Abstract

The twisted Gaussian Schell-model (GSM) beams, recently introduced by Simon and Mukunda [ J. Opt. Soc. Am. A 9, 95 ( 1993)], are interpreted in physical-optics terms by decomposition of such beams into weighted superpositions of overlapping, mutually uncorrelated but spatially coherent component fields. The decomposition provides considerable physical insight into the propagation characteristics of the twisted GSM beams and also suggests convenient practical methods for generating these novel wave fields. Key properties of the twisted GSM beams are demonstrated experimentally by use of an acousto-optic coherence control technique to supply the necessary partially coherent fields.

© 1994 Optical Society of America

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References

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  1. R. Simon, N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10, 95–109 (1993).
    [CrossRef]
  2. A. C. Schell, “A technique for the determination of the radiation pattern of a partially coherent aperture,” IEEE Trans. Antennas Propag. AP-15, 187–188 (1967).
    [CrossRef]
  3. F. Gori, C. Palma, “Partially coherent sources which give rise to highly directional light beams,” Opt. Commun. 27, 185–188 (1978).
    [CrossRef]
  4. F. Gori, “Directionality and spatial coherence,” Opt. Acta 27, 1025–1034 (1980).
    [CrossRef]
  5. F. Gori, G. Guattari, “A new type of optical fields,” Opt. Commun. 48, 7–12 (1983).
    [CrossRef]
  6. J. Turunen, E. Tervonen, A. T. Friberg, “Acousto-optic control and modulation of optical coherence by electronically synthesized holographic gratings,” J. Appl. Phys. 67, 49–59 (1990).
    [CrossRef]
  7. P. DeSantis, F. Gori, G. Guattari, C. Palma, “Synthesis of partially coherent fields,” J. Opt. Soc. Am. A 3, 1258–1262 (1986).
    [CrossRef]
  8. F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
    [CrossRef]
  9. A. Starikov, E. Wolf, “Coherent-mode representation of Gaussian Schell-model sources and their radiation fields,” J. Opt. Soc. Am. 72, 923–927 (1982).
    [CrossRef]
  10. E. Wolf, “New theory of partial spatial coherence in the space-frequency domain. Part I: spectra and cross spectra of steady-state sources,” J. Opt. Soc. Am. 72, 343–351 (1982).
    [CrossRef]
  11. E. Tervonen, A. T. Friberg, J. Turunen, “Gaussian Schell-model beams generated with synthetic acousto-optic holograms,” J. Opt. Soc. Am. A 9, 796–803 (1991).
    [CrossRef]
  12. L. Mandel, E. Wolf, “Spectral coherence and the concept of cross-spectral purity,” J. Opt. Soc. Am. 66, 529–535 (1976).
    [CrossRef]
  13. Sometimes an invariant quantity of the form α= σ(z)/w(z), called the global degree of coherence, is introduced to characterize the transverse coherence properties of the GSM beam. However, it is evident that this quantity is not normalized. Note also that the parameter β defined in Eq. (2) is not the same as the parameter β of Ref. 1; the latter corresponds to our 1/2α.
  14. A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986), Sec. 15.6.
  15. A. T. Friberg, J. Turunen, “Imaging of Gaussian Schell-model sources,” J. Opt. Soc. Am. A 5, 713–720 (1988).
    [CrossRef]
  16. The calculations can be performed with the help of the known quadratic integrals, given, e.g., by Eq. (3.16) of Ref. 1. However, because of the required matrix inversion, this is a straightforward but laborious undertaking, and therefore the intermediate steps are not given. The results can also be checked conveniently with use of the 4×4 dimensional variance matrices V and V′ of the input and the output beams, respectively, and by the fact that in any first-order system 5 these matrices transform as V→V′=SVST, where the superscript Tdenotes the transposition [see Eqs. (3.27), (3.32), and (10.3) of Ref. 1]. Note, however, that the cross spectral density of Ref. 1 is the complex conjugate of that employed here.

1993 (1)

1991 (1)

1990 (1)

J. Turunen, E. Tervonen, A. T. Friberg, “Acousto-optic control and modulation of optical coherence by electronically synthesized holographic gratings,” J. Appl. Phys. 67, 49–59 (1990).
[CrossRef]

1988 (1)

1986 (1)

1983 (1)

F. Gori, G. Guattari, “A new type of optical fields,” Opt. Commun. 48, 7–12 (1983).
[CrossRef]

1982 (2)

1980 (2)

F. Gori, “Directionality and spatial coherence,” Opt. Acta 27, 1025–1034 (1980).
[CrossRef]

F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[CrossRef]

1978 (1)

F. Gori, C. Palma, “Partially coherent sources which give rise to highly directional light beams,” Opt. Commun. 27, 185–188 (1978).
[CrossRef]

1976 (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, 187–188 (1967).
[CrossRef]

DeSantis, P.

Friberg, A. T.

Gori, F.

P. DeSantis, F. Gori, G. Guattari, C. Palma, “Synthesis of partially coherent fields,” J. Opt. Soc. Am. A 3, 1258–1262 (1986).
[CrossRef]

F. Gori, G. Guattari, “A new type of optical fields,” Opt. Commun. 48, 7–12 (1983).
[CrossRef]

F. Gori, “Directionality and spatial coherence,” Opt. Acta 27, 1025–1034 (1980).
[CrossRef]

F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[CrossRef]

F. Gori, C. Palma, “Partially coherent sources which give rise to highly directional light beams,” Opt. Commun. 27, 185–188 (1978).
[CrossRef]

Guattari, G.

Mandel, L.

Mukunda, N.

Palma, C.

P. DeSantis, F. Gori, G. Guattari, C. Palma, “Synthesis of partially coherent fields,” J. Opt. Soc. Am. A 3, 1258–1262 (1986).
[CrossRef]

F. Gori, C. Palma, “Partially coherent sources which give rise to highly directional light beams,” Opt. Commun. 27, 185–188 (1978).
[CrossRef]

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, 187–188 (1967).
[CrossRef]

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986), Sec. 15.6.

Simon, R.

Starikov, A.

Tervonen, E.

E. Tervonen, A. T. Friberg, J. Turunen, “Gaussian Schell-model beams generated with synthetic acousto-optic holograms,” J. Opt. Soc. Am. A 9, 796–803 (1991).
[CrossRef]

J. Turunen, E. Tervonen, A. T. Friberg, “Acousto-optic control and modulation of optical coherence by electronically synthesized holographic gratings,” J. Appl. Phys. 67, 49–59 (1990).
[CrossRef]

Turunen, J.

Wolf, E.

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, 187–188 (1967).
[CrossRef]

J. Appl. Phys. (1)

J. Turunen, E. Tervonen, A. T. Friberg, “Acousto-optic control and modulation of optical coherence by electronically synthesized holographic gratings,” J. Appl. Phys. 67, 49–59 (1990).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (4)

Opt. Acta (1)

F. Gori, “Directionality and spatial coherence,” Opt. Acta 27, 1025–1034 (1980).
[CrossRef]

Opt. Commun. (3)

F. Gori, G. Guattari, “A new type of optical fields,” Opt. Commun. 48, 7–12 (1983).
[CrossRef]

F. Gori, C. Palma, “Partially coherent sources which give rise to highly directional light beams,” Opt. Commun. 27, 185–188 (1978).
[CrossRef]

F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[CrossRef]

Other (3)

Sometimes an invariant quantity of the form α= σ(z)/w(z), called the global degree of coherence, is introduced to characterize the transverse coherence properties of the GSM beam. However, it is evident that this quantity is not normalized. Note also that the parameter β defined in Eq. (2) is not the same as the parameter β of Ref. 1; the latter corresponds to our 1/2α.

A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986), Sec. 15.6.

The calculations can be performed with the help of the known quadratic integrals, given, e.g., by Eq. (3.16) of Ref. 1. However, because of the required matrix inversion, this is a straightforward but laborious undertaking, and therefore the intermediate steps are not given. The results can also be checked conveniently with use of the 4×4 dimensional variance matrices V and V′ of the input and the output beams, respectively, and by the fact that in any first-order system 5 these matrices transform as V→V′=SVST, where the superscript Tdenotes the transposition [see Eqs. (3.27), (3.32), and (10.3) of Ref. 1]. Note, however, that the cross spectral density of Ref. 1 is the complex conjugate of that employed here.

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Figures (9)

Fig. 1
Fig. 1

Astigmatic optical lens system used for converting an anisotropic GSM beam into a twisted GSM beam.

Fig. 2
Fig. 2

a, Coherent-beam decomposition of an anisotropic GSM source, which is fully coherent in the υ direction: overlapping elliptical beams that propagate in different directions. b, Coherent-beam decomposition of a twisted GSM source with η = −1: spatially shifted elliptical beams that propagate in different directions.

Fig. 3
Fig. 3

Propagation of a twisted GSM beam with η = −1. The elliptical coherent-beam decomposition in planes a, z = 0; b, z = zR; and c, z → ∞.

Fig. 4
Fig. 4

a, Coherent-beam decomposition of a twisted GSM source with η = −1/2 and β = 1 / 5. b, Same as a but with η = 0; i.e., the twist phase vanishes.

Fig. 5
Fig. 5

Experimental arrangement: AOD, acousto-optic deflector; L1 and L2, spherical lenses; C1–C6, cylindrical lenses; S, spatial filter.

Fig. 6
Fig. 6

a, Phase profile and b, the approximately Gaussian power spectrum of an optimized binary-phase grating.

Fig. 7
Fig. 7

Theoretical simulation (left-hand column) and experimental demonstration (right-hand column) of the rotation of the twisted GSM beam: three individual components in the coherent-beam decomposition are shown on propagation in planes (from top to bottom) z = 0, 150, 300, 450, and 5000 mm.

Fig. 8
Fig. 8

Measured distributions of the optical intensity (a, b, and c) and the complex degree of spatial coherence (d, e, and f) at the waist of the twisted GSM beam for input GSM beam eccentricities E = 2.0, 2.7, and 3.4, respectively. The filled and the open circles represent the measurements in the x and y directions, respectively, and the solid lines show the theoretical results.

Fig. 9
Fig. 9

Twisted GSM beam profiles: the filled and the open circles in a–c give the measured beam widths on propagation along the x and y directions for the three sets of experimental parameters given in Table 1. The three solid curves (in decreasing order of divergence) show the theoretical beam profiles of a twisted GSM beam, an ordinary GSM beam, and a coherent laser beam.

Tables (1)

Tables Icon

Table 1 Theoretical Values for the Experimental Parameters

Equations (50)

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W ( ρ 1 , ρ 2 , z ) = I 0 ( 2 / π ) w 2 ( z ) × exp [ ( ρ 1 2 + ρ 2 2 ) / w 2 ( z ) ] × exp [ ( ρ 1 ρ 2 ) 2 / 2 σ 2 ( z ) ] × exp [ i k ( ρ 1 2 ρ 2 2 ) / 2 R ( z ) ] × exp [ i k ρ 1 ρ 2 u ( z ) ] ,
β = { 1 + [ w ( z ) / σ ( z ) ] 2 } 1 / 2 ,
η = k σ 2 ( z ) u ( z ) .
w ( z ) = w ( 0 ) [ 1 + ( z / z R ) 2 ] 1 / 2 ,
σ ( z ) = σ ( 0 ) [ 1 + ( z / z R ) 2 ] 1 / 2 ,
R ( z ) = z [ 1 + ( z R / z ) 2 ] ,
u ( z ) = u ( 0 ) [ 1 + ( z / z R ) 2 ] 1 ,
z R = π w 2 ( 0 ) λ β [ 1 + η 2 ( 1 β 2 2 β ) 2 ] 1 / 2
ψ tan ψ = lim z w ( z ) / z = w ( 0 ) / z R = λ π w ( 0 ) β [ 1 + η 2 ( 1 β 2 2 β ) 2 ] 1 / 2 .
( Δ ψ ) twist = [ 1 + η 2 ( 1 β 2 2 β ) 2 ] 1 / 2 1 ;
[ ρ 0 ρ 0 ] = [ A B C D ] [ ρ s ρ s ] ,
U ( ρ 0 , 0 ) = i k 2 π ( det B ) 1 / 2 exp ( ikL ) × U s ( ρ s , z 0 ) exp [ i k ( ρ 0 D B 1 ρ 0 2 ρ s B 1 ρ 0 + ρ s B 1 A ρ s ) / 2 ] d 2 ρ s ,
W ( ρ 1 , ρ 2 , z ) = U * ( ρ 1 , z ) U ( ρ 2 , z )
T = 1 2 [ 1 0 0 Ω 0 1 Ω 0 0 Ω 1 1 0 Ω 1 0 0 1 ] ,
D ( d ) = [ 1 0 d 0 0 1 0 d 0 0 1 0 0 0 0 1 ] ,
( f x , f y ) = [ 1 0 0 0 0 1 0 0 f x 1 0 1 0 0 f y 1 0 1 ] ,
( f ) = D ( f / 2 ) L ( , f / 2 ) D ( f / 2 ) L ( f , ) × D ( f / 2 ) L ( , f / 2 ) D ( f / 2 ) = [ 1 0 f 0 0 1 0 0 f 1 0 0 0 0 0 0 1 ] .
( θ ) = [ cos θ sin θ 0 0 sin θ cos θ 0 0 0 0 cos θ sin θ 0 0 sin θ cos θ ] ,
T ( f ) = ( f ) ( π / 4 ) ( f ) = 1 2 [ 1 0 0 f 0 1 f 0 0 f 1 1 0 f 1 0 0 1 ] .
W ( x 1 , y 1 , x 2 , y 2 , 0 ) = ( 2 / f 2 λ 2 ) exp ( i k ( x 1 y 1 x 2 y 2 ) / f ) × W s ( u 1 , υ 1 , u 2 , υ 2 , 4 f ) × exp [ i k 2 ( υ 1 x 1 u 1 y 1 υ 2 x 2 + u 2 y 2 ) / f ] × exp [ i k ( u 1 υ 1 u 2 υ 2 ) / f ] d u 1 d υ 1 d u 2 d υ 2 ,
W ( u 1 , υ 1 , u 2 , u 2 , 4 f ) = I s exp [ ( u 1 2 + u 2 2 ) / w u 2 ] exp [ ( υ 1 2 + υ 2 2 ) / w υ 2 ] × exp [ ( u 1 u 2 ) 2 / 2 σ u 2 ] exp [ ( υ 1 + υ 2 ) 2 / 2 σ υ 2 ] .
f = Ω = z R = π w u 2 β u / λ = π w υ 2 β υ / λ
β = β u β υ ,
η = β u β υ 1 β u β υ
w 2 ( 0 ) = ½ ( w u 2 + w υ 2 ) ,
u ( 0 ) = [ k w 2 ( 0 ) ] 1 ( β u β υ ) / β u β υ .
σ 2 ( 0 ) = w 2 ( 0 ) β u β υ / ( 1 β u β υ ) ,
W ( u 1 , υ 1 , u 2 , υ 2 ) = P ( θ u , θ υ ) U * ( u 1 , υ 1 ; θ u , θ υ ) × U ( u 2 , υ 2 ; θ u , θ υ ) d θ u d θ υ ,
U ( u , υ ; θ u , θ υ ) = I s exp ( u 2 / w u 2 ) exp ( υ 2 / w υ 2 ) × exp ( i k θ u u ) exp ( i k θ υ υ ) ,
P ( θ u , θ υ ) = ( 2 π ) 1 k 2 σ u σ υ exp [ ½ ( k σ u ) 2 θ u 2 ] × exp [ ½ ( k σ υ ) 2 θ υ 2 ] ,
U ( x , y , θ u , θ υ ) = A exp [ ( x x 0 ) 2 / w x 2 ] exp [ ( y y 0 ) 2 / w y 2 ] × exp [ i c ( x x 0 ) ( y y 0 ) ] × exp [ i k θ x x + θ y y ] ,
A = i 2 I s [ 1 + ( 2 f / k w u w υ ) 2 ] 1 / 2 × exp ( i k L ) exp ( i k f θ u θ υ / 2 ) ,
w x 2 = w u 2 [ 1 + ( 2 f / k w u w υ ) 2 ] / 2 ,
w y 2 = w υ 2 [ 1 + ( 2 f / k w u w υ ) 2 ] / 2 ,
c = ( k / f ) [ 1 ( 2 f / k w u w υ ) 2 ] / [ 1 + ( 2 f / k w u w υ ) 2 ] ;
( x 0 , y 0 ) = ( f θ υ / 2 , f θ u / 2 ) ,
( θ x , θ y ) = ( θ u / 2 , θ υ / 2 ) .
w x 2 = z R k 2 ( 1 + β 2 ) [ η 2 ( 1 β 2 ) 2 + 4 β 2 ] 1 / 2 + η ( 1 β 2 ) ,
w y 2 = z R k 2 ( 1 + β 2 ) [ η 2 ( 1 β 2 ) 2 + 4 β 2 ] 1 / 2 η ( 1 β 2 ) ,
c = ( k / z R ) ( 1 β 2 ) / ( 1 + β 2 ) ,
[ x 0 ( z ) , y 0 ( z ) ] = ( x 0 + θ x z , y 0 + θ y z ) .
( w min / w max ) 2 = [ 1 1 g ( z ) ] / [ 1 + 1 g ( z ) ] ,
g ( z ) = 4 β 2 ( 1 + β 2 ) 2 [ 1 + ( 1 + β 2 ) 2 η 2 ( 1 β 2 ) 2 + 4 β 2 ( z 2 z R 2 2 z R z ) 2 ] × [ 1 + ( z 2 z R 2 2 z R z ) ] 1 ,
tan 2 ϕ = ( 2 z R z z 2 z R 2 ) [ η 2 ( 1 β 2 ) 2 + 4 β 2 ] 1 / 2 η ( 1 + β 2 ) .
w min / w max = β = β u ,
tan 2 ϕ = 2 z R z z 2 z R 2 .
W ( u 1 , u 2 , υ 1 , υ 2 ) = P ( θ ) U * ( u 1 , υ 1 ; θ ) U ( u 2 , υ 2 ; θ ) d θ ,
U ( u , υ ; θ ) = I s exp ( u 2 / w u 2 ) × exp ( υ 2 / w υ 2 ) exp ( i k θ u ) ,
P ( θ ) = ( 2 π ) 1 / 2 k σ exp [ ½ ( k σ ) 2 θ 2 ] .
P m = exp ( 2 m 2 / Δ 2 ) [ n = exp ( 2 n 2 / Δ 2 ) ] 1 ,

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