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76
OPTICS LETTERS / Vol. 16, No. 2 / January 15,1991
Spatial coherence of laboratory soft-x-ray lasers
M. D. Feit and J. A. Fleck, Jr.
Lawrence Livermore National Laboratory, Livermore, California 94550
Received June 26,1990; accepted October 5,1990; manuscript in hand November 9,1990
Employing a time-dependent model with counterpropagating beams and saturable gain, we calculate the degree of
coherence for a range of x-ray lasers. From these results we infer a scaling law that can be used to estimate the
degree of coherence of a general class of x-ray lasers.
Rapid progress in the development of laboratory softx-ray lasers lends hope that these devices can serve as
intense sources of coherent x rays.1- 3 A possible application is the production of x-ray holograms4 for
studying biological processes on a microscopic scale.
Current laboratory x-ray lasers function as single-pass
amplifiers without the benefit of feedback mirrors and
thus behave as amplifiers of incoherent spontaneous
emission radiation. Thus spatial coherence of the
emitted light comes from aperturing by the boundaries of the gain medium, just as coherent light results
from aperturing of an incoherent source such as an
incandescent lamp.
The description of amplified spontaneous emission
(ASE) devices is, however, more complex, in part because the source and the aperture are longitudinally
distributed and because the ASE device contains a
gain medium, which may be saturable, and a spacedependent refractive index.
Determination of the spatial coherence of the laser
field requires both amplitude and phase information,
which makes it necessary to solve a wave equation.
London et al. 5 have treated coherence by solving a
paraxial wave equation in terms of normal modes,
which is valid only for unsaturated gain. For saturated gain it is necessary to treat counterpropagating
beams in a saturable gain medium by using numerical
beam-propagation techniques. In this Letter we apply such a model to the evaluation of the degree of
coherence of some representative ASE devices and
show how the introduction of a dimensionless transverse position variable leads to a wide generalization
of the results.
The equation describing the counterpropagating
beams is6
E = E+ exp[i(wt
-
kz)] + E- exp[i(cot + kz)],
(1)
where the complex field amplitudes E+ and E- satisfy
the paraxial wave equations
2ikl daE + 9E) = V1 2E+ + iakE+
2
+ k (n
2ik
(
c
aEL
at
2
+ ikS+(x, z),
(2a)
1)E- + ikS-(x, z).
(2b)
-1)E+
aE
+ k 2 (n2
-
0146-9592/91/020076-03$5.00/0
Here V I2 = 02/8x2 + a2 /8y2, a is a saturable gain
coefficient, k = wlc, n is the refractive index, and S+
and S- are stochastic sources defined by
J-1 K-1
S'(x, y, z, t) = A
N12
N/2
, E, E
E
6(t - jAt)
j=O k=O m=-N/2+1 n=-N/2+1
kAz)}
X expli[2ir(mx + ny)IL + jkmn]1, (3)
X {u[z
-
(k + 1)Az] - u(z
-
where u(z) = 1 if z 2 0 and 0 if z < 0. Each M-function
contribution is represented by a finite or band-limited
Fourier series whose components have equal amplitudes but random phases Ojkmnevaluated from pseudorandom numbers. The Fourier series are constructed by sampling on an N X N square grid of side L
and grid spacing Ax = Ay = LIN. The field equations
(2a) and (2b) are solved by using the fast-Fouriertransform-based split-operator algorithm.6
The saturable gain coefficient can be determined
from the rate equations governing the laser transition.
For a simple four-level model that includes only radiative transitions and neglects the density of lower laser
state atoms, the saturable gain coefficient has the
form 6
ao
1 + I/Isat'
I = IE+I2 + E-12 ,
(4)
where ao is the small-signal gain coefficient and the
saturation intensity is given by
Isat=2APO,
2
(5)
v0
with Avo representing the atomic linewidth. The
spontaneous emission normalization constant A in Eq.
(3) can be determined from the laser transition rate
equations and a transfer equation governing the laser
radiation. The result is6
2a"1/2
N(Az) 1/2
AIVL AU 1/2
ljsat
Avo 4,r )
(6)
where N2 is the number of sampling points on the twodimensional calculational grid and AvLis the frequency bandwidth of the laser radiation. If the ratio Avo/
AVL is equated to unity, the gain will saturate in ap© 1991 Optical Society of America
January 15, 1991 / Vol. 16, No. 2 / OPTICS LETTERS
proximately 15 small-signal gain lengths for typical
configurations.
The spatial coherence of an optical beam is defined
by the complex degree of coherence 7 :
quantitatively
A12(X1
X2)
(E*(x 1 ,
=
t)E(X
2,
(7
t)
Z12 (x 1, x 2 ) = [(IE(xl, t)12 )(IE(X 2, t)12)]1/ 2 '
(7)
where for simplicity we consider a two-dimensional
system. Normally the angle brackets in Eq. (7) would
signify ensemble averages, but for a stationary random
process one can substitute time averages. Thus we
can use solutions generated by Eqs. (2) and (3) to
compute A12Clearly Eqs. (1) and (2) apply to a variety of ASE
systems, but to gain understanding we restrict our
attention to a simple system, namely, a two-dimen-
77
ted in Fig. 2 as a function of xa/XL, showing that the
degree of coherence agrees quite nicely through the
first lobe for all values of L. Beyond the first lobe the
patterns are quite diverse. In any case, agreement
over the first lobe is important because the first lobe
defines the degree of coherence of the laser.
Defining the coherence length Xcoh as the distance
from the origin over which the degree of coherence is at
least 0.5, we find from Fig. 2 that
(10)
XL/4a.
Xcoh-
For the coherence length to correspond to the entire
radius of the laser, condition (10) gives
a2
1
AL
4'
(11)
sional slab 51.6 ,m in width, having lengths of 1, 2, 4,
and 8 cm. We further assume that X = 20 nm and ao =
5 cm-'.
The spontaneous emission source is normal-
ized with AVo/AVL= 1. Thus the 1- and the 2-cm lasers
should exhibit unsaturated gain, the 4-cm laser moderate gain saturation, and the 8-cm laser strong gain
saturation. For simplicity we ignore refraction, although it can be important for exploding foil lasers,
which exhibit free-electron density gradients. The
effect of refraction is discussed in Ref. 6.
Figure 1 shows the degree of coherence, IAL12(0, x)I,
plotted against x/a, where a is the half-width of the
slab, for the laser lengths listed above. The coherence
patterns resemble diffraction patterns, suggesting
that one invoke the van Cittert-Zernike theorem7 to
gain understanding. This theorem allows the complex degree of coherence to be computed on a plane
some distance away from an incoherent source of light.
For one-transverse-dimensional geometry the result is
e(
ex
A 12 (O, x, z) =
f
X
j
ikx2
dx'(IE(x', 0)12)
dx'(IE(x', 0)12) exp iZ)-
~~~~z
(8)
Here L12( X, z) is the complex degree of coherence
between the field at the origin and at the point x,
evaluated in a plane situated a longitudinal distance z
from the source plane. If, for simplicity, we assume
that the time-averaged intensity (IE(x, 0)12) is uniform over the source plane and extends from -a to a,
the resulting complex degree of coherence from Eq. (8)
2
/ikx
= exp
-i2z)
sinkxa)
kxa
0.0
0.2
0.4
0.6
0.8
1.0
1.2
x/a
Fig. 1. Plot of the degree of coherence as a function of
fractional distance x/a from the center of the laser, for four
laser lengths.
-2z
_ ,0
A 12(0, X,z)
N4
=L
1.00
0.80
0.60
cN
=&
0.40
0.20
(9)
Some of the plots in Fig. 1 suggest the functional
behavior (sin y/y)2 , where y is an appropriately scaled
variable. Equation (9) suggests that y = kxa/L may
be the appropriate variable, where a and L are the
width and the length, respectively, of the laser. To
test this scaling the degree of coherence has been plot-
0.00
-, 9.
0
r":=
1
2
3
xa/XL
Fig. 2. Degree of coherence for the four laser lengths of Fig.
1 plotted as a function of the dimensionless parameter xa/
AL. The curve with points is for L = 1 cm.
78
OPTICS LETTERS / Vol. 16, No. 2 / January 15, 1991
It is of interest to determine the effective source for
the output of the laser quantitatively by making use of
Eq. (8). One must first perform a least-squares fit of
the phase of the complex degree of coherence to determine the position of the source. After removing the
phase from the right-hand side of Eq. (8) one can
determine the intensity distribution of the effective
source by performing an inverse Fourier transform on
the remaining factor. The resulting effective source
positions and intensity distributions are displayed in
Fig. 3. Unfortunately the effective source distributions deviate substantially from the expected flat profile and thus do not explain why conditions (10) and
3.0
2.0
1
1.0
Co
0
O
X
1.0
._
>, 0.5
a)
(11) work so well. It should be remembered, however,
that conditions (10) and (11) apply to the main lobe of
the degree of coherence, whereas the effective source
distributions in Fig. 3 depend on the degree of coherence over its full range of variation. The Gaussianlike shape of the effective source intensity and the
0
-S2.0
degree of coherence as shown in Fig. 1 for L = 8 cm
1.0
0-
-40
-20
0
20
x( Pm)
40
Fig. 3. Effective source intensity, calculated with the van
Cittert-Zernike formula, as a function of transverse position, for three of the laser lengths considered in Fig. 1. The z
coordinate measures the distance of the effective source
position from the.back end of the laser.
where /Vis the Fresnel number of the laser. We have
calculated the degree of coherence for a variety of
other slabs ranging from 7.5 to 200 /im in radius, characterized by both unsaturated and saturated gain. In
all cases condition
(10) was fulfilled.
We conclude
that conditions (10) and (11) provide a general description of the coherence of ASE devices in the absence of transverse gain and refractive-index gradients.
For a Ne-like Se x-ray laser8 typical parameters
would be a = 200 tum,L = 4 cm, and X = 20 nm. In this
case condition (10) gives
Xcoh =
1 ttm, indicating poor
coherence properties. Although this result is changed
slightly by including transverse variation of refraction
and gain,6 the basic conclusion regarding coherence
remains the same.
Figure 2 and condition (10) suggest that the degree
of coherence of the laser output can be approximated
by the degree of coherence of a uniformly illuminated
incoherent slit of half-width a/n located a distance L/n
behind the output face, where n is an arbitrary divisor.
suggest that the output beam is characterized by a
single mode.
To summarize, relations (10) and (11) provide a
simple criterion for determining the coherence of an xray laser configuration. According to these relations,
currently available x-ray lasers are not likely to be
highly coherent devices.
This research was performed under the auspices of
the U.S. Department of Energy by the Lawrence Livermore National Laboratory under contract W-7405ENG-48.
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