Abstract

An implementation of the ray-tracing method for the study of radiation properties, such as intensity and spatial coherence, of an amplified spontaneous emission system is presented. Simple relations that express the controlling effect of the refraction on the output radiation properties are derived. We use analytical expressions, along with detailed numerical calculations, in order to analyze the intensity and the spatial coherence of the output radiation measured in x-ray laser experiments.

© 1993 Optical Society of America

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  1. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).
  2. G. Hazak and A. Bar-Shalom, Phys. Rev. A 40, 7055–7064 (1989); R. A. London, M. Strauss, and M. D. Rosen, Phys. Rev. Lett. 65, 563–566 (1990).
    [CrossRef] [PubMed]
  3. M. D. Feit and J. A. Fleck, J. Opt. Soc. Am. B 7, 2048–2060 (1990).
    [CrossRef]
  4. R. A. London, Phys. Fluids 31, 184–192 (1988).
    [CrossRef]
  5. B. Boswell, D. Shvarts, T. Boehly, and B. Yaakobi, Phys. Fluids B 2, 436–444 (1990).
    [CrossRef]
  6. G. Hazak and A. Bar-Shalom, Phys. Rev. A 38, 1300–1308 (1988).
    [CrossRef] [PubMed]
  7. M. Gross and S. Haroche, Phys. Rep. 93, 301–396 (1982).
    [CrossRef]
  8. L. Allen and J. H. Eberly, Optical Resonance and Two Level Atoms(Wiley, New York, 1975).
  9. H. Haken, Rev. Mod. Phys. 47, 67–119 (1975).
    [CrossRef]
  10. Yu. A. Kravtsov and Yu. I. Orlov, Geometrical Optics of Inhomogeneous Media(Springer-Verlag, Berlin, 1990).
    [CrossRef]
  11. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, U.K., 1980).
  12. M. H. Sher, J. J. Macklin, J. F. Young, and S. E. Harris, Opt. Lett. 12, 891–893 (1987); C. P. J. Barty, D. A. King, G. Y. Yin, K. H. Hahn, J. E. Field, J. F. Young, and S. E. Harris, Phys. Rev. Lett. 61, 2201–2204 (1988).
    [CrossRef] [PubMed]
  13. M. D. Feit and J. A. Fleck, Opt. Lett. 16, 76–78 (1991).
    [CrossRef] [PubMed]
  14. O. Zahavi, Z. Zinamon, and G. Hazak, J. Opt. Soc. Am. A 9, 1807–1811 (1992). For nonrefractive media the degree of spatial coherence is given, according to the van Cittert–Zernike theorem, by the normalized Fourier transform of the source intensity distribution (see Ref. 11), i.e., |μ(x, 0)| = F^[S0(K0)]. Here, F^ denotes the normalized Fourier transform and K0= (2π/λL)x is the transform variable. From a geometrical point of view, this transform variable can be related to the optical length difference between two rays that start at a source point and reach two different observation points, x and 0. In the presence of refraction this property is maintained, although the rays are no longer straight and can be shown to yield a more general expression for the transform variable. Using geometrical optics, in which the phase of the field is related to the optical length of the ray, one can show that the generalized transform variable K is given by (2π/λ)a02x, where a02≡ ∂2ϕ/∂x∂x0≡ ∂k/∂x0. A rigorous derivation of this result is given in that paper.
    [CrossRef]
  15. J. E. Trebes, K. A. Nugent, S. Mrowka, R. A. London, T. W. Barbee, M. R. Carter, J. A. Koch, B. J. Maegowan, D. L. Matthews, L. B. Da Silva, G. F. Stone, and M. D. Feit, Phys. Rev. Lett. 68, 588–591 (1992).
    [CrossRef] [PubMed]
  16. E. Wolf, Phys. Rev. Lett. 56, 1370–1372 (1986); G. M. Morris and D. Falkis, Opt. Commun. 62, 5–11 (1987); H. C. Kandpal, J. S. Vaishya, and K. C. Joshi, Phys. Rev. A 41, 4541–4542 (1990).
    [CrossRef] [PubMed]

1992 (2)

1991 (1)

1990 (2)

B. Boswell, D. Shvarts, T. Boehly, and B. Yaakobi, Phys. Fluids B 2, 436–444 (1990).
[CrossRef]

M. D. Feit and J. A. Fleck, J. Opt. Soc. Am. B 7, 2048–2060 (1990).
[CrossRef]

1989 (1)

G. Hazak and A. Bar-Shalom, Phys. Rev. A 40, 7055–7064 (1989); R. A. London, M. Strauss, and M. D. Rosen, Phys. Rev. Lett. 65, 563–566 (1990).
[CrossRef] [PubMed]

1988 (2)

R. A. London, Phys. Fluids 31, 184–192 (1988).
[CrossRef]

G. Hazak and A. Bar-Shalom, Phys. Rev. A 38, 1300–1308 (1988).
[CrossRef] [PubMed]

1987 (1)

1986 (1)

E. Wolf, Phys. Rev. Lett. 56, 1370–1372 (1986); G. M. Morris and D. Falkis, Opt. Commun. 62, 5–11 (1987); H. C. Kandpal, J. S. Vaishya, and K. C. Joshi, Phys. Rev. A 41, 4541–4542 (1990).
[CrossRef] [PubMed]

1982 (1)

M. Gross and S. Haroche, Phys. Rep. 93, 301–396 (1982).
[CrossRef]

1975 (1)

H. Haken, Rev. Mod. Phys. 47, 67–119 (1975).
[CrossRef]

Allen, L.

L. Allen and J. H. Eberly, Optical Resonance and Two Level Atoms(Wiley, New York, 1975).

Barbee, T. W.

J. E. Trebes, K. A. Nugent, S. Mrowka, R. A. London, T. W. Barbee, M. R. Carter, J. A. Koch, B. J. Maegowan, D. L. Matthews, L. B. Da Silva, G. F. Stone, and M. D. Feit, Phys. Rev. Lett. 68, 588–591 (1992).
[CrossRef] [PubMed]

Bar-Shalom, A.

G. Hazak and A. Bar-Shalom, Phys. Rev. A 40, 7055–7064 (1989); R. A. London, M. Strauss, and M. D. Rosen, Phys. Rev. Lett. 65, 563–566 (1990).
[CrossRef] [PubMed]

G. Hazak and A. Bar-Shalom, Phys. Rev. A 38, 1300–1308 (1988).
[CrossRef] [PubMed]

Boehly, T.

B. Boswell, D. Shvarts, T. Boehly, and B. Yaakobi, Phys. Fluids B 2, 436–444 (1990).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, U.K., 1980).

Boswell, B.

B. Boswell, D. Shvarts, T. Boehly, and B. Yaakobi, Phys. Fluids B 2, 436–444 (1990).
[CrossRef]

Carter, M. R.

J. E. Trebes, K. A. Nugent, S. Mrowka, R. A. London, T. W. Barbee, M. R. Carter, J. A. Koch, B. J. Maegowan, D. L. Matthews, L. B. Da Silva, G. F. Stone, and M. D. Feit, Phys. Rev. Lett. 68, 588–591 (1992).
[CrossRef] [PubMed]

Da Silva, L. B.

J. E. Trebes, K. A. Nugent, S. Mrowka, R. A. London, T. W. Barbee, M. R. Carter, J. A. Koch, B. J. Maegowan, D. L. Matthews, L. B. Da Silva, G. F. Stone, and M. D. Feit, Phys. Rev. Lett. 68, 588–591 (1992).
[CrossRef] [PubMed]

Eberly, J. H.

L. Allen and J. H. Eberly, Optical Resonance and Two Level Atoms(Wiley, New York, 1975).

Feit, M. D.

J. E. Trebes, K. A. Nugent, S. Mrowka, R. A. London, T. W. Barbee, M. R. Carter, J. A. Koch, B. J. Maegowan, D. L. Matthews, L. B. Da Silva, G. F. Stone, and M. D. Feit, Phys. Rev. Lett. 68, 588–591 (1992).
[CrossRef] [PubMed]

M. D. Feit and J. A. Fleck, Opt. Lett. 16, 76–78 (1991).
[CrossRef] [PubMed]

M. D. Feit and J. A. Fleck, J. Opt. Soc. Am. B 7, 2048–2060 (1990).
[CrossRef]

Fleck, J. A.

Gross, M.

M. Gross and S. Haroche, Phys. Rep. 93, 301–396 (1982).
[CrossRef]

Haken, H.

H. Haken, Rev. Mod. Phys. 47, 67–119 (1975).
[CrossRef]

Haroche, S.

M. Gross and S. Haroche, Phys. Rep. 93, 301–396 (1982).
[CrossRef]

Harris, S. E.

Hazak, G.

Koch, J. A.

J. E. Trebes, K. A. Nugent, S. Mrowka, R. A. London, T. W. Barbee, M. R. Carter, J. A. Koch, B. J. Maegowan, D. L. Matthews, L. B. Da Silva, G. F. Stone, and M. D. Feit, Phys. Rev. Lett. 68, 588–591 (1992).
[CrossRef] [PubMed]

Kravtsov, Yu. A.

Yu. A. Kravtsov and Yu. I. Orlov, Geometrical Optics of Inhomogeneous Media(Springer-Verlag, Berlin, 1990).
[CrossRef]

London, R. A.

J. E. Trebes, K. A. Nugent, S. Mrowka, R. A. London, T. W. Barbee, M. R. Carter, J. A. Koch, B. J. Maegowan, D. L. Matthews, L. B. Da Silva, G. F. Stone, and M. D. Feit, Phys. Rev. Lett. 68, 588–591 (1992).
[CrossRef] [PubMed]

R. A. London, Phys. Fluids 31, 184–192 (1988).
[CrossRef]

Macklin, J. J.

Maegowan, B. J.

J. E. Trebes, K. A. Nugent, S. Mrowka, R. A. London, T. W. Barbee, M. R. Carter, J. A. Koch, B. J. Maegowan, D. L. Matthews, L. B. Da Silva, G. F. Stone, and M. D. Feit, Phys. Rev. Lett. 68, 588–591 (1992).
[CrossRef] [PubMed]

Matthews, D. L.

J. E. Trebes, K. A. Nugent, S. Mrowka, R. A. London, T. W. Barbee, M. R. Carter, J. A. Koch, B. J. Maegowan, D. L. Matthews, L. B. Da Silva, G. F. Stone, and M. D. Feit, Phys. Rev. Lett. 68, 588–591 (1992).
[CrossRef] [PubMed]

Mrowka, S.

J. E. Trebes, K. A. Nugent, S. Mrowka, R. A. London, T. W. Barbee, M. R. Carter, J. A. Koch, B. J. Maegowan, D. L. Matthews, L. B. Da Silva, G. F. Stone, and M. D. Feit, Phys. Rev. Lett. 68, 588–591 (1992).
[CrossRef] [PubMed]

Nugent, K. A.

J. E. Trebes, K. A. Nugent, S. Mrowka, R. A. London, T. W. Barbee, M. R. Carter, J. A. Koch, B. J. Maegowan, D. L. Matthews, L. B. Da Silva, G. F. Stone, and M. D. Feit, Phys. Rev. Lett. 68, 588–591 (1992).
[CrossRef] [PubMed]

Orlov, Yu. I.

Yu. A. Kravtsov and Yu. I. Orlov, Geometrical Optics of Inhomogeneous Media(Springer-Verlag, Berlin, 1990).
[CrossRef]

Sher, M. H.

Shvarts, D.

B. Boswell, D. Shvarts, T. Boehly, and B. Yaakobi, Phys. Fluids B 2, 436–444 (1990).
[CrossRef]

Siegman, A. E.

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

Stone, G. F.

J. E. Trebes, K. A. Nugent, S. Mrowka, R. A. London, T. W. Barbee, M. R. Carter, J. A. Koch, B. J. Maegowan, D. L. Matthews, L. B. Da Silva, G. F. Stone, and M. D. Feit, Phys. Rev. Lett. 68, 588–591 (1992).
[CrossRef] [PubMed]

Trebes, J. E.

J. E. Trebes, K. A. Nugent, S. Mrowka, R. A. London, T. W. Barbee, M. R. Carter, J. A. Koch, B. J. Maegowan, D. L. Matthews, L. B. Da Silva, G. F. Stone, and M. D. Feit, Phys. Rev. Lett. 68, 588–591 (1992).
[CrossRef] [PubMed]

Wolf, E.

E. Wolf, Phys. Rev. Lett. 56, 1370–1372 (1986); G. M. Morris and D. Falkis, Opt. Commun. 62, 5–11 (1987); H. C. Kandpal, J. S. Vaishya, and K. C. Joshi, Phys. Rev. A 41, 4541–4542 (1990).
[CrossRef] [PubMed]

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, U.K., 1980).

Yaakobi, B.

B. Boswell, D. Shvarts, T. Boehly, and B. Yaakobi, Phys. Fluids B 2, 436–444 (1990).
[CrossRef]

Young, J. F.

Zahavi, O.

Zinamon, Z.

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

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

Opt. Lett. (2)

Phys. Fluids (1)

R. A. London, Phys. Fluids 31, 184–192 (1988).
[CrossRef]

Phys. Fluids B (1)

B. Boswell, D. Shvarts, T. Boehly, and B. Yaakobi, Phys. Fluids B 2, 436–444 (1990).
[CrossRef]

Phys. Rep. (1)

M. Gross and S. Haroche, Phys. Rep. 93, 301–396 (1982).
[CrossRef]

Phys. Rev. A (2)

G. Hazak and A. Bar-Shalom, Phys. Rev. A 40, 7055–7064 (1989); R. A. London, M. Strauss, and M. D. Rosen, Phys. Rev. Lett. 65, 563–566 (1990).
[CrossRef] [PubMed]

G. Hazak and A. Bar-Shalom, Phys. Rev. A 38, 1300–1308 (1988).
[CrossRef] [PubMed]

Phys. Rev. Lett. (2)

J. E. Trebes, K. A. Nugent, S. Mrowka, R. A. London, T. W. Barbee, M. R. Carter, J. A. Koch, B. J. Maegowan, D. L. Matthews, L. B. Da Silva, G. F. Stone, and M. D. Feit, Phys. Rev. Lett. 68, 588–591 (1992).
[CrossRef] [PubMed]

E. Wolf, Phys. Rev. Lett. 56, 1370–1372 (1986); G. M. Morris and D. Falkis, Opt. Commun. 62, 5–11 (1987); H. C. Kandpal, J. S. Vaishya, and K. C. Joshi, Phys. Rev. A 41, 4541–4542 (1990).
[CrossRef] [PubMed]

Rev. Mod. Phys. (1)

H. Haken, Rev. Mod. Phys. 47, 67–119 (1975).
[CrossRef]

Other (4)

Yu. A. Kravtsov and Yu. I. Orlov, Geometrical Optics of Inhomogeneous Media(Springer-Verlag, Berlin, 1990).
[CrossRef]

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, U.K., 1980).

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

L. Allen and J. H. Eberly, Optical Resonance and Two Level Atoms(Wiley, New York, 1975).

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Figures (11)

Fig. 1
Fig. 1

Temporal buldup of the on-axis radiation intensity with gain–dephasing ratio G0L/Γ = 10 ps.

Fig. 2
Fig. 2

Temporal buildup of the near-field intensity pattern with gain–dephasing ratio G0L/Γ = 10 ps. The medium parameters are a = 50 μm, L = 1 cm, λ = 18.3 nm, and ne = 5 × 1020 cm−3.

Fig. 3
Fig. 3

Near-field intensity pattern for (a) a uniform and (b) a nonuniform medium. The medium parameters are a = 50 μm, L = 1 cm, λ = 18.3 nm, ne = 5 × 1020 cm−3, and G0 = 5 cm−1.

Fig. 4
Fig. 4

Far-field intensity pattern for (a) a uniform and (b) a nonuniform medium. The medium parameters are the same as those for Fig. 3.

Fig. 5
Fig. 5

Set of rays in a refractive medium with a far-field condition [Eq. (13)]. The medium parameters are a = 50 μm, L = 1 cm, λ = 18.3 nm, and ne = 5 × 1020 cm−3

Fig. 6
Fig. 6

Amplitude of the radiation field, a0, calculated along the rays of Fig. 5 for different values of observation angle θ. The medium parameters are the same as those for Fig. 5.

Fig. 7
Fig. 7

Three-dimensional plot of a set of rays in a refractive cylindrical medium. The medium parameters are the same as those for Fig. 5.

Fig. 8
Fig. 8

Temporal buildup of the degree of spatial coherence with gain–dephasing ratio G0/Γ = 10. The medium parameters are a = 60 μm, L = 2 cm, λ = 20.3 nm, G0 = 5 cm−1 and ne = 0.

Fig. 9
Fig. 9

Degree of spatial coherence for different values of electron density. The medium parameters are a = 60 μm, L = 3 cm, λ = 20.3 nm, and G0 = 5 cm−1.

Fig. 10
Fig. 10

Degree of spatial coherence in the far-field region near the two peaks of the intensity (see Fig. 4(b)]. The medium parameters are the same as those for Fig. 4.

Fig. 11
Fig. 11

Degree of spatial coherence of a three-dimensional non-refractive medium: a comparison between numerical calculations and the analytical expression [Eq. (17)]. The medium parameters are a = 50 μm, L = 1 cm, λ = 18.3 nm, and ne = 0 cm−3.

Equations (27)

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i k 0 ( 1 c t + z ) E ( r T , z , t ) + 1 2 r T 2 E ( r T , z , t ) + 1 2 k 0 2 [ 1 - η 2 ( r T , z ) ] E ( r T , z , t ) = 2 π d k 0 2 [ P ( r T , z , t ) ] ave N 2 ( r T , z ) .
i t P ( r T , z , t ) = ( Δ - i γ ) P ( r T , z , t ) - 2 d W ( r T , z , t ) E ( r T , z , t ) + δ F ( r T , z , t ) ,
t W ( r T , z , t ) = 0.
[ δ F ( r T , z , t ) ] ave [ δ F ( r T , z , t ) ] ave = 2 ( Γ / N 2 ) δ 2 ( r T - r T ) × δ ( z - z ) δ ( t - t ) , [ P ( r T , z , 0 ) ] ave [ P ( r T , z , 0 ) ] ave = ( 1 / N 2 ) δ 2 ( r T - r T ) × δ ( z - z ) ,
C = 0 t d t 0 d 2 r T 0 0 L d z 0 K ( r T , z , t ; r T 0 , z 0 , t 0 ) × K * ( r T , z , t ; r T 0 , z 0 , t 0 ) S ( R T 0 , z 0 , t 0 ) .
S ( r T 0 , z 0 , t 0 ) = ( 4 π d ) 2 [ 2 Γ + δ ( t 0 ) ] N 2 ( r T 0 , z 0 ) .
d r T d z = r T ϕ k .
d d z z + d r T d z · r T .
d d z k = 1 2 r T [ η 2 ( r T , z ) ] .
d d z ϕ = 1 2 k 2 - 1 2 [ 1 - η 2 ( r T , z ) ] ,             ϕ ( z 0 ) = 0 ,
d d z g = v 2 ( r T , z ) ,             g ( z 0 ) = 0 ,
d d z a 0 = - 1 2 ( r T · k ) a 0 ,             a 0 ( z 0 ) = a 00 ,
d d z ( r T k ) = 1 2 r T r T [ η 2 ( r T , z ) ] - ( r T k ) · ( r T k ) , r T k ( z 0 ) = ( r T k ) 0 .
K ( r T , z , t ; r T 0 , z 0 , t 0 ) = a 0 ( r T , z ; r T 0 , z 0 ) exp [ i k 0 ϕ ( r T , z ; r T 0 , z 0 ) ] exp ( - Γ τ ) × I 0 [ 4 g ( r T , z ; r T 0 , z 0 ) Γ τ ] 1 / 2 Θ ( τ ) ,
I ( r T , z , t ) C ( r T , z , t ; r T , z , t ) = ( 4 π d ) 2 d 2 r T 0 0 L d z 0 N 2 ( r T 0 , z 0 ) K K * | t 0 = 0 + 2 Γ ( 4 π d ) 2 0 t d t 0 d 2 r T 0 × 0 L d z 0 N 2 ( r T 0 , z 0 ) K K * ,
I ( r T , z , ) = ( const . ) × d 2 r T 0 0 L d z 0 N 2 ( r T 0 , z 0 ) a 0 2 ( r T , z , r T 0 , z 0 ) × exp [ g ( r T , z , r T 0 , z 0 ) ] I 0 [ g ( r T , z , r T 0 , z 0 ) ] .
μ ( r T , z , t ; r T , z , t ) C ( r T , z , t ; r T , z , t ) [ I ( r T , z , t ) I ( r T , z , t ) ] 1 / 2 ,
μ ( r T , z , ; r T , z , ) = const . [ I ( r T , z ) I ( r T , z ) ] 1 / 2 d 2 r T 0 0 L d z 0 N 2 ( r T 0 , z 0 ) × a 0 ( r T , z ; r T 0 , z 0 ) a 0 ( r T , z ; r T 0 , z 0 ) exp { i k 0 [ ϕ ( r T , z ; r T 0 , z 0 ) - ϕ ( r T , z ; r T 0 , z 0 ) ] } × exp [ g ( r T , z ; r T 0 , z 0 ) + g ( r T , z ; r T 0 , z 0 ) 2 ] × I 0 [ g ( r T , z ; r T 0 , z 0 ) g ( r T , z ; r T 0 , z 0 ) ] 1 / 2 .
ω p 2 ( x ) = { ω p 0 2 [ 1 - ( x / a ) 2 ] - a x a 0 elsewhere .
v 2 ( x = { G 0 [ 1 - ( x / a ) 2 ] - a x a 0 elsewhere ,
x ( z ) = 1 cosh [ ( α / L ) ( z ¯ - z 0 ) ] { x 0 cosh [ α L ( z ¯ - z ) ] + ( θ L α ) sinh [ α L ( z - z 0 ) ] } ,
a 0 face ( z 0 , L ; θ ) = { cosh [ α L ( L - z 0 ) ] + [ 1 α ( D / L ) ] sinh [ α L ( L - z 0 ) ] } - 1 / 2 1 D 1 / 2 ,
a 0 side ( z 0 , z ¯ ; θ ) = { cosh [ α L ( z ¯ - z 0 ) ] - [ α ( α / L ) θ - 1 α ( D / L ) ] sinh [ α L ( z ¯ - z 0 ) ] } - 1 / 2 1 D 1 / 2 ,
ω p 2 ( r ) = { ω p 0 2 [ 1 - ( r / R ) 2 ] 0 r R 0 elsewhere .
μ ( x , L ; 0 , L ) = const . [ I ( x , L ) I ( 0 , L ) ] 1 / 2 - a + a d x 0 0 L d z 0 N 2 ( x 0 , z 0 ) × a 0 ( x , L , x 0 , z 0 ) a 0 ( 0 , L ; x 0 , z 0 ) × exp { i 2 π λ [ ϕ , L ; x 0 , z 0 ) - ϕ ( 0 , L ; x 0 , z 0 ) ] } × exp [ g ( x , L ; x 0 z 0 ) + g ( 0 , L ; x 0 , z 0 ) 2 ] × I 0 [ g ( x , L ; x 0 , z 0 ) g ( 0 , L ; x 0 , z 0 ) ] 1 / 2 .
μ ( x , L ; 0 , L ) ( sin ( K x ) / K x ,
μ ( r , L ; 0 , L ) = 2 J 1 ( κ r ) / κ r ,

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