x1, x2, z),
Γ(x1, x2, z)= Γ˜0(ν1, ν2)exp{2πi[ν1x1-ν2x2+(z/λ)(1-λ2ν12-1-λ2ν22)]}dν1dν2,
Γ(x1, x2, z)=Γ0(x1, x2)P(x1, x2, z),
P(x1, x2, z)= exp{2πi[ν1x1-ν2x2+(z/λ)(1-λ2ν12-1-λ2ν22)]}dν1dν2.
v(x, z)=exp(ikz)iλzv0(x)exp[iπ(x-x)2/λz]dx,
v˜(ν, z)=v˜0(ν)exp(ikz)exp(-iπλzν2).
Γ0(x1, x2)=v0(x1)v0*(x2)Γ(x1, x2, z),
Γ(x1, x2, z)=1(λz)2Γ0(x1, x2)×expiπλz[(x1-x1)2-(x2-x2)2]dx1dx2.
Γ˜(ν1, ν2, z)=Γ˜0(ν1, ν2)exp[-iπz(ν12-ν22)].

Metrics