Abstract

The Wigner distribution function (WDF) is used to study the propagation of partially coherent beams through optical systems. The width and divergence of multimode beams are defined in terms of moments of the WDF. It is shown that these and other parameters of partially coherent beams of arbitrary form may be calculated using geometrical ray traces. Generalized transformation laws of these beam parameters by an ABCD optical system are obtained. These laws are tested experimentally for multimode beams and thin lenses. The applications of the ray tracing method can be extended to a non-ABCD system with mild aberrations.

© 1988 Optical Society of America

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References

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  2. S. A. Self, “Focusing of Spherical Gaussian Beams,” Appl. Opt. 22, 658 (1983).
    [Crossref] [PubMed]
  3. S. Nemoto, T. Makimoto, “Generalized Spot Size for a Higher Order Beam Mode,” J. Opt. Soc. Am. 69, 578 (1979).
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  4. S. Lavi, S. Gabay, E. Miron, G. Erez, “Self-Calibrating Technique for Beam Divergence Measurements of cw and Pulsed Lasers,” Appl. Opt. 20, 1145 (1981).
    [Crossref] [PubMed]
  5. E. B. Rockower, “Laser Beam-Quality/Aperture-Shape Scaling Relation,” Appl. Opt. 25, 1394 (1986).
    [Crossref] [PubMed]
  6. E. Collett, E. Wolf, “Is Complete Spatial Coherence Necessary for the Generation of Highly Directional Light Beams?” Opt. Lett. 2, 27 (1978).
    [Crossref] [PubMed]
  7. J. Turunen, A. T. Friberg, “Matrix Representation of Gaussian Schell Model Beams in Optical Systems,” Opt. Laser Technol. 18, 259 (1986).
    [Crossref]
  8. H. Kogelnik, T. Li, “Laser Beams and Resonators,” Proc. IEEE 54, 1312 (1966).
    [Crossref]
  9. E. Wigner, “On the Quantum Correction for Thermodynamic Equilibrium,” Phys. Rev. 40, 749 (1932).
    [Crossref]
  10. M. J. Bastiaans, “The Wigner Distribution Function Applied to Optical Signals and Systems,” Opt. Commun. 25, 26 (1978).
    [Crossref]
  11. M. J. Bastiaans, “Application of the Wigner Distribution Function to Partially Coherent Light,” J. Opt. Soc. Am. A 3, 1227 (1986).
    [Crossref]
  12. M. J. Bastiaans, “The Wigner Distribution Function and Its Application to First Order Optics,” J. Opt. Soc. Am. 69, 1710 (1979).
    [Crossref]
  13. I. M. Besieris, F. D. Tappert, “Stochastic Wave-Kinetic Theory in the Liouville Approximation,” J. Math. Phys. 17, 734 (1976).
    [Crossref]
  14. A. Papoulis, “Ambiguity Function in Fourier Optics,” J. Opt. Soc. Am. 64, 779 (1974).
    [Crossref]
  15. D. Marcuse, Light Transmission Optics (Van Nostrand, New York, 1982), Chaps. 3 and 11.
  16. L. Mandel, E. Wolf, “Spectral Coherence and the Concept of Cross-Spectral Purity,” J. Opt. Soc. Am. 66, 529 (1976).
    [Crossref]
  17. A. C. Schell, “A Technique for the Determination of the Radiation Pattern of a Partially Coherent Aperture,” IEEE Trans. Antennas Propag. AP-15, 187 (1967).
    [Crossref]
  18. F. Zernike, “Title,” Physica 5, 785 (1938).
    [Crossref]
  19. A. E. Siegman, Lasers (University Science Books, Mill Valley, CA, 1986), Chaps. 15–20.
  20. A. Yariv, Quantum Electronics (Wiley, New York, 1978), Chap. 7.
  21. W. Koechner, Solid State Laser Engineering (Springer-Verlag, New York, 1976), p. 187.
  22. Z. Karny, S. Lavi, O. Kafri, “Direct Determination of the Number of Transverse Modes of a Light Beam,” Opt. Lett. 8, 409 (1983).
    [Crossref] [PubMed]
  23. E. W. Marchand, Gradient Index Optics (Academic, New York, 1978), Chap. 8.

1986 (4)

1983 (2)

1981 (1)

1979 (2)

1978 (2)

E. Collett, E. Wolf, “Is Complete Spatial Coherence Necessary for the Generation of Highly Directional Light Beams?” Opt. Lett. 2, 27 (1978).
[Crossref] [PubMed]

M. J. Bastiaans, “The Wigner Distribution Function Applied to Optical Signals and Systems,” Opt. Commun. 25, 26 (1978).
[Crossref]

1976 (2)

I. M. Besieris, F. D. Tappert, “Stochastic Wave-Kinetic Theory in the Liouville Approximation,” J. Math. Phys. 17, 734 (1976).
[Crossref]

L. Mandel, E. Wolf, “Spectral Coherence and the Concept of Cross-Spectral Purity,” J. Opt. Soc. Am. 66, 529 (1976).
[Crossref]

1974 (1)

1967 (1)

A. C. Schell, “A Technique for the Determination of the Radiation Pattern of a Partially Coherent Aperture,” IEEE Trans. Antennas Propag. AP-15, 187 (1967).
[Crossref]

1966 (1)

H. Kogelnik, T. Li, “Laser Beams and Resonators,” Proc. IEEE 54, 1312 (1966).
[Crossref]

1938 (1)

F. Zernike, “Title,” Physica 5, 785 (1938).
[Crossref]

1932 (1)

E. Wigner, “On the Quantum Correction for Thermodynamic Equilibrium,” Phys. Rev. 40, 749 (1932).
[Crossref]

Bastiaans, M. J.

Besieris, I. M.

I. M. Besieris, F. D. Tappert, “Stochastic Wave-Kinetic Theory in the Liouville Approximation,” J. Math. Phys. 17, 734 (1976).
[Crossref]

Collett, E.

Erez, G.

Friberg, A. T.

J. Turunen, A. T. Friberg, “Matrix Representation of Gaussian Schell Model Beams in Optical Systems,” Opt. Laser Technol. 18, 259 (1986).
[Crossref]

Gabay, S.

Herman, R. M.

Kafri, O.

Karny, Z.

Koechner, W.

W. Koechner, Solid State Laser Engineering (Springer-Verlag, New York, 1976), p. 187.

Kogelnik, H.

H. Kogelnik, T. Li, “Laser Beams and Resonators,” Proc. IEEE 54, 1312 (1966).
[Crossref]

Lavi, S.

Li, T.

H. Kogelnik, T. Li, “Laser Beams and Resonators,” Proc. IEEE 54, 1312 (1966).
[Crossref]

Makimoto, T.

Mandel, L.

Marchand, E. W.

E. W. Marchand, Gradient Index Optics (Academic, New York, 1978), Chap. 8.

Marcuse, D.

D. Marcuse, Light Transmission Optics (Van Nostrand, New York, 1982), Chaps. 3 and 11.

Miron, E.

Nemoto, S.

Papoulis, A.

Rockower, E. B.

Schell, A. C.

A. C. Schell, “A Technique for the Determination of the Radiation Pattern of a Partially Coherent Aperture,” IEEE Trans. Antennas Propag. AP-15, 187 (1967).
[Crossref]

Self, S. A.

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, Mill Valley, CA, 1986), Chaps. 15–20.

Tappert, F. D.

I. M. Besieris, F. D. Tappert, “Stochastic Wave-Kinetic Theory in the Liouville Approximation,” J. Math. Phys. 17, 734 (1976).
[Crossref]

Turunen, J.

J. Turunen, A. T. Friberg, “Matrix Representation of Gaussian Schell Model Beams in Optical Systems,” Opt. Laser Technol. 18, 259 (1986).
[Crossref]

Wiggins, T. A.

Wigner, E.

E. Wigner, “On the Quantum Correction for Thermodynamic Equilibrium,” Phys. Rev. 40, 749 (1932).
[Crossref]

Wolf, E.

Yariv, A.

A. Yariv, Quantum Electronics (Wiley, New York, 1978), Chap. 7.

Zernike, F.

F. Zernike, “Title,” Physica 5, 785 (1938).
[Crossref]

Appl. Opt. (4)

IEEE Trans. Antennas Propag. (1)

A. C. Schell, “A Technique for the Determination of the Radiation Pattern of a Partially Coherent Aperture,” IEEE Trans. Antennas Propag. AP-15, 187 (1967).
[Crossref]

J. Math. Phys. (1)

I. M. Besieris, F. D. Tappert, “Stochastic Wave-Kinetic Theory in the Liouville Approximation,” J. Math. Phys. 17, 734 (1976).
[Crossref]

J. Opt. Soc. Am. (4)

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

Opt. Commun. (1)

M. J. Bastiaans, “The Wigner Distribution Function Applied to Optical Signals and Systems,” Opt. Commun. 25, 26 (1978).
[Crossref]

Opt. Laser Technol. (1)

J. Turunen, A. T. Friberg, “Matrix Representation of Gaussian Schell Model Beams in Optical Systems,” Opt. Laser Technol. 18, 259 (1986).
[Crossref]

Opt. Lett. (2)

Phys. Rev. (1)

E. Wigner, “On the Quantum Correction for Thermodynamic Equilibrium,” Phys. Rev. 40, 749 (1932).
[Crossref]

Physica (1)

F. Zernike, “Title,” Physica 5, 785 (1938).
[Crossref]

Proc. IEEE (1)

H. Kogelnik, T. Li, “Laser Beams and Resonators,” Proc. IEEE 54, 1312 (1966).
[Crossref]

Other (5)

A. E. Siegman, Lasers (University Science Books, Mill Valley, CA, 1986), Chaps. 15–20.

A. Yariv, Quantum Electronics (Wiley, New York, 1978), Chap. 7.

W. Koechner, Solid State Laser Engineering (Springer-Verlag, New York, 1976), p. 187.

D. Marcuse, Light Transmission Optics (Van Nostrand, New York, 1982), Chaps. 3 and 11.

E. W. Marchand, Gradient Index Optics (Academic, New York, 1978), Chap. 8.

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Figures (3)

Fig. 1
Fig. 1

Schematic of the experimental system: A, beam absorber; BS, 99%R beam splitter at 1.06 μm; S,S′, locations of input and output waists.

Fig. 2
Fig. 2

Beam profile at the waist S: (a) beam 1: open aperture, 300 W, F0 = 300 mm, z0 = 400 mm; (b) beam 2: 4-mm diam aperture, 75 W, F0 = 500 mm, z0 = 825 mm.

Fig. 3
Fig. 3

Transformation of beam parameters by a lens. Normalized output waist distance (a) and magnification (b) vs normalized input waist distance. The different curves correspond to various values for the normalized Rayleigh range.

Tables (1)

Tables Icon

Table I Measured Beam Parameters for the Two Beams

Equations (56)

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Γ ( x , s , z ) = f ( x + s / 2 , z ) f * ( x - s / 2 , z ) = [ I ( x + s / 2 , z ) I ( x - s / 2 , z ) ] 1 / 2 γ ( x , s , z ) ,
γ ( x , 0 , z ) = 1 ;             Γ ( x , 0 , z ) = I ( x , z ) .
I ( x , z ) = P 0 / [ ( 2 π ) 1 / 2 W ( z ) ] · exp [ - x 2 / 2 W 2 ( z ) ] ,
γ ( x , s , z ) = exp [ - Φ 2 ( z ) k 2 s 2 / 2 ] ,
h ( x , u , z ) = - d s exp ( i k u s ) Γ ( x , s , z ) ,
I ( x , z ) = k / 2 π - d u h ( x , u , z ) .
h ( x , u , z ) = h [ x - u ( z - z ) , u , z ] ,
h ( x , u , z ) = - d x δ [ x - x + u ( z - z ) ] h ( x , u , z ) .
Γ ( x , s , z ) = k / ( 2 π ) · - d u exp ( - i k u s ) h ( x , u , z ) ,
Γ ( x , s , z ) = 1 / λ ( z - z ) · - d x - d s exp [ - i k ( x - x ) ( s - s ) / ( z - z ) ] · Γ ( x , s , z ) .
x = A x + B u ;             u = C x + D u .
h ( x , u , z ) = h ( A x + B u , C x + D u , z ) = h ( x , u , z ) .
Γ ( x , s , z ) = 1 / λ B · - d x - d s Γ ( x , s , z ) × exp { - i k / B · [ s ( D x - x ) - s ( x - A x ) ] } .
Γ ( x , s , z ) = 1 / λ · - d u - d s Γ ( D x - B u , s , z ) × exp { - i k [ s u - s ( A u - C x ) ] } .
G ( u , z ) = - d x h ( x , u , z ) ;
W 2 ( z ) = ( 1 / P 0 ) - d x x 2 I ( x , z ) ;
Φ 2 ( z ) = ( k / 2 π P 0 ) - d u u 2 G ( u , z ) .
G ( u , z ) = g ( u , z ) g * ( u , z ) ,
g ( u , z ) = - d x exp ( i k u x ) f ( x , z ) .
W ( z ) Φ ( z ) 1 / 2 k = λ / 4 π .
W s Φ s = N / 2 k ,
z R = W s / Φ s = S / λ N ,
W 2 = ( k / 2 π P 0 ) - d x - d u h ( x , u ) x 2 ,
x x u u - x u u x = AD - BC = 1.
W 2 = ( k / 2 π P 0 ) - d x - d u ( A x + B u ) 2 h ( x , u ) .
W 2 = A 2 W s 2 + B 2 Φ s 2 .
Φ 2 = C 2 W s 2 + D 2 Φ s 2 .
W 2 = W s 2 + z 2 Φ s 2 = W s 2 [ 1 + ( z / z R ) 2 ] ,
- d x - d u ( A s x + B s u ) ( C s x + D s u ) h ( x , u ) = A s C s W s 2 + B s D s Φ s 2 = 0.
m 2 = W s 2 / W s 2 = A s 2 + ( B s / z R ) 2 .
A s = D s / ( D s 2 + C s 2 z R 2 )
m = ( D s 2 + C s 2 z R 2 ) - 1 / 2 .
Φ s W s = Φ s W s ,
z R = m 2 z R .
A = 1 - z 2 / F ; B = z 1 + z 2 - z 1 z 2 / F , C = - 1 / F ; D = 1 - z 1 / F ,
z R 2 = ( 1 - D s 2 ) / C s 2 = - B s / C s ,
A s = D s .
z R = 2 k W s 2 / N .
N = W s 2 / W 0 2 .
n = Ω S / λ 2 ,
n = N 2 ,
Φ s = W F / F .
i k f ( x , z ) z = - 1 2 2 f ( x , z ) x 2 - k 2 n ( x , z ) f ( x , z ) .
i k Γ z = - 2 Γ x s - k 2 Γ ( x , s , z ) [ n ( x + s / 2 ) - n ( x - s / 2 ) ] .
i k Γ z = - 2 Γ x s - 2 k 2 Γ ( x , s , z ) [ s 2 n x + 1 3 ! ( s 2 ) 3 3 n x 3 + 1 5 ! ( s 2 ) 5 5 n x 5 + ] .
h ( x , u , z ) z = - u h x - h u n x + 1 3 ! ( λ 4 π ) 2 3 h u 3 3 n x 3 - 1 5 ! ( λ 4 π ) 4 5 h u 5 5 n x 5 + .
d d z [ h ( x , u , z ) ] = h z + h x h z + h u u z = 0 ,
z = z ( u , x ) .
u z = n x ;             u = x z .
s 1 2 24 n x / 3 n x 3 .
z x m = m u x m - 1 ,
z u m = j = 0 [ m - 1 2 ] ( - λ 4 π ) 2 j ( m 2 j + 1 ) u m - 2 j - 1 2 j + 1 n x 2 j + 1 ,
( m 2 j + 1 )
z - - d u x 2 h ( x , u , z ) = - - d x - d u x 2 u h x = 2 - d x - d u x u h ( x , u , z ) .
- - d x - d u x 2 h u n x = - - d x x 2 n x h ( x , u , z ) - = 0.
z u x = x n x + u 2 .

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