INDEPENDENT PHASE AND AMPLITUDE CONTROL OF A LASER BEAM USING A SINGLE -PHASE -ONLY SPATIAL LIGHT MODULATOR
LLE Review, Volume 96
225
Laser-beam shaping is a rapidly developing field of research
driven by both technological improvements of beam-shaping
devices and the ever-increasing demands of applications. In
high-energy laser chains, efficient beam shaping is successfully
achieved in the front ends by passive methods such as
beam apodization1 or intracavity mode shaping;2 however,
these static techniques are unable to correct dynamic laser-
beam profiles caused by alignment drifts or thermal problems.
Spatial light modulators (SLM's) are versatile devices that
can modulate the polarization or the phase of laser beams at
high refresh rates. It has been demonstrated that a SLM can be
used to compensate for the thermal phase distortion occurring
in high-energy glass amplifiers.3 Similarly, SLM's have been
used in high-energy laser applications, such as intracavity
beam shaping4 or focal-spot control.5 In all these applications,
only the phase-modulation capability of the SLM was used;
however, there are numerous applications where phase-only
modulation can be achieved differently. For instance, deformable
mirrors are more attractive when it comes to wavefront
correction of a large, high-energy, laser beam. Their scale and
damage threshold allow them to be used within the power
amplifier, while SLM's are confined to the front end because
of their modest size and low damage threshold. Nevertheless,
a corrective device that would address both phase and amplitude
simultaneously may be successfully used in high-energy
lasers to significantly reduce the alignment procedure time, to
improve the amplifier fill factor by injecting a more-adapted
beam shape, to reduce the risk of damage in the laser chain by
removing hot spots, and to improve the on-target characteristics
of the beam by better control of the phase.
Several techniques have been proposed to produce complex
modulation of an electromagnetic field with an SLM for
encoding computer-generated holograms.6,7 In both cases,
two neighboring pixels with a single-dimension modulation
capability are coupled to provide the two degrees of freedom
required for independent phase and amplitude modulation. In
our work, we use a similar approach, but our requirements
Independent Phase and Amplitude Control of a Laser Beam
Using a Single-Phase-Only Spatial Light Modulator
differ from that of the hologram generation. First, the number
of modulation points across the beam does not need to be high
because spatial filtering imposes a low-pass limit on the spatial
frequencies allowed in the system. Second, a high-efficiency
modulation process is required to minimize passive losses.
Lastly, the required amplitude-modulation accuracy should be
better than measured shot-to-shot beam fluctuations for the
correction to be fully beneficial.
In this article, we propose a new method to modulate both
the phase and amplitude of a laser beam, with a single-phase-
only SLM using a carrier spatial frequency and a spatial filter.
As a result, the local intensity in the beam spatial profile is
related to the amplitude of the carrier modulation, while its
phase is related to the mean phase of the carrier. In the first part
of this article, we show the simple relation between the transmitted
intensity and the phase-modulation amplitude, and in
the second part, we experimentally verify this scheme and use
it to demonstrate beam shaping in a closed-loop configuration.
The principle of the modulation is depicted in Fig. 96.19 for
the case of a plane wave. The SLM is used as a phase-only
device that applies a one-dimensional phase grating to the
electric field. As a consequence, the two-dimensional propagation
integral reduces to a one-dimensional one. In such a
case, the electric field transmitted—or reflected—through the
modulator accumulates a phase φ given by
′=
()
[]
EE
jx
0
exp
,
φ
(1)
where E
0
can be complex and φ is a periodic square phase
modulation, of period Λ, oscillating between the values φ
1
and
φ
2
. After propagation through a lens, in an f - f configuration
and under the Fraunhofer approximation, the electromagnetic
field distribution at the focus of the lens is proportional to the
Fourier transform of Eq. (1). To calculate the Fourier transform
of E , one can consider, for instance, the initial electromagnetic
field as a sum of two square waves defined by
INDEPENDENT PHASE AND AMPLITUDE CONTROL OF A LASER BEAM USING A SINGLE -PHASE -ONLY SPATIAL LIGHT MODULATOR
226
LLE Review, Volume 96
EE
xn
x
j
n
10
2
1
=-
()
ï¿»
() ()
ℜ
δφΛ
Λ
rect
exp
,
(2)
EE
xn
x
j
n
20
2
2
2
=-
()
ï¿»-
() ()
ℜ
δφΛΛ
Λ
rect
exp
,
(3)
where δ is the Dirac function and denotes the convolution
product. After some algebra, the electromagnetic field distribution
at the focus, given as the sum of the Fourier transforms
of E
1
and E
2
, can be written as
˜
sin
cos
exp
,
EEc
jn
n
∝
⊇
⊄
ï¿»
↓
+
⊇
⊄
ï¿»
↓
∞
+
-
⊇
⊄
ï¿»
↓
∪
∈
⊆
ù
û
ú
-
()
ℜ
0
12
22
2
22
π
νφ
π
ν
φφ
π
ν
δν
ΛΔΛ
Λ
Λ
(4)
where n is the spatial frequency, and the Dirac comb function
represents the diffraction pattern created by the SLM phase
grating. Removing the higher-order terms
n >
()
0
in this
diffraction pattern with a spatial filter results in an electric field
given by Eq. (5), where the amplitude is determined by the
phase difference Δφ and the phase is equal to the average of φ
1
and φ
2
:
˜
cos
exp
.
EE
j
0
22
0
12
() ∝
⊇
⊄
ï¿»
↓
+
⊇
⊄
ï¿»
↓
Δφφφ
(5)
This result is still true for finite beams, provided the amplitude
and phase of the initial beam slowly vary with respect to
the modulation frequency. If the electromagnetic field spatially
varies at higher frequencies, then the imaging system
will act as a spatial filter and will modify the spatial distribution
of light regardless of the application of a phase modulation on
the SLM.
In the experimental setup shown in Fig. 96.20, the light
source is a pulsed, 300-Hz laser, the beam of which is up-
collimated so that it overfills the SLM area and is linearly
polarized. It is reflected off the SLM and then imaged on an
8-bit charge-coupled-device (CCD) camera (Cohu 4910
series) or to a Hartmann-Shack wavefront sensor (Wavefront
Sciences CLAS-HP). The camera was used mainly for system
alignment and diagnostic or whenever high-spatial-resolution
beam amplitude measurement was required. The wavefront
sensor was used for simultaneous phase and amplitude measurements.
We use a non-pixelated, 256-level, phase-only
SLM from Hamamatsu (Model X8267) with a 20 ∞
20-mm2 active area, optically addressed by a 768 ∞ 768-pixel
liquid crystal display (LCD) screen. Thanks to a slight defocus
of the imaging system, a continuous phase modulation can be
achieved on the SLM at the expense of a slight reduction in
the resolution.
SLM
SF
TC6277
E
0
A
x
0
xx
x
2
E
0
cos
f
f
f
2
f
1
Λ
E
0
A
x
A
x
f
< f >
Δ f
f
f
Figure 96.19
Independent phase- and amplitude-
modulation scheme. The input beam is
modulated in phase by the phase-only
SLM and then propagates through a
spatial filter (SF). The SLM is placed at
a focal distance from the SF lens so the
electromagnetic field distribution at the
SF pinhole is proportional to the Fou-
rier transform of the electromagnetic
field distribution at the SLM location.
INDEPENDENT PHASE AND AMPLITUDE CONTROL OF A LASER BEAM USING A SINGLE -PHASE -ONLY SPATIAL LIGHT MODULATOR
LLE Review, Volume 96
227
Although the SLM was designed for normal-incidence use,
we believe that a small angle of incidence does not affect the
system's performance. Only a 768 ∞ 768 matrix is used in the
imaging. The advantages of using a non-pixelated SLM are
(1) its absence of loss due to the fill factor and diffraction of
pixelated SLM and (2) its high damage threshold, which we
tested to be 680 (±130) mJ/cm2 with a 1-ns, 1053-nm Gaussian
beam. This value is nearly two times better than that of the
pixelated SLM that we tested.
To be relevant to beam shaping in a high-energy laser
facility, such as the OMEGA laser, the pass band of the beam-
shaping spatial filter must be at least as large as the spatial
filters in the main laser power amplifier, which are as large as
30 times the diffraction limit. To ensure removal of the SLM
carrier spatial frequency, the minimum spatial frequency must
then be at least 30 times the fundamental spatial frequency of
the beam. Practically, this means the minimum number of
pixels required for that application is 60 across the beam (two
per period). The beam f number in the spatial filter is 25,
which means that the diffraction-limited focal spot is roughly
25 μm and the pinhole should be at least 750 μm in diameter.
We used a 1-mm pinhole and a modulation frequency 64 times
that of the fundamental beam frequency. In the SLM plane, this
corresponds to a period of 12 pixels. For lower numbers of
pixels per period, the finite slope between two nearby pixels
degrades the modulation profile. For larger periods, up to 24
pixels, the beam is efficiently modulated by the SLM, but the
system becomes more sensitive to the pinhole alignment.
Figure 96.21 demonstrates a linear amplitude-modulation
scheme, as well as high contrast and arbitrary spatial shaping.
In the upper part, Eq. (5) is inversed to obtain the phase-
modulation amplitude corresponding to a linear amplitude
modulation, while the lower part of the image shows nearly
complete extinction for a π-rad modulation and ~100% transmission
when the phase modulation is 0 rad. The low-contrast
speckle, seen in Fig. 96.21, limits the achievable extinction
ratio that we measured varying from 50:1 to 10:1 across the
beam. It should be noted that the extinction is achieved while
only half of the dynamic range of the SLM was used (128
levels), which leaves at least half of the SLM dynamic range
free for phase modulation, in the worst case.
SLM
H-S
CCD
SF
TC6278
Polarizer
Laser
Figure 96.20
Experimental setup. SF: spatial-filter pinhole; H-S:
Hartmann-Shack wavefront sensor. The SLM is used in
reflection and a flip-in mirror is used to measure either the
intensity or phase profiles.
TC6279
0.0
0.2
0.4
0.6
0.8
1.0
T
ransmission
(%)
Figure 96.21
A modulated beam demonstrates the amplitude control offered
by the combined SLM/spatial-filter system. The lineout
in the upper portion demonstrates the effective transmission
function, while the lower part demonstrates high-contrast
modulation with as much as a 50:1 extinction ratio.
INDEPENDENT PHASE AND AMPLITUDE CONTROL OF A LASER BEAM USING A SINGLE -PHASE -ONLY SPATIAL LIGHT MODULATOR
228
LLE Review, Volume 96
Figure 96.22 demonstrates simultaneous amplitude and
wavefront-shaping performance of this system by summing
two one-dimensional patterns with spatial frequencies below
and above the cutoff frequency of the spatial filter, as shown by
the mask in Fig. 96.22(a). Figure 96.22(b) shows the beam
amplitude measured with the Hartmann-Shack sensor, while
Fig. 96.22(c) illustrates the measured wavefront. Both images
display data dropout near the center because the measured
intensity falls below the detection threshold of the wavefront
sensor. Little phase-to-amplitude coupling is observed, demonstrating
that the phase information is conserved through the
filter while the intensity modulation is achieved.
Figure 96.23 illustrates the performance of this beam-
shaping scheme in an iterative, closed-loop configuration. A
single convergence scheme is applied in which less amplitude
modulation is applied where not enough transmission is
achieved and more where too much is measured. For demonstration
purposes, we propose to correct the pixels for which
the measured intensity on a 8-bit gray scale is higher than 80
counts. After mapping the SLM to the CCD using a fiducial
image, the required transmission at each location of the SLM
and the corresponding phase-modulation amplitude are calculated.
The first step correction result is shown by the image in
Fig. 96.23(b) along with its corresponding lineout, which
TC6280
0.0
0.2
0.4
0.6
0.8
1.0
-3
-2
-1
0
1
Radians
(a) Phase mask
(b) Intensity
(c) Phase
Figure 96.22
Independent phase and amplitude modulation
is demonstrated. The mask (a) leads
to a beam that exhibits simultaneous amplitude
(b) and phase (c) modulation.
0
40
80
120
160
(a)
(b)
(c)
(d)
TC6281
a
b
c
Figure 96.23
Dynamic amplitude beam control. The initial beam (a) is shaped
into top-hat beams (b) and (c). The lineouts show the typical error
to the intensity goal.
INDEPENDENT PHASE AND AMPLITUDE CONTROL OF A LASER BEAM USING A SINGLE -PHASE -ONLY SPATIAL LIGHT MODULATOR
LLE Review, Volume 96
229
shows that most of the correction factor has been underestimated
since the average intensity is above 80 counts. Similarly,
the correction does not lead to a uniform beam because of the
spatially dependent transfer function of the SLM. Nevertheless,
the error, defined as the difference between the real
intensity and the goal intensity, in an rms sense, for those points
initially higher than 80, is reduced from 60% to 16%. Using
image (b), the error signal is reduced by changing the modulation
to achieve the goal. The result of a second correction is
shown in Fig. 96.23(c), where the error signal has been reduced
to 8.5%, which is dominated by the speckle noise
discussed earlier.
We have shown a dynamic modulation scheme that addresses
simultaneously both the phase and the amplitude of a
laser beam. By modulating the phase of a laser beam at high
spatial frequencies, one can couple the phase-modulation
amplitude to the transmission of a spatial filter in a straightforward
way. Following that, we have demonstrated that this
scheme can be used for beam correction.
ACKNOWLEDGMENT
This work was supported by the U.S. Department of Energy Office of
Inertial Confinement Fusion under Cooperative Agreement No. DE-FC03-
92SF19460, the University of Rochester, and the New York State Energy
Research and Development Authority. The support of DOE does not constitute
an endorsement by DOE of the views expressed in this article.
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