Abstract

It is demonstrated that the members of different classes of two-dimensional fields that have the same intensity distributions everywhere in free space but different coherence properties can be identified by measuring the fields’ intensities after passing the fields through an anamorphic optical system, such as a cylindrical lens. In this way the ambiguity in coherence determination from intensity measurements alone, present for the case of two-dimensional fields, is removed.

© 2003 Optical Society of America

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References

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  1. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).
  2. W. H. Carter, “Difference in the definitions of coherence in the space–time and in the space–frequency domain,” J. Mod. Opt. 39, 1461–1470 (1992).
    [CrossRef]
  3. S. S. K. Titus, A. Wasan, J. S. Vaishya, H. C. Kandpal, “Determination of phase and amplitude of degree of coherence from spectroscopic measurements,” Opt. Commun. 173, 45–49 (2000).
    [CrossRef]
  4. J. C. Barreiro, J. Ojeda-Castañeda, “Degree of coherence: a lensless measuring technique,” Opt. Lett. 18, 302–304 (1993).
    [CrossRef] [PubMed]
  5. G. T. Herman, Image Reconstruction from Projections: The Fundamentals of Computerized Tomography (Academic, New York, 1980).
  6. F. Gori, M. Santarsiero, G. Guattari, “Coherence and the spatial distribution of intensity,” J. Opt. Soc. Am. A 10, 673–679 (1993).
    [CrossRef]
  7. V. Bagini, F. Gori, M. Santarsiero, G. Guattari, G. Schirripa Spagnolo, “Space intensity distribution and projections of the cross spectral density,” Opt. Commun. 102, 495–504 (1993).
    [CrossRef]
  8. M. G. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
    [CrossRef] [PubMed]
  9. T. Alieva, F. Agulló-López, “Reconstruction of the optical correlation function in a quadratic refractive index medium,” Opt. Commun. 114, 161–169 (1995).
    [CrossRef]
  10. B. Eppich, C. Gao, H. Weber, “Determination of the ten second order intensity moments,” Opt. Laser Technol. 30, 337–340 (1998).
    [CrossRef]

2000 (1)

S. S. K. Titus, A. Wasan, J. S. Vaishya, H. C. Kandpal, “Determination of phase and amplitude of degree of coherence from spectroscopic measurements,” Opt. Commun. 173, 45–49 (2000).
[CrossRef]

1998 (1)

B. Eppich, C. Gao, H. Weber, “Determination of the ten second order intensity moments,” Opt. Laser Technol. 30, 337–340 (1998).
[CrossRef]

1995 (1)

T. Alieva, F. Agulló-López, “Reconstruction of the optical correlation function in a quadratic refractive index medium,” Opt. Commun. 114, 161–169 (1995).
[CrossRef]

1994 (1)

M. G. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef] [PubMed]

1993 (3)

1992 (1)

W. H. Carter, “Difference in the definitions of coherence in the space–time and in the space–frequency domain,” J. Mod. Opt. 39, 1461–1470 (1992).
[CrossRef]

Agulló-López, F.

T. Alieva, F. Agulló-López, “Reconstruction of the optical correlation function in a quadratic refractive index medium,” Opt. Commun. 114, 161–169 (1995).
[CrossRef]

Alieva, T.

T. Alieva, F. Agulló-López, “Reconstruction of the optical correlation function in a quadratic refractive index medium,” Opt. Commun. 114, 161–169 (1995).
[CrossRef]

Bagini, V.

V. Bagini, F. Gori, M. Santarsiero, G. Guattari, G. Schirripa Spagnolo, “Space intensity distribution and projections of the cross spectral density,” Opt. Commun. 102, 495–504 (1993).
[CrossRef]

Barreiro, J. C.

Beck, M.

M. G. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef] [PubMed]

Carter, W. H.

W. H. Carter, “Difference in the definitions of coherence in the space–time and in the space–frequency domain,” J. Mod. Opt. 39, 1461–1470 (1992).
[CrossRef]

Eppich, B.

B. Eppich, C. Gao, H. Weber, “Determination of the ten second order intensity moments,” Opt. Laser Technol. 30, 337–340 (1998).
[CrossRef]

Gao, C.

B. Eppich, C. Gao, H. Weber, “Determination of the ten second order intensity moments,” Opt. Laser Technol. 30, 337–340 (1998).
[CrossRef]

Gori, F.

V. Bagini, F. Gori, M. Santarsiero, G. Guattari, G. Schirripa Spagnolo, “Space intensity distribution and projections of the cross spectral density,” Opt. Commun. 102, 495–504 (1993).
[CrossRef]

F. Gori, M. Santarsiero, G. Guattari, “Coherence and the spatial distribution of intensity,” J. Opt. Soc. Am. A 10, 673–679 (1993).
[CrossRef]

Guattari, G.

F. Gori, M. Santarsiero, G. Guattari, “Coherence and the spatial distribution of intensity,” J. Opt. Soc. Am. A 10, 673–679 (1993).
[CrossRef]

V. Bagini, F. Gori, M. Santarsiero, G. Guattari, G. Schirripa Spagnolo, “Space intensity distribution and projections of the cross spectral density,” Opt. Commun. 102, 495–504 (1993).
[CrossRef]

Herman, G. T.

G. T. Herman, Image Reconstruction from Projections: The Fundamentals of Computerized Tomography (Academic, New York, 1980).

Kandpal, H. C.

S. S. K. Titus, A. Wasan, J. S. Vaishya, H. C. Kandpal, “Determination of phase and amplitude of degree of coherence from spectroscopic measurements,” Opt. Commun. 173, 45–49 (2000).
[CrossRef]

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).

McAlister, D. F.

M. G. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef] [PubMed]

Ojeda-Castañeda, J.

Raymer, M. G.

M. G. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef] [PubMed]

Santarsiero, M.

F. Gori, M. Santarsiero, G. Guattari, “Coherence and the spatial distribution of intensity,” J. Opt. Soc. Am. A 10, 673–679 (1993).
[CrossRef]

V. Bagini, F. Gori, M. Santarsiero, G. Guattari, G. Schirripa Spagnolo, “Space intensity distribution and projections of the cross spectral density,” Opt. Commun. 102, 495–504 (1993).
[CrossRef]

Spagnolo, G. Schirripa

V. Bagini, F. Gori, M. Santarsiero, G. Guattari, G. Schirripa Spagnolo, “Space intensity distribution and projections of the cross spectral density,” Opt. Commun. 102, 495–504 (1993).
[CrossRef]

Titus, S. S. K.

S. S. K. Titus, A. Wasan, J. S. Vaishya, H. C. Kandpal, “Determination of phase and amplitude of degree of coherence from spectroscopic measurements,” Opt. Commun. 173, 45–49 (2000).
[CrossRef]

Vaishya, J. S.

S. S. K. Titus, A. Wasan, J. S. Vaishya, H. C. Kandpal, “Determination of phase and amplitude of degree of coherence from spectroscopic measurements,” Opt. Commun. 173, 45–49 (2000).
[CrossRef]

Wasan, A.

S. S. K. Titus, A. Wasan, J. S. Vaishya, H. C. Kandpal, “Determination of phase and amplitude of degree of coherence from spectroscopic measurements,” Opt. Commun. 173, 45–49 (2000).
[CrossRef]

Weber, H.

B. Eppich, C. Gao, H. Weber, “Determination of the ten second order intensity moments,” Opt. Laser Technol. 30, 337–340 (1998).
[CrossRef]

Wolf, E.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).

J. Mod. Opt. (1)

W. H. Carter, “Difference in the definitions of coherence in the space–time and in the space–frequency domain,” J. Mod. Opt. 39, 1461–1470 (1992).
[CrossRef]

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

Opt. Commun. (3)

V. Bagini, F. Gori, M. Santarsiero, G. Guattari, G. Schirripa Spagnolo, “Space intensity distribution and projections of the cross spectral density,” Opt. Commun. 102, 495–504 (1993).
[CrossRef]

T. Alieva, F. Agulló-López, “Reconstruction of the optical correlation function in a quadratic refractive index medium,” Opt. Commun. 114, 161–169 (1995).
[CrossRef]

S. S. K. Titus, A. Wasan, J. S. Vaishya, H. C. Kandpal, “Determination of phase and amplitude of degree of coherence from spectroscopic measurements,” Opt. Commun. 173, 45–49 (2000).
[CrossRef]

Opt. Laser Technol. (1)

B. Eppich, C. Gao, H. Weber, “Determination of the ten second order intensity moments,” Opt. Laser Technol. 30, 337–340 (1998).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. Lett. (1)

M. G. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef] [PubMed]

Other (2)

G. T. Herman, Image Reconstruction from Projections: The Fundamentals of Computerized Tomography (Academic, New York, 1980).

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).

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Figures (4)

Fig. 1
Fig. 1

Intensities of the Vn+ (left) and Vn- (center) coherent modes, and their ratio (right) after passing through a cylindrical lens that focalizes along x for (a) n=1, (b) n=2, (c) n=3, and (d) n=4.

Fig. 2
Fig. 2

Same as in Fig. 1, but for a cylindrical lens that makes and angle of 30° with respect to the x axis.

Fig. 3
Fig. 3

Intensities of superpositions of the coherent modes in Fig. 1(a), with (a) α=0, (b) α=2.5, (c) α=5, (d) α=7.5, and (e) α=10 (in all cases β=10-α).

Fig. 4
Fig. 4

Intensities of the V+ (left) and V- (center) coherent modes and their ratio (right) after the fields are passed through a cylindrical lens that focalizes along x.

Equations (19)

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Vn±(x, y; 0)=G(r)(x±iy)n,
V(x, y; 0)=c+Vn+(x, y; 0)+c-Vn-(x, y; 0),
W(x1, y1, x2, y2 ; 0)=G(r1)G*(r2)[αξn+βξ*n],
I(x, y; 0)=W(x1=x, y1=y, x2=x, y2=y; 0)=(α+β)|G(r)|2r2n,
μ(x1, y1, x2, y2)=W(x1, y1, x2, y2)/[I(x1, y1)I(x2, y2)]1/2
μ(x1, y1, x2, y2)=G(r1)G*(r2)|G(r1)||G(r2)|×αξξ*n/2+βξ*ξn/2
I(x, y; z)=W(x1=x, y1=y, x2=x, y2=y; 0)=(α+β)|Gn(r; z)|2,
|Gn(r; z)|2=4π2λ2z20G(ρ)ρn×expik2zρ2Jnkrρzρdρ2,
Vn±(x, y; z)=KG(r)exp[-ik(x-x)2/2z-ik(y-y)2/2z-ikx2/2 f]×(x±iy)ndxdy,
Vn±(x, y; z)/K=k=1n-1Cnk(±i)n-k×Dk-1(x0, a+ib)Dn-k-1(y0, a+ic)2(a+ib)(a+ic)+Dn-1(x0, a+ib)(a+ib)expy024(a+ic)+(±i)nDn-1(y0, a+ic)(a+ic)expx024(a+ib),
K=πK exp[-ik(x2+y2)/2z]/2[(a+ib)(a+ic)]1/2,
Dk(x, s)=dk[x exp(x2/4s)]/dxk,
Cnk=n!/[k!(n-k)!],a=1/w2,
b=k/2z+k/2 f,c=k/2z,
x0=ikx/z,y0=iky/z.
W(x1, y1, x2, y2; z)=W(x1, y1, x2, y2; 0)Ξ*(x1, y1, x1, y1)×Ξ(x2, y2, x2, y2)dx1dy1dx2dy2,
V(x, y; z)=V(x, y; 0)Ξ(x, y, x, y)dxdy,
I(x, y; z)=αI++βI-,
V(x, y; 0)=c+V+(x, y; 0)+c-V-(x, y; 0)=[c+(ax+iby)+c-(ax-iby)]G(r),

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