Speckle statistics and transverse coherence of an x-ray laser with fluctuations in its active medium

doi:10.1364/OE.21.003225

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Speckle statistics and transverse coherence of an x-ray laser with fluctuations in its active medium

K. A. Janulewicz, C. M. Kim, and H. Stiel »View Author Affiliations

Optics Express, Vol. 21, Issue 3, pp. 3225-3234 (2013)
http://dx.doi.org/10.1364/OE.21.003225

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▸ Article Outline
– Abstract
1. Introduction
2. The model
3. Method
4. Results and discussion
5. Conclusion
– References and links

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Abstract

It is shown that the statistics of the intensity distribution in the output beam of a collisional X-ray laser, analysed in terms of the degree of freedom or equivalently the number of the coherence modes in the beam cross-section, has non-Gaussian character. The non-Gaussian character seems to be caused by the small-scale plasma/medium fluctuations. It was assumed that these overlap the modal structure imposed by the geometry of the medium and considered as equivalent to a large-scale inhomogeneity. Thus, the fluctuations decide about the character of the output beam transverse coherence. It is also shown that the relevant to this model compound statistics of the intensity fluctuations in the output beam is well described by the m-m-distribution, a specific form of the K-distribution. The deviation from the Gaussian statistics was confirmed by the field correlation function at the laser exit plane, retrieved from the experimental data.

1. Introduction

One of the fundamental conclusions of the modal theory of X-ray lasers (XRLs) was the necessity of increasing the aspect ratio of the plasma column to improve modal selectivity of the medium and, in the final effect, to increase the transverse coherence [1

P. D. Drummond and J. H. Eberly, “Transverse coherence and scaling in four dimensional simulations of superfluorescence,” Phys. Rev. A 25, 3446 (1982) [CrossRef]

–5

R. P. Ratowsky and R. A. London, “Propagation of mutual coherence in refratctive x-ray lasers using a WKB method,” Phys. Rev. A 51, 2361 (1995) [CrossRef] [PubMed]

]. This conclusion can be extended to all mirrorless lasers showing a very high gain level (50–80 cm⁻¹ in the case of XRLs) or to the passive or active guiding structures. As a consequence of a short-lived gain, the amplified spontaneous emission (ASE) is used in the modern XRLs as the energy extraction mechanism [6

G. J. Pert, “Output characteristics of amplified-stimulated-emission lasers,” J. Opt. Soc. Am. B 11, 1425 (1994) [CrossRef]

]. One of the distinguishing effects of the ASE-based lasers is the presence of speckle - a deep and random modulation of the intensity distribution in the output beam, giving a kind of granular structure. This is caused mainly by lack of the mode-selecting cavity. As a result, the lasers of this type need description in the terms of the multi-mode propagation. Interestingly, gain-guiding and refractive anti-guiding belong inherently to the working regime of the XRLs using a slab target [3

R. A. London, M. Strauss, and M. D. Rosen, “Modal analysis of x-ray laser coherence,” Phys. Rev. Lett. 65, 563 (1990) [CrossRef] [PubMed]

, 4

P. Amendt, R. London, and M. Strauss, “Optimization of single-stage- x-ray laser coherence,” Phys. Rev. A 47, 4348 (1993) [CrossRef] [PubMed]

The conclusion of the modal theory has never been verified experimentally in a systematic way, although some of the reported experimental observations contradicted it [7

J. E. Trebes, K. A. Nugent, S. Mrowka, R. A. London, T. W. Barbee, M. R. Carter, J. A. Koch, B. J. MacGowan, D. L. Matthews, L. B. Da Silva, G. F. Stone, and M. D. Feit, “Measurement of the spatial coherence of a soft-x-ray laser,” Phys. Rev. Lett. 68, 588 (1992) [CrossRef] [PubMed]

, 8

Y. Liu, Y. Wang, M. A. Larotonda, B. M. Luther, J. J. Rocca, and D. T. Attwood, “Spatial coherence measurements of a 13.2 nm transient nickel-like cadmium soft X-ray laser pumped at grazing incidence,” Opt. Express 14, 12872–9 (2006) [CrossRef] [PubMed]

]. Dynamics of the gain-guided system with a very high gain level allows for pre-dominance of the few modes of the highest gain. Moreover, usually the noise in the gain-guided systems is significantly correlated. The gain-guided modes determine the possible transverse profiles of the intensity and coherence. The natural multi-mode structure is the reason of the partial transverse coherence that can be quantified by a ratio of the equivalent coherence area A_coh (the area transverse to the propagation direction over which the field is correlated), and the beam cross-section A_beam (e.g. at 1/e²-level). Typically, the laser-pumped XRLs show a coherence level equivalent to about 1% of the coherent photons in the beam. The approximated transverse coherence length is typically given as

L_{coh} ≅ \sqrt{A_{coh}}

While short-wavelength speckle was frequently discussed in relation to the scattering experiments, it was hardly analyzed in the context of the lasers using gain-guiding. Further on, we will focus on the case of plasma-based XRLs. Recently, a simple model of the speckle origin in the X-ray laser output beam has been proposed in [9

O. Gilbaud, A. Klisnick, K. Cassou, S. Kazamias, D. Ros, G. Jamelot, D. Joyeux, and D. Phalippou, “Origin of microstructures in picosecond X-ray laser beam,” Europhys. Lett. 74, 823–829 (2006) [CrossRef]

]. This model used description in the terms of the classic and partially developed speckle [10

J. W. Goodman, Statistical Optics , chapt.9.1, (A Wiley-Interscience Publication, John Wiley and Sons, Inc., 1985)

] and this is, as we are going to show in this paper, usually an inadequate description. The question arises, whether the speckle itself could be a reasonable source of the additional information on the modal structure and transverse coherence, in the situation when Young’s interferometry suffers due to presence of speckle and systematically underestimates the coherence level [8

In this paper we deliver, by the statistical analysis of the recorded speckle patterns, the experimental evidence that the modal theory of XRLs, subject to supplementing it by plasma fluctuations [11

P. Amendt, M. Strauss, and R. A. London, “Plasma fluctuations and X-ray laser transverse coherence,” Phys. Rev. A 53, R23 (1996) [CrossRef] [PubMed]

], describes in a reasonable way XRL transverse coherence. The evidence includes the reproduced field correlation function at the laser exit. The analysis and proposed model assume a non-Gaussian character of the random walk approximating the mode build-up process; the method never used in the context of X-ray lasers. The experimental data is taken from the X-ray laser working in the specific scheme of the grazing incidence pumping (GRIP).

2. The model

An active system initiated by a random signal (spontaneous emission) is in the best way described by expanding the field in a set of coherent but non-self-adjoint (power non-orthogonal) eigenmodes with random coefficients [2

G. Hazak and A. Bar-Shalom, “Mode-selecting effects and coherence in hot-plasma x-ray lasers,” Phys. Rev. A 40, 7055 (1989) [CrossRef] [PubMed]

–5

R. P. Ratowsky and R. A. London, “Propagation of mutual coherence in refratctive x-ray lasers using a WKB method,” Phys. Rev. A 51, 2361 (1995) [CrossRef] [PubMed]

]. These, put into the wave equation, allow for the variable separation and hence separate descriptions of the longitudinal and transverse effects. The paramount parameters of such a modal approach are effective Fresnel number N_Fe = kg₀a² (with a length L in the standard form of the expression for the Fresnel number replaced by the gain scale-length 1/g₀; k is the wave number, while a is the radius of the active medium) and the normalised density parameter η = h₀/g₀, where h₀ = ω_pe²/kc². These parameters, when calculated for our GRIP X-ray laser with a length of 7 mm and a width of the active medium 2a equal to 32 μm, are equal to 610 and 2.6, respectively. These values are significantly reduced in comparison to those characteristic for the quasi-steady-state (QSS) XRLs, for which the modal theory was originally formulated [4

P. Amendt, R. London, and M. Strauss, “Optimization of single-stage- x-ray laser coherence,” Phys. Rev. A 47, 4348 (1993) [CrossRef] [PubMed]

]. The reduced values confirm that the development trend of X-ray lasers was in accordance with the recommendations of the modal theory [3

R. A. London, M. Strauss, and M. D. Rosen, “Modal analysis of x-ray laser coherence,” Phys. Rev. Lett. 65, 563 (1990) [CrossRef] [PubMed]

]. However, the increase in the aspect ratio (equivalent to increase in the medium length at a constant medium width) also did not bring the increase in the coherence level, expected from the scaling rules of the modal theory.

Amendt et al. attempted to explain this failing for QSS XRLs by incorporating plasma fluctuations into the modal theory [11

P. Amendt, M. Strauss, and R. A. London, “Plasma fluctuations and X-ray laser transverse coherence,” Phys. Rev. A 53, R23 (1996) [CrossRef] [PubMed]

]. It was assumed that the turbulences caused by the plasma fluctuations induce mode coupling. Weaker effect should give and an interplay between the source motion normally to the propagation axis and the amplification parameters changing randomly but with a certain correlation length l_s along the propagation direction. These effects were found to be responsible for the significant reduction (saturation) in the increase rate of the coherence level along the medium length (see Fig. 3 in [11

P. Amendt, M. Strauss, and R. A. London, “Plasma fluctuations and X-ray laser transverse coherence,” Phys. Rev. A 53, R23 (1996) [CrossRef] [PubMed]

]). A similar behaviour is expected in the model proposed below. In the slab target arrangement of the GRIP X-ray lasers, plasma fluctuations can arise due to imperfect uniformity (speckle) of the pump laser beam. The spatial intensity modulation of the preforming beam (long duration) will generate modulation of the ablation front (ablation pressure p_a ∼ P_L^7/9 increases with the laser power P_L[12

G. J. Pert, “Two-dimensional hydrodynamic models of laser-produced plasmas,” J. Plasma Physics 41 (part 2), 263 (1989) [CrossRef]

]) resembling the Rayleigh-Taylor instability [13

R. G. Evans, A.J. Bennett, and G.J. Pert, “Rayleigh-Taylor instabilities in laser accelerated targets,” Phys. Rev. Lett. 49, 1639 (1982) [CrossRef]

]. Such a modulation of the main heating beam would form the plasma (electron) pressure structure p_e = n_ek_BT_e likely increasing the fluctuations intensity. Limited smoothness of the target surface may also contribute to the modulation of the ablation front [13

R. G. Evans, A.J. Bennett, and G.J. Pert, “Rayleigh-Taylor instabilities in laser accelerated targets,” Phys. Rev. Lett. 49, 1639 (1982) [CrossRef]

]. Intuitively and very roughly, this can be illustrated as a gain medium in a form of a sliced fibre, with the slices randomly moving perpendicularly to the fibre axis [14

B. Crosignani and A. Yariv, “Statistical properties of modal noise in fiber-laser systems,” J. Opt. Soc. Am. 73, 1022 (1983) [CrossRef]

]. The theoretical findings of [11

P. Amendt, M. Strauss, and R. A. London, “Plasma fluctuations and X-ray laser transverse coherence,” Phys. Rev. A 53, R23 (1996) [CrossRef] [PubMed]

] should be also valid for our GRIP XRL. It was conservatively estimated that the reference parameter σ_f = 4l_s〈(dx₀/dz)²〉/g₀a, quantifying strength of the fluctuations, should not differ by more than a factor of 4 from that estimated in [11

P. Amendt, M. Strauss, and R. A. London, “Plasma fluctuations and X-ray laser transverse coherence,” Phys. Rev. A 53, R23 (1996) [CrossRef] [PubMed]

] (σ_f ≅0.1) for the first XRL laser. Here angular brackets mean averaging over the ensemble and x₀(z) is the transverse displacement of the active medium caused by the non-uniform imprint of the laser beam.

Some additional randomization factor in the speckle distribution appears due to differences in the group velocity [15

L. Casperson and A. Yariv, “Pulse propagation in a high-gain medium,” Phys. Rev. Lett. 26, 293 (1971) [CrossRef]

] of the modes traversing different gain area. Considering two modes sampling the gain areas of 80 cm⁻¹ and 50 cm⁻¹, the corresponding group velocities υ_gr will give a delay of ∼2.3 ps over a length of 5 mm. This is much more than the usual coherence time τ_coh and such modes become uncorrelated or statistically independent. The gain maximum moves also slightly with time [16

G. J. Pert, “Optimizing the performance of nickel-like collisionally pumped x-ray lasers,” Phys. Rev. A 73, 033809 (2006) [CrossRef]

, 17

K. A. Janulewicz and C. M. Kim, “Role of the precursor in a triple-pulse pumping scheme of a nickel-like silver soft-x-ray laser in the grazing-incidence-pumping geometry,” Phys. Rev. E 82, 056405 (2010) [CrossRef]

]. These effects have, however, limited influence on the random process of the modes build-up as they are related to the large scale (conventional) modes and the number of those is limited in the considered geometry. The fluctuations and other randomizing effects cause the statistics of the intensity distribution to be non-Gaussian and this shows some analogy to the scintillation theory applied to the radiation traversing a medium under turbulent (fluctuating) conditions [18

E. Jakeman and P. N. Pusey, “Significance of K distributions in scattering experiments,” Phys. Rev. Lett. 40, 546 (1978) [CrossRef]

–20

M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser bem propagating through turbulent media,” Opt. Eng. 40, 1554 (2001) [CrossRef]

The XRL-speckle analysis presented in [9

] was based on the assumptions valid for the classic speckle. These are frequently described by a random walk with the resultant phasor having the components (amplitude and phase) ruled by the Gaussian joint probability density function (PDF). This model is applicable to the systems free of fluctuations and with a modal structure determined by the gain medium geometry. It assumes that the probability to obtain a given integrated intensity level E within the intensity interval of E, E + dE at an observation plane, is determined by the negative exponential probability density distribution for a single mode laser. As a consequence, the multimode beam will obey to a good approximation the gamma (called also m−) statistics (Eq. (1)), being a compound statistics of some number of statistically independent fully coherent individual modes.

p_{M} (E / \bar{E}) = M^{M} {(E / \bar{E})}^{M - 1} \exp (- M E / \bar{E}) / [Γ (M) \bar{E}]

(1)

Here, M is the number of statistically independent but equally intense (average value) coherence modes or equivalently, the number of the degrees of freedom within the beam diameter (M = A_beam/A_coh). Hence, bearing in mind what was said in the introduction, 1/M can be used as a measure of the transverse coherence. Γ is the standard gamma function. The speckle contrast is equal to

K_{s p} = σ_{E} / \bar{E} = 1 / \sqrt{M}

, where σ_E is the standard deviation of the intensity fluctuation in the speckle pattern. Thus, the intensity distribution in the speckle could be a source of information on the modal structure and coherence level of the radiation. For the sake of consistency with the experimental practice we have used E/Ē - the normalized time-integrated pulse intensity instead of the normalized peak intensity I/Ī. This multi-mode radiation creates so-called partially developed speckle.

3. Method

3.1. Data acquisition and processing

The experimental data processed below were obtained from the far-field images of the output beam of a Ni-like silver soft X-ray laser working at 13.9 nm. The images were recorded in the single-shot regime, during the experiment described in detail in [21

H. T. Kim, C.-M Kim, I. W. Choi, H. C. Kang, N Hafz, S. G. Lee, J. H. Sung, T. J. Yu, K.-H. Hong, T. M. Jeong, Y.-C. Noh, D.-K. Ko, J. Tümmler, P. V. Nickles, W. Sandner, K. A. Janulewicz, and J. Lee, “Characteristics of Ni-like silver x-ray laser pumped by single profiled laser pulse,” J. Opt. Soc. Am B 25, B76–B84 (2008) [CrossRef]

]. The observation plane was placed 55 cm away from the laser exit. The plasma column of 7 mm in length was obtained by irradiation of a slab target in the typical GRIP geometry (double-pulse scheme with a long pulse incident normally to the target). A small-signal-gain coefficient was estimated to be of 61 cm⁻¹ and an active medium radius deduced from the field correlation function (CF_f = 〈A(ξ₁, η₁)A^*(ξ₂, η₂)〉) at the medium exit was 16 μm. A(ξ_i, η_i) is the complex field amplitude at the laser exit and i=1,2.

The average time-integrated intensity Ē and the corresponding standard deviation σ_E were the main parameters of the statistical output. An example of the speckle pattern obtained in the experiment and the corresponding histogram are given in Fig. 1. The histogram of the image intensity has been created with the intensity binning including 50 levels. The analysed rectangular part of the beam was selected within a window drawn to cover the maximum possible area of the irregular beam spot and minimize the contribution from the non-irradiated but noisy vicinity of the recorded spot. These low-level contributions still distort slightly the histogram at the origin, even if the background was subtracted from the analyzed pattern.

Fig. 1 a) The far-field image of an X-ray laser shot in a double-pulse GRIP geometry and b) the corresponding probability density distribution. The speckle image corresponds to the analysis area.

3.2. Simple estimates

The mean value of the integrated intensity obtained from processing the speckle pattern was put in Eq. (1) with M as the fitting parameter, the procedure suggested by Goodman as a reasonable one [22

J. W. Goodman, “Statistical properties of laser speckle patterns” in Laser Speckle and Related Phenomena , J. C. Dainty, Ed., (Springer Verlag, 2nd. edition, 1984)

]. A value of M=3.23≅3.2 was obtained in the fitting process of a high goodness (standard error ≤10⁻²). We used the formulae for polarized radiation but bearing in mind that the values of the degree of freedom have to be halved in the case of undefined polarisation. In consecutive four shots the derived values of M were between 2.4 and 6.5. From now on we concentrate on the speckle pattern shown in Fig. 1(a). The obtained result confirms that the photon statistics is definitely non-Poisonian (in this case the contrast value K_sp = σ_E/Ē would be equal to 1 [22

J. W. Goodman, “Statistical properties of laser speckle patterns” in Laser Speckle and Related Phenomena , J. C. Dainty, Ed., (Springer Verlag, 2nd. edition, 1984)

]). The contrast value extracted from the data was equal to 0.56. On the other hand, the surprisingly small number of modes determined in the fitting process would suggest a noticeable level of coherence. In fact, the estimated value of 30 % (equal to 1/M) of the coherent photons in the beam significantly overestimates the experimental values of the transverse coherence level. Even correction due to the limited analysis window (see [23

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications , (Roberts & Company Publishers, Englewood, Colorado 2006)

]), giving M_ef ≅4.8 for the full beam, reduces the contribution of the fully coherent photons only to a level of 14 %. These values are far out of the range quoted in any experimental report.

A circular, uniform, incoherently radiating disc (approximated by δ(x)-like CF_f) can be used as an additional reference of the coherence level in the observation plane. The generalised van Cittert-Zernike theorem [10

J. W. Goodman, Statistical Optics , chapt.9.1, (A Wiley-Interscience Publication, John Wiley and Sons, Inc., 1985)

, 24

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

] was applied to calculate the transverse coherence length in such a case. The latter depends on the intensity distribution in the source plane I(ξ, η), and for the flat-top (uniform) intensity profile it is expressed as A_coh = λ²/ΔΩ_s. Here, ΔΩ_s is the solid angle subtended by the source area (2a=32 μm) from the observation point. This formula gave a coherence length of 259.6 μm. More realistic radial profile I(r) = I₀cosh⁻²(r/a), approximating parabolic shape of the intensity distribution, gave a coherence length of 159.5 μm. This result is closer to the typical experimentally observed values (L_coh ≃ d_beam/18 ÷ d_beam/20) and corresponds to 0.25 % of fully coherent photons in the beam.

4. Results and discussion

In fact, two major assumptions of the used gamma statistics (Eq. (1)), namely equal average intensity and full statistical independence, are not satisfied in the X-ray laser medium. Not all modes are equally excited as the spontaneous emission level is dependent on time and space. Violation of the condition of statistical independence could bring artificial reduction in the number of the observed coherence modes [23

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications , (Roberts & Company Publishers, Englewood, Colorado 2006)

, 25

R. Barakat, “The brightness distribution of the sum of two coherent speckle patterns,” Opt. Commun. 8, 14–16 (1973) [CrossRef]

] but not on a scale of the huge existing difference between the model and the experimental output.

It is known that there exists a mutual dependence between the intensity distribution (optionally field correlation) at the source plane (near field, NF) and the field correlation (optionally, intensity distribution) in the far-field (FF) [23

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications , (Roberts & Company Publishers, Englewood, Colorado 2006)

]. These images are related by a generalized van Cittert-Zernike theorem or, in other words, by a Fourier transform [23

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications , (Roberts & Company Publishers, Englewood, Colorado 2006)

, 24

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

]. The 2D-Fourier inverse transform of the intensity distribution at the observation plane in FF was used to get the field correlation function (CF_f ) at the plasma column exit. The result of the transformation is shown in Fig. 2. It is seen that CF_f is slightly asymmetric suggesting different beam spot diameters and divergences along both Cartesian coordinates. The resulting CF_f at the source plane is narrow in the main peak with a width of d_cor ≤1 μm at FWHM. It is significantly broadened at a low level (1/e² level, i.e. 13.5 % of the maximum value) to about 3 – 4 μm. At a very low level the non-zero “correlation spots” are observed even within much larger circle. It was assumed that the points where C_f drops from to zero and show no noisy structure determine radii of the beam at the active medium exit (Fig. 2(b)). From that, the effective radius of the circular spot of the same area as the beam of elliptic shape was estimated to be equal to ≅16 μm. This value has been assumed to be the approximated gain-area-radius. The shape and a width of the main peak of CF_f are noticeably different from those obtained by the assumption of the uniform incoherent source of the same diameter (CF_f width equal to 0.5 μm).

Fig. 2 a) The contour plot of the 2D correlation function (CF_f ) of the field at the laser exit reconstructed by the Fourier transform of the intensity autocorrelation function in the far-field (speckle pattern); the saturation in the central part of the image is introduced deliberately to show details of the low-level part of CF_f at a very high dynamics of the figure; b) cross-sections of the reconstructed correlation function along x and y axes.

The shape of this experimental correlation function, especially presence of the residual wings/pedestal, compares favourably with the mutual coherence function calculated for the fluctuation parameter σ_f =0.1 in Ref. [11

P. Amendt, M. Strauss, and R. A. London, “Plasma fluctuations and X-ray laser transverse coherence,” Phys. Rev. A 53, R23 (1996) [CrossRef] [PubMed]

] (see Fig. 3 there). The residual correlation tail in the shape of CF_f is a clear indication of the non-Gaussian statistics of the field components and determines the angular spread of the speckle pattern.

Fig. 3 Fit of the histogram presented in Fig. 1(b) applying K-distribution in its specific form (m-m distribution) given by Eq. (3). M1=3.2, M2=62. The dashed-line-curve is an m distribution with M=3.23.

Jakeman and Pusey [18

E. Jakeman and P. N. Pusey, “Significance of K distributions in scattering experiments,” Phys. Rev. Lett. 40, 546 (1978) [CrossRef]

, 26

E. Jakeman, “Speckle statistics with a small number of scatterers,” Opt. Eng. 23, 453 (1984) [CrossRef]

] pioneered interest in the non-Gaussian statistics applicable to the optical scattering problems in turbulent media by proposing so-called K-distribution. It was shown that a specific form of the random walk with the fluctuating number of contributions is the limiting process leading to this distribution. They found that the negative binomial distribution (NBD) well fulfils this role, and the normalised variance remains finite even if the number of steps tends to infinity. The continuous version of NBD with non-negative steps (e.g. for intensity) is the gamma distribution, discussed previously.

Scintillation index is the commonly used measure of the turbulence strength in the scintillation theory [27

L. C. Andrews, R. L. Phillips, C. Y. Hopen, and M. A. Al-Habash, “Theory of optical scintillation,” J. Opt. Soc. Am 16, 1417 (1999) [CrossRef]

]. It is equal to the normalised irradiance variance

{\tilde{σ}}_{E}^{2}

and hence, to the squared speckle contrast

K_{s p}^{2}

, considered earlier. In the scintillation theory a turbulence with

{\tilde{σ}}_{E}^{2} = 0.31

belongs to a weak one as the scintillation index is significantly lower than 1. The description within the standard log-norm statistics, typical for this class of fluctuations, shows a noticeable level of uncertainty for this level of

{\tilde{σ}}_{E}^{2}

. It was also found that for our experimental data the log-norm distribution shows maximum at E/Ē slightly below 0.5, in contrast to our experimental PDF peaking about E/Ē=1. It is assumed that the non-stationary character of the optical turbulence gives a spatio-temporally dependent spontaneous emission (source) and induces mode coupling. Both effects result the non-Gaussian statistics of the optical field after traversing such a medium. These fields used to be modelled by a conditional random process describing a compound or multiply stochastic field.

A spherical coherent wave traversing a weak turbulence is well described by the Rice-Nakagami distribution, well approximated by the m-distribution, and this is identical to the gamma distribution [19

V. S. Rao Gudimetla and J. Fred Holmes, “Probability density function of the intensity for a laser-generated speckle field after propagation through the turbulent atmosphere,” J. Opt. Soc. Am. 72, 1213 (1982) [CrossRef]

]. In our model, the spontaneous noise, initially incoherent but with a some correlation level (result of gain-guiding) will be amplified (increase of coherence) over the distance l_s, defined earlier. After that, it will dissipate part of its energy in the scattering process on a small-scale (weak) turbulence. One can consider turbulences as the scatterers being a collection of specular points (facet model) associated with small wavelets (modes generated by diffraction on this turbulence). This process is carried on the top of a larger scale structure (discussed earlier modes determined by “smooth” or “non-turbulent” propagation conditions, i.e. medium geometry, gain and refraction). The standard mode structure can be treated as a result of a large-scale perturbation to which applies the m (gamma) statistics described earlier. This acts like a weak lens containing radiation within the medium. During propagation, the stimulated emission responsible for increase in coherence of the amplified modes competes with the deteriorating effect of diffraction on the turbulent cells. The total scattering process acts like a modulation of small-scale fluctuations by large-scale fluctuations. Translating this to the statistical description we can assume that the primary process with the m statistics is driving another process, which is also described by the m statistics (that of the standard multimode structure). In other words, the mean value of the first conditional process (condition that the mean has a specific value) is given by the m distribution, and this has to be multiplied by PDF of the second process (modes) and integrated according to the formula

p (E) = \int_{0}^{\infty} p_{1} (E | x) p_{2} (x) d x)

to get the unconditional PDF (p(E)) of the compound process. In our case this will lead effectively to a double-stochastic process. Here p₁(E | x) is the conditional probability (small-scale scattering) and p₂(x) is the PDF of the second process (standard modes). This model assumes tacitly that the large-scale scattering process (geometrical modes) has the same statistics as it would have without turbulences but with the constant mean value replaced by a random one, determined by the small-scale turbulent process.

This form of the compound statistics, called the m-m distribution (or equivalently gamma-gamma distribution), was introduced in [19

], in the form we have used. However its origin back-dates to much earlier time [28

M. Nakagami, “The m-distribution - a general formula of intensity distribution of rapid fading,” in Statistical Methods in Radio Wave Propagation , W. C. Hoffman, ed. (Pergamon, 1960), p.6–36

]. Further on we shall use the term m-m distribution. It is simply a more generalized form of the K-distribution proposed in [18

E. Jakeman and P. N. Pusey, “Significance of K distributions in scattering experiments,” Phys. Rev. Lett. 40, 546 (1978) [CrossRef]

] and seems to be suitable to describe correctly statistics of the non-Gaussian random processes like those in a quantum amplifier. We applied it in the form [19

]

p_{M - M} (E / \bar{E}) = \frac{2 {(M_{1} M_{2})}^{(M_{1} + M_{2}) / 2}}{E Γ (M_{1}) Γ (M_{2})} {(\frac{E}{E})}^{(M_{1} + M_{2} - 2) / 2} \times K_{| M_{2} - M_{1} |} (2 \sqrt{M_{1} M_{2} \frac{E}{E}})

(2)

where K_{|M₂−M₁|} is the modified Bessel function of the second kind and of |M₂ − M₁| order. M₂ and M₁ denote the number of modes in each of the contributing processes. This type of the distribution results in the speckle contrast (a ratio of the intensity standard deviation and the mean intensity) of the form [23

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications , (Roberts & Company Publishers, Englewood, Colorado 2006)

]

\frac{σ_{E}}{E} = \sqrt{\frac{Γ (M_{1}) Γ (2 + M_{1}) Γ (M_{2}) Γ (2 + M_{2})}{Γ^{2} (1 + M_{1}) Γ^{2} (1 + M_{2})} - 1}

(3)

The experimental data presented in Fig. 1 were fitted with the m-m distribution given by Eq. (2) and the result is shown in Fig. 3. The fit is of equally high goodness as that with the Eq. (1), especially at higher intensity level. A contrast ratio σ_E/Ē calculated by applying Eq. (3) with the obtained in the fitting process parameters M₁ and M₂ was equal to 0.58 and compared favourably with a value of 0.56 available directly from the experimental data. Taking into account the compound character of the statistics applied (equivalently the multiplicative character of the small and large-scale scattering) and the fact that in the proposed mode buildup process large-scale process modulates the small-scale process,we can use the values M₁ and M₂ derived from fitting the histogram and formulate the ratio A_beam/A_coh ≅ M₁ × M₂ as a new measure of beam transverse coherence. This relation is a direct analogue of the expression A_beam/A_coh ≅ M formulated earlier, with M derived from Eq. (1) in the fitting process. In the final effect, each “large superscatterer” (geometrical mode) contains some number of small scatterers according to the spirit of the idea of the K-distribution. It is worth noting that the M₁ retrieved from Eq. (3) is equal to the value of the M parameter obtained by using Eq. (1). From this it follows that L_coh ≅ 263.5 μm. This corresponds to 1/14 of the total beam diameter and agrees reasonably well with a value of 1/18 obtained in [8

] by applying Young’s interferometry.

5. Conclusion

In summary, we have found that even if the statistical analysis of a speckle pattern in the X-ray laser output beam of a partially coherent source fulfils the m-distribution being a consequence of the classic Gaussian statistics of the involved variates, this is only an apparent effect. More exact inspection of the results shows that, in general, the coherence level expressed by the number of the degrees of freedom (modes) in the beam cross-section, derived from the fitting process based on the m-distribution, is noticeably overestimated. Additionally, the intensity distribution in the FF transferred by the inverse Fourier transform to the XRL exit plane gives a field correlation function at the laser exit suggesting non-Gaussian character of the processes leading to speckle creation. Following the idea of strong influence of the small-scale plasma spatio-temporal fluctuations on the coherence level given in [11

P. Amendt, M. Strauss, and R. A. London, “Plasma fluctuations and X-ray laser transverse coherence,” Phys. Rev. A 53, R23 (1996) [CrossRef] [PubMed]

], we have formulated a model including these fluctuations in a form of weak turbulences being modulated by the large-scale structure of the conventional modes controlled by the static parameters of the gain medium. This combination of both effects leads effectively to a double-stochastic process, in which each of the contributing processes is ruled by the m-statistics. Thus, the compound process can be described by the m-m distribution giving effectively increased number of the modes. This describes reasonably the coherence level of the laser output. Transverse coherence of a GRIP XRL, estimated in this way, agrees well with that determined by other methods.

Spontaneous character of the mode excitation, medium inhomogeneities and the build-up mechanisms of the photon flux from the spontaneous noise give very broad correlation areas but only at a low level (partial correlation). We estimated a transverse coherence length corresponding to 1/14–1/15 of the the beam diameter at the plasma column exit, depending on the intensity profile. The problem of the fully coherent (single mode) signal injected into an amplifying medium with fluctuations is out of scope of this work but its solution is essential for consistency of the proposed scenario.

The obtained result extends also the class of the propagation phenomena where K-distribution proved to be a very effective analysis method of turbulent media by the active optical systems with gain- and refraction index-guiding accompanied by low-frequency fluctuations (mechanically unstable fiber lasers, active fibers or simply randomly distributed medium defects).

Acknowledgments

This work was supported by the Ministry of Education, Science and Technology of Korea through Basic Science Research Program (No. R15-2008-006-03001-0); the Korea-Germany collaboration program of Korean National Research Foundation (No. 2010-00633) and by Gwangju Institute of Science and Technology through DASAN grant and Photonics 2020 project.

References and links

1.	P. D. Drummond and J. H. Eberly, “Transverse coherence and scaling in four dimensional simulations of superfluorescence,” Phys. Rev. A 25, 3446 (1982) [CrossRef]
2.	G. Hazak and A. Bar-Shalom, “Mode-selecting effects and coherence in hot-plasma x-ray lasers,” Phys. Rev. A 40, 7055 (1989) [CrossRef] [PubMed]
3.	R. A. London, M. Strauss, and M. D. Rosen, “Modal analysis of x-ray laser coherence,” Phys. Rev. Lett. 65, 563 (1990) [CrossRef] [PubMed]
4.	P. Amendt, R. London, and M. Strauss, “Optimization of single-stage- x-ray laser coherence,” Phys. Rev. A 47, 4348 (1993) [CrossRef] [PubMed]
5.	R. P. Ratowsky and R. A. London, “Propagation of mutual coherence in refratctive x-ray lasers using a WKB method,” Phys. Rev. A 51, 2361 (1995) [CrossRef] [PubMed]
6.	G. J. Pert, “Output characteristics of amplified-stimulated-emission lasers,” J. Opt. Soc. Am. B 11, 1425 (1994) [CrossRef]
7.	J. E. Trebes, K. A. Nugent, S. Mrowka, R. A. London, T. W. Barbee, M. R. Carter, J. A. Koch, B. J. MacGowan, D. L. Matthews, L. B. Da Silva, G. F. Stone, and M. D. Feit, “Measurement of the spatial coherence of a soft-x-ray laser,” Phys. Rev. Lett. 68, 588 (1992) [CrossRef] [PubMed]
8.	Y. Liu, Y. Wang, M. A. Larotonda, B. M. Luther, J. J. Rocca, and D. T. Attwood, “Spatial coherence measurements of a 13.2 nm transient nickel-like cadmium soft X-ray laser pumped at grazing incidence,” Opt. Express 14, 12872–9 (2006) [CrossRef] [PubMed]
9.	O. Gilbaud, A. Klisnick, K. Cassou, S. Kazamias, D. Ros, G. Jamelot, D. Joyeux, and D. Phalippou, “Origin of microstructures in picosecond X-ray laser beam,” Europhys. Lett. 74, 823–829 (2006) [CrossRef]
10.	J. W. Goodman, Statistical Optics , chapt.9.1, (A Wiley-Interscience Publication, John Wiley and Sons, Inc., 1985)
11.	P. Amendt, M. Strauss, and R. A. London, “Plasma fluctuations and X-ray laser transverse coherence,” Phys. Rev. A 53, R23 (1996) [CrossRef] [PubMed]
12.	G. J. Pert, “Two-dimensional hydrodynamic models of laser-produced plasmas,” J. Plasma Physics 41 (part 2), 263 (1989) [CrossRef]
13.	R. G. Evans, A.J. Bennett, and G.J. Pert, “Rayleigh-Taylor instabilities in laser accelerated targets,” Phys. Rev. Lett. 49, 1639 (1982) [CrossRef]
14.	B. Crosignani and A. Yariv, “Statistical properties of modal noise in fiber-laser systems,” J. Opt. Soc. Am. 73, 1022 (1983) [CrossRef]
15.	L. Casperson and A. Yariv, “Pulse propagation in a high-gain medium,” Phys. Rev. Lett. 26, 293 (1971) [CrossRef]
16.	G. J. Pert, “Optimizing the performance of nickel-like collisionally pumped x-ray lasers,” Phys. Rev. A 73, 033809 (2006) [CrossRef]
17.	K. A. Janulewicz and C. M. Kim, “Role of the precursor in a triple-pulse pumping scheme of a nickel-like silver soft-x-ray laser in the grazing-incidence-pumping geometry,” Phys. Rev. E 82, 056405 (2010) [CrossRef]
18.	E. Jakeman and P. N. Pusey, “Significance of K distributions in scattering experiments,” Phys. Rev. Lett. 40, 546 (1978) [CrossRef]
19.	V. S. Rao Gudimetla and J. Fred Holmes, “Probability density function of the intensity for a laser-generated speckle field after propagation through the turbulent atmosphere,” J. Opt. Soc. Am. 72, 1213 (1982) [CrossRef]
20.	M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser bem propagating through turbulent media,” Opt. Eng. 40, 1554 (2001) [CrossRef]
21.	H. T. Kim, C.-M Kim, I. W. Choi, H. C. Kang, N Hafz, S. G. Lee, J. H. Sung, T. J. Yu, K.-H. Hong, T. M. Jeong, Y.-C. Noh, D.-K. Ko, J. Tümmler, P. V. Nickles, W. Sandner, K. A. Janulewicz, and J. Lee, “Characteristics of Ni-like silver x-ray laser pumped by single profiled laser pulse,” J. Opt. Soc. Am B 25, B76–B84 (2008) [CrossRef]
22.	J. W. Goodman, “Statistical properties of laser speckle patterns” in Laser Speckle and Related Phenomena , J. C. Dainty, Ed., (Springer Verlag, 2nd. edition, 1984)
23.	J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications , (Roberts & Company Publishers, Englewood, Colorado 2006)
24.	L. Mandel and E. Wolf, Optical Coherence and Quantum Optics , (Cambridge University Press, 1995)
25.	R. Barakat, “The brightness distribution of the sum of two coherent speckle patterns,” Opt. Commun. 8, 14–16 (1973) [CrossRef]
26.	E. Jakeman, “Speckle statistics with a small number of scatterers,” Opt. Eng. 23, 453 (1984) [CrossRef]
27.	L. C. Andrews, R. L. Phillips, C. Y. Hopen, and M. A. Al-Habash, “Theory of optical scintillation,” J. Opt. Soc. Am 16, 1417 (1999) [CrossRef]
28.	M. Nakagami, “The m-distribution - a general formula of intensity distribution of rapid fading,” in Statistical Methods in Radio Wave Propagation , W. C. Hoffman, ed. (Pergamon, 1960), p.6–36

OCIS Codes
(030.1640) Coherence and statistical optics : Coherence
(140.7240) Lasers and laser optics : UV, EUV, and X-ray lasers

ToC Category:
Coherence and Statistical Optics

History
Original Manuscript: November 6, 2012
Revised Manuscript: January 18, 2013
Manuscript Accepted: January 20, 2013
Published: February 1, 2013

Citation
K. A. Janulewicz, C. M. Kim, and H. Stiel, "Speckle statistics and transverse coherence of an x-ray laser with fluctuations in its active medium," Opt. Express 21, 3225-3234 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-3-3225

Sort: Author | Year | Journal | Reset

References

P. D. Drummond and J. H. Eberly, “Transverse coherence and scaling in four dimensional simulations of superfluorescence,” Phys. Rev. A25, 3446 (1982) [CrossRef]
G. Hazak and A. Bar-Shalom, “Mode-selecting effects and coherence in hot-plasma x-ray lasers,” Phys. Rev. A40, 7055 (1989) [CrossRef] [PubMed]
R. A. London, M. Strauss, and M. D. Rosen, “Modal analysis of x-ray laser coherence,” Phys. Rev. Lett.65, 563 (1990) [CrossRef] [PubMed]
P. Amendt, R. London, and M. Strauss, “Optimization of single-stage- x-ray laser coherence,” Phys. Rev. A47, 4348 (1993) [CrossRef] [PubMed]
R. P. Ratowsky and R. A. London, “Propagation of mutual coherence in refratctive x-ray lasers using a WKB method,” Phys. Rev. A51, 2361 (1995) [CrossRef] [PubMed]
G. J. Pert, “Output characteristics of amplified-stimulated-emission lasers,” J. Opt. Soc. Am. B11, 1425 (1994) [CrossRef]
J. E. Trebes, K. A. Nugent, S. Mrowka, R. A. London, T. W. Barbee, M. R. Carter, J. A. Koch, B. J. MacGowan, D. L. Matthews, L. B. Da Silva, G. F. Stone, and M. D. Feit, “Measurement of the spatial coherence of a soft-x-ray laser,” Phys. Rev. Lett.68, 588 (1992) [CrossRef] [PubMed]
Y. Liu, Y. Wang, M. A. Larotonda, B. M. Luther, J. J. Rocca, and D. T. Attwood, “Spatial coherence measurements of a 13.2 nm transient nickel-like cadmium soft X-ray laser pumped at grazing incidence,” Opt. Express14, 12872–9 (2006) [CrossRef] [PubMed]
O. Gilbaud, A. Klisnick, K. Cassou, S. Kazamias, D. Ros, G. Jamelot, D. Joyeux, and D. Phalippou, “Origin of microstructures in picosecond X-ray laser beam,” Europhys. Lett.74, 823–829 (2006) [CrossRef]
J. W. Goodman, Statistical Optics, chapt.9.1, (A Wiley-Interscience Publication, John Wiley and Sons, Inc., 1985)
P. Amendt, M. Strauss, and R. A. London, “Plasma fluctuations and X-ray laser transverse coherence,” Phys. Rev. A53, R23 (1996) [CrossRef] [PubMed]
G. J. Pert, “Two-dimensional hydrodynamic models of laser-produced plasmas,” J. Plasma Physics41 (part 2), 263 (1989) [CrossRef]
R. G. Evans, A.J. Bennett, and G.J. Pert, “Rayleigh-Taylor instabilities in laser accelerated targets,” Phys. Rev. Lett.49, 1639 (1982) [CrossRef]
B. Crosignani and A. Yariv, “Statistical properties of modal noise in fiber-laser systems,” J. Opt. Soc. Am.73, 1022 (1983) [CrossRef]
L. Casperson and A. Yariv, “Pulse propagation in a high-gain medium,” Phys. Rev. Lett.26, 293 (1971) [CrossRef]
G. J. Pert, “Optimizing the performance of nickel-like collisionally pumped x-ray lasers,” Phys. Rev. A73, 033809 (2006) [CrossRef]
K. A. Janulewicz and C. M. Kim, “Role of the precursor in a triple-pulse pumping scheme of a nickel-like silver soft-x-ray laser in the grazing-incidence-pumping geometry,” Phys. Rev. E82, 056405 (2010) [CrossRef]
E. Jakeman and P. N. Pusey, “Significance of K distributions in scattering experiments,” Phys. Rev. Lett.40, 546 (1978) [CrossRef]
V. S. Rao Gudimetla and J. Fred Holmes, “Probability density function of the intensity for a laser-generated speckle field after propagation through the turbulent atmosphere,” J. Opt. Soc. Am.72, 1213 (1982) [CrossRef]
M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser bem propagating through turbulent media,” Opt. Eng.40, 1554 (2001) [CrossRef]
H. T. Kim, C.-M Kim, I. W. Choi, H. C. Kang, N Hafz, S. G. Lee, J. H. Sung, T. J. Yu, K.-H. Hong, T. M. Jeong, Y.-C. Noh, D.-K. Ko, J. Tümmler, P. V. Nickles, W. Sandner, K. A. Janulewicz, and J. Lee, “Characteristics of Ni-like silver x-ray laser pumped by single profiled laser pulse,” J. Opt. Soc. Am B25, B76–B84 (2008) [CrossRef]
J. W. Goodman, “Statistical properties of laser speckle patterns” in Laser Speckle and Related Phenomena, J. C. Dainty, Ed., (Springer Verlag, 2nd. edition, 1984)
J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications, (Roberts & Company Publishers, Englewood, Colorado2006)
L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, (Cambridge University Press, 1995)
R. Barakat, “The brightness distribution of the sum of two coherent speckle patterns,” Opt. Commun.8, 14–16 (1973) [CrossRef]
E. Jakeman, “Speckle statistics with a small number of scatterers,” Opt. Eng.23, 453 (1984) [CrossRef]
L. C. Andrews, R. L. Phillips, C. Y. Hopen, and M. A. Al-Habash, “Theory of optical scintillation,” J. Opt. Soc. Am16, 1417 (1999) [CrossRef]
M. Nakagami, “The m-distribution - a general formula of intensity distribution of rapid fading,” in Statistical Methods in Radio Wave Propagation, W. C. Hoffman, ed. (Pergamon, 1960), p.6–36

Al-Habash, M. A.
Amendt, P.
Andrews, L. C.
Attwood, D. T.
Barakat, R.
Barbee, T. W.
Bar-Shalom, A.
Bennett, A.J.
Carter, M. R.
Casperson, L.
Cassou, K.
Choi, I. W.
Crosignani, B.
Da Silva, L. B.
Drummond, P. D.
Eberly, J. H.
Evans, R. G.
Feit, M. D.
Fred Holmes, J.
Gilbaud, O.
Goodman, J. W.
Hafz, N
Hazak, G.
Hong, K.-H.
Hopen, C. Y.
Jakeman, E.
Jamelot, G.
Janulewicz, K. A.
Jeong, T. M.
Joyeux, D.
Kang, H. C.
Kazamias, S.
Kim, C. M.
Kim, C.-M
Kim, H. T.
Klisnick, A.
Ko, D.-K.
Koch, J. A.
Larotonda, M. A.
Lee, J.
Lee, S. G.
Liu, Y.
London, R.
London, R. A.
Luther, B. M.
MacGowan, B. J.
Mandel, L.
Matthews, D. L.
Mrowka, S.
Nakagami, M.
Nickles, P. V.
Noh, Y.-C.
Nugent, K. A.
Pert, G. J.
Pert, G.J.
Phalippou, D.
Phillips, R. L.
Pusey, P. N.
Rao Gudimetla, V. S.
Ratowsky, R. P.
Rocca, J. J.
Ros, D.
Rosen, M. D.
Sandner, W.
Stone, G. F.
Strauss, M.
Sung, J. H.
Trebes, J. E.
Tümmler, J.
Wang, Y.
Wolf, E.
Yariv, A.
Yu, T. J.

Europhys. Lett.

O. Gilbaud, A. Klisnick, K. Cassou, S. Kazamias, D. Ros, G. Jamelot, D. Joyeux, and D. Phalippou, “Origin of microstructures in picosecond X-ray laser beam,” Europhys. Lett.74, 823–829 (2006) [CrossRef]

J. Opt. Soc. Am

L. C. Andrews, R. L. Phillips, C. Y. Hopen, and M. A. Al-Habash, “Theory of optical scintillation,” J. Opt. Soc. Am16, 1417 (1999) [CrossRef]

J. Opt. Soc. Am B

H. T. Kim, C.-M Kim, I. W. Choi, H. C. Kang, N Hafz, S. G. Lee, J. H. Sung, T. J. Yu, K.-H. Hong, T. M. Jeong, Y.-C. Noh, D.-K. Ko, J. Tümmler, P. V. Nickles, W. Sandner, K. A. Janulewicz, and J. Lee, “Characteristics of Ni-like silver x-ray laser pumped by single profiled laser pulse,” J. Opt. Soc. Am B25, B76–B84 (2008) [CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. B

G. J. Pert, “Output characteristics of amplified-stimulated-emission lasers,” J. Opt. Soc. Am. B11, 1425 (1994) [CrossRef]

J. Plasma Physics

G. J. Pert, “Two-dimensional hydrodynamic models of laser-produced plasmas,” J. Plasma Physics41 (part 2), 263 (1989) [CrossRef]

Opt. Commun.

R. Barakat, “The brightness distribution of the sum of two coherent speckle patterns,” Opt. Commun.8, 14–16 (1973) [CrossRef]

Opt. Eng.

E. Jakeman, “Speckle statistics with a small number of scatterers,” Opt. Eng.23, 453 (1984) [CrossRef]
M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser bem propagating through turbulent media,” Opt. Eng.40, 1554 (2001) [CrossRef]

Opt. Express

Y. Liu, Y. Wang, M. A. Larotonda, B. M. Luther, J. J. Rocca, and D. T. Attwood, “Spatial coherence measurements of a 13.2 nm transient nickel-like cadmium soft X-ray laser pumped at grazing incidence,” Opt. Express14, 12872–9 (2006) [CrossRef] [PubMed]

Phys. Rev. A

P. Amendt, M. Strauss, and R. A. London, “Plasma fluctuations and X-ray laser transverse coherence,” Phys. Rev. A53, R23 (1996) [CrossRef] [PubMed]
G. J. Pert, “Optimizing the performance of nickel-like collisionally pumped x-ray lasers,” Phys. Rev. A73, 033809 (2006) [CrossRef]
P. D. Drummond and J. H. Eberly, “Transverse coherence and scaling in four dimensional simulations of superfluorescence,” Phys. Rev. A25, 3446 (1982) [CrossRef]
G. Hazak and A. Bar-Shalom, “Mode-selecting effects and coherence in hot-plasma x-ray lasers,” Phys. Rev. A40, 7055 (1989) [CrossRef] [PubMed]
P. Amendt, R. London, and M. Strauss, “Optimization of single-stage- x-ray laser coherence,” Phys. Rev. A47, 4348 (1993) [CrossRef] [PubMed]
R. P. Ratowsky and R. A. London, “Propagation of mutual coherence in refratctive x-ray lasers using a WKB method,” Phys. Rev. A51, 2361 (1995) [CrossRef] [PubMed]

Phys. Rev. E

K. A. Janulewicz and C. M. Kim, “Role of the precursor in a triple-pulse pumping scheme of a nickel-like silver soft-x-ray laser in the grazing-incidence-pumping geometry,” Phys. Rev. E82, 056405 (2010) [CrossRef]

Phys. Rev. Lett.

E. Jakeman and P. N. Pusey, “Significance of K distributions in scattering experiments,” Phys. Rev. Lett.40, 546 (1978) [CrossRef]
L. Casperson and A. Yariv, “Pulse propagation in a high-gain medium,” Phys. Rev. Lett.26, 293 (1971) [CrossRef]
R. G. Evans, A.J. Bennett, and G.J. Pert, “Rayleigh-Taylor instabilities in laser accelerated targets,” Phys. Rev. Lett.49, 1639 (1982) [CrossRef]
R. A. London, M. Strauss, and M. D. Rosen, “Modal analysis of x-ray laser coherence,” Phys. Rev. Lett.65, 563 (1990) [CrossRef] [PubMed]
J. E. Trebes, K. A. Nugent, S. Mrowka, R. A. London, T. W. Barbee, M. R. Carter, J. A. Koch, B. J. MacGowan, D. L. Matthews, L. B. Da Silva, G. F. Stone, and M. D. Feit, “Measurement of the spatial coherence of a soft-x-ray laser,” Phys. Rev. Lett.68, 588 (1992) [CrossRef] [PubMed]

Other

J. W. Goodman, Statistical Optics, chapt.9.1, (A Wiley-Interscience Publication, John Wiley and Sons, Inc., 1985)
J. W. Goodman, “Statistical properties of laser speckle patterns” in Laser Speckle and Related Phenomena, J. C. Dainty, Ed., (Springer Verlag, 2nd. edition, 1984)
J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications, (Roberts & Company Publishers, Englewood, Colorado2006)
L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, (Cambridge University Press, 1995)
M. Nakagami, “The m-distribution - a general formula of intensity distribution of rapid fading,” in Statistical Methods in Radio Wave Propagation, W. C. Hoffman, ed. (Pergamon, 1960), p.6–36

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Speckle statistics and transverse coherence of an x-ray laser with fluctuations in its active medium

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Abstract

1. Introduction

2. The model

3. Method

3.1. Data acquisition and processing

3.2. Simple estimates

4. Results and discussion

5. Conclusion

Acknowledgments

References and links

References

Europhys. Lett.

J. Opt. Soc. Am

J. Opt. Soc. Am B

J. Opt. Soc. Am.

J. Opt. Soc. Am. B

J. Plasma Physics

Opt. Commun.

Opt. Eng.

Opt. Express

Phys. Rev. A

Phys. Rev. E

Phys. Rev. Lett.

Other

Cited By

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