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Letter

Spatio-temporal mechanism for rapid speckle
supression in laser sources
RONEN CHRIKI,1,* SIMON MAHLER,1,2 CHENE TRADONSKY,1 VISHWA PAL,1
ASHER A. FRIESEM1 AND NIR DAVIDSON1
1Department
2Univ.

of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 7610001, Israel
Paris Sud, Université Paris Saclay, 91405 Orsay, France

*Corresponding author: ronen.chriki@weizmann.ac.il
Received XX Month XXXX; revised XX Month, XXXX; accepted XX Month XXXX; posted XX Month XXXX (Doc. ID XXXXX); published XX Month XXXX

Many applications that involve bright illumination
sources, such as lasers, suffer from undesired coherent
artifacts of speckle noise. The speckle noise is typically
suppressed by superimposing many uncorrelated speckle
realizations, and summing their intensities incoherently.
This is possible for long time scales, where the speckle
contrast in multimode lasers is known to depend only on
the number of transverse (spatial) laser modes.
Unfortunately, it is difficult to accumulate a large number
of independent speckle realizations on short time scales.
In this letter we propose and demonstrate a new spatiotemporal mechanism for rapid speckle suppression in a
multimode laser source. We show that in the regime of
short time scales, speckle contrast is suppressed as a
function of the number of longitudinal laser modes. The
mechanism is exemplified for rapid speckle suppression
in full-field laser imaging of a fast moving object. © 2017
Optical Society of America
OCIS codes: (140.3460) Lasers; (030.6140) Speckle; (030.1640) Coherence.
http://dx.doi.org/10.1364/optica.99.099999

INTRODUCTION
Speckle patterns appear when coherent light beams propagate through
or are reflected by scattering media [1,2]. They are formed as a result of
interference between waves, where the phase differences between
them is random, and occur in different physical wave regimes, such as
radio waves [3], microwaves [4], optical waves [1], x-rays [5], and even
matter waves [6]. In imaging systems, speckles cause random intensity
variations that reduce signal to noise ratio (SNR) and corrupt the output
image. In order to reduce speckle contrast, and thereby speckle noise,
several methods have been developed. These involve the generation of
many uncorrelated speckle realizations from the same object and then
summing their intensities incoherently (for instance, by moving a
diffuser that is placed between the laser source and the object) [1].
Recently, several novel multimode lasers with extremely low spatial
coherence have been used for speckle suppression [7–10]. In
multimode lasers, each transverse mode generates an independent
uncorrelated speckle realization, thereby leading to reduction of the

speckle contrast. Specifically, for a laser of N transverse modes in the
regime of long integration times (i.e. long exposure times of the
detecting device), the total intensity is the incoherent sum of the
intensities of all transverse modes, and the speckle contrast C is N-1/2 [8].
More generally, the speckle contrast is expected to vary with
exposure time of the detecting device, and should depend on the
temporal dynamics of the laser source. To see this intuitively, consider
for example the light of N transverse modes, separated by some
frequency spacing Δ𝜈, that illuminates a diffuser. Each mode will
generate an independent uncorrelated speckle realization. For long
exposure time Δτ≫Δ𝜈 −1 , the total intensity is an incoherent sum of the
intensities of all modes, and the speckle contrast is C=N-1/2, as noted
above. For short exposure time Δτ≪Δ𝜈 −1 , the light of transverse modes
interfere with one another and generate a new speckle realization with
speckle contrast of unity, C=1. Clearly, the speckle contrast depends on
exposure time.
It is generally believed that the number of longitudinal modes M in a
multimode laser does not play any role in speckle suppression. This
general assertion is a manifestation of a fundamental difference
between spatial and temporal incoherence, whereby spatial
incoherence is considered to be far more efficient for speckle
suppression than temporal incoherence [1]. In this paper we
concentrate on short exposure time scales (typically in the nanosecond
regime) and present a new spatio-temporal mechanism for speckle
suppression, in which the longitudinal modes play a dominant role. In
this regime of exposure times and for lasers with a large number of
transverse modes (N≫M), we show experimentally and theoretically
that speckle contrast depends only on the number of longitudinal
modes, and not on the number of transverse modes. We show that
longitudinal modes that are close in frequency behave as super-modes
and contribute to speckle contrast reduction in short time scales that are
inversely proportional to the free spectral range (FSR) of the laser
cavity. Explicitly, for the short time scale regime, the speckle contrast of
C=M-1/2 is obtained with a laser of M longitudinal modes. For long time
scales that are inversely proportional to the frequency spacing between
transverse modes, the speckle contrast is according to the traditional
relation of C=N-1/2.
Speckles are involved in fundamental research [11–17] and in many
applications [18–22], so a better understanding of their temporal
dynamics and suppression are of great interest. For example, speckle
suppression for short exposure times can be advantageous for ultra-fast

full field imaging applications , such as flow dynamic measurements in
aerodynamics [23,24], study of bubble formation in liquids [25,26] and
study of ablation [27].

EXPERIMENTAL ARRANGEMENT AND RESULTS
As noted above, speckle contrast in the regime of long exposure times
depends on the number of transverse modes as N-1/2. So, if significant
speckle suppression is required (i.e. speckle contrast of few percent), a
highly incoherent laser source with thousands of transverse modes is
needed. For such a source we resorted to the degenerate cavity laser
(DCL) [28,29] and investigated the temporal dynamics of speckle
suppression. The DCL provides extremely large tunability of the
number of transverse modes (from 1-300,000 transverse modes [8]),
and was used for efficient speckle suppression [8,30,31].
Our DCL, depicted in Fig. 1(a) was comprised of a high reflective
mirror, Nd:YAG gain medium, two lenses with f=25cm in a 4f telescope
configuration and an output coupler. The gain medium was pumped by
a double Xenon flash lamp with quasi-CW 100µsec long pulses, and the
reflectivity of the output coupler was R=80%. The 4f telescope inside the
cavity assured that any field distribution is accurately imaged onto itself
after every round trip, so any field distribution is a degenerate eigenmode of the cavity [28]. The diameter of the gain medium was 0.95cm,
much larger than the diffraction spot size of the telescope, so the cavity
supported a huge number of transverse modes (N=320,000 [8]), i.e., it
was a highly spatially incoherent source. Figure 1(b) shows two
representative speckle images for long exposure times, one obtained
with a spatially coherent Nd:YAG laser and the other with the DCL. As
evident, the speckle contrast is greatly reduced with the DCL, indicating
that the DCL indeed supports a very large number of transverse modes.
The number of lasing longitudinal modes in the DCL was determined
by the ratio between the total bandwidth of the laser (measured by
means of a Michelson-interferometer as 32GHz) and the free spectral
range of the cavity (measured from the beating frequencies in the laser
output as 123MHz, see supplement 1), to yield M=260. Our laser thus
fulfills the condition N≫M.
To measure temporal dynamics of the speckle contrast from the DCL,
a thin optical diffuser was placed outside the cavity, and a fast
photodetector, much smaller than the typical speckle size, measured the
intensity time dynamics of a single point in the speckle field, as shown in
right part of Fig. 1(a). The speckle contrast was measured by performing
100 uncorrelated time series measurements with a fixed rotation of the

Fig. 1. Laser and detection arrangements and representative speckle
images. (a) Experimental arrangements of the DCL and of the speckle
contrast detection. OC – output coupler; BS – beam splitter; PD –
photodetector. (b) Representative speckle images with a spatially
coherent source (left) and with the DCL (right) for long exposure times.

diffuser (and hence rotation of the speckle field) before each
measurement. The speckle field measurements were normalized by the
total output intensity, measured by a second fast photodetector as
shown in the right part of Fig 1(a).
The experimental results of speckle time dynamics are presented in
Fig. 2. Figure 2(a) shows representative time series measurements for
seven randomly selected points in the speckle field. Figure 2(b) shows
the corresponding seven time average intensities 𝐼 (̅ Δτ) =
𝛥𝜏
1
∫ 𝐼(𝑡)𝑑𝑡 simulating the effect of finite exposure time Δτ. Based on
𝛥𝜏 0
such measurements, the speckle contrast 𝐶(Δτ) was then calculated at
different exposure times using 𝐶(Δτ) = [〈𝐼 (̅ Δτ)2 〉/〈𝐼 (̅ Δτ)〉2 − 1]1/2 ,
where ⟨·⟩ denotes an ensemble average over 100 different points in the
speckle field. For long exposure times, the speckle contrast
measurements were also performed by full field imaging of the speckle
field with a CMOS camera, verifying the validity of our method for
measuring speckle contrast (see supplement 1).
The resulting speckle contrasts as a function of exposure times are
presented in Fig. 2(c). Three temporal regimes are clearly identified:
(a) long exposure time regime (Δτ>3μsec) where the speckle contrast
is reduced to C=0.014, slightly higher than that expected due to nonuniformity of the illuminating beam, i.e. N-1/2 =0.002 [8]; (b) short
exposure time regime (10ns< Δτ<1μsec), where speckle contrast is
C~0.056, in good agreement with the expected value M-1/2 =0.062 (see
next section); and (c) instantaneous exposure time regime (Δτ<1ns),

Fig. 2. Experimental speckles temporal dynamics. (a) Intensity time
series of seven randomly selected points in the speckles field.
(b) Corresponding average intensity as a function of exposure time for
the selected seven points. (c) Speckle contrast as a function of exposure
time. Dashed lines in (b) and (c) denote transitions between the
different exposure time regimes.

where the speckle contrast increases inversely with exposure time, and
is expected to reach its maximal value of C=1 for exposure time shorter
than 31ps (the inverse of the lasing spectral bandwidth of 32GHz). Such
instantaneous time scale was beyond the resolution of our
photodetector. Transitions between the different exposure time
regimes are denoted by dashed lines in Fig. 2(b-c). They correspond to
the calculated and measured free spectral range (123MHz), and to the
expected minimal broadening of a single mode due to mechanical
instabilities (~300KHz) [32].
We repeated the measurements of speckle contrast as a function of
exposure time for lasing with various numbers of transverse modes,
and found similar graphs with the same three temporal regimes for all
measurements, see supplement 1.

MODAL FREQUENCY SPECTRUM AND SIMULATION
RESULTS
The temporal dynamics of the speckle contrast C are strongly related to
the spectral frequencies of the laser. For an ideal degenerate cavity, the
frequency 𝜐𝑞,𝑚,𝑛 for the qth longitudinal mode and for the transverse
mode of order (m, n) is
𝜐𝑞,𝑚,𝑛 =

𝑐
8𝑓

(𝑞 + 𝑚 + 𝑛 + 1) ≡

𝑐
8𝑓

𝐾,

(1)

where c is the speed of light, c/8f is the FSR and K≡q+m+n+1 (see
supplement 1). Equation (1) indicates full degeneracy in K, whereby all
transverse modes in the degenerate cavity are accurately degenerate in
frequency. Any aberrations in the cavity would break this degeneracy.
For example, isotropic aberrations would result in
𝜐𝑞,𝑚,𝑛 =

𝑐
8𝑓

[𝐾 + ε ⋅ (𝑚 + 𝑛 + 1)],

(2)

where ε depends on the type and degree of the aberrations (see
supplement 1). For the realistic case of low aberrations, 𝜀≪1, the
frequency spacing Δνε between modes with the same value of K is much
smaller than the FSR, as illustrated in Fig. 3(a).
The modal frequency spectrum in Fig. 3(a) resembles that of the wellknown near concentric cavity laser (see supplement 1). We found a
direct quantitative connection between the near degeneracy of the
transverse modes in gain and loss (enabling support of huge number of
transverse modes even in the presence of mode competition [8,30]) and
their near degeneracy in frequency. While we cannot provide a rigorous
proof, we believe this to be generally true.
Based on this modal frequency spectrum, it is possible to analyze and
explain the mechanism of speckle suppression for short exposure times.
We performed numerical calculations to determine the speckle contrast
as a function of exposure time and number of transverse and
longitudinal modes. The frequency spacing between modes were taken
to be Δ𝜈𝜀 =10kHz and Δ𝜈𝐹𝑆𝑅 =100MHz, i.e. Δ𝜈𝜀 ≪ Δ𝜈𝐹𝑆𝑅 as in our
experiments. Figures 3(b) and 3(c) present results for the simple cases
of 𝑀≫N=1 and 𝑁≫M=1, respectively, while Fig. 3(d) presents the
results for 𝑁≫𝑀≫1 that corresponds to our experiment.
Figure 3(b) shows the calculated speckle contrast as a function of
exposure time for a laser with M=100 longitudinal modes and a single
transverse mode N=1. As evident, the speckle contrast is equal to unity
for all exposure times of the detecting device. This demonstrates that
spatial diversity is a dominant component in speckle suppression with
typical laser sources and thin diffusers. Figure 3(c) shows the calculated
speckle contrast as a function of exposure time for a laser with a single
longitudinal mode, M=1, and N=100 closely spaced transverse modes.
As evident, the speckle contrast is suppressed to C=N-1/2=0.1, but only
for exposure times Δτ≫Δ𝜈𝜀−1 , when all the transverse modes are
mutually incoherent. For Δτ≪Δ𝜈𝜀−1 , the transverse modes interfere

Fig. 3. Numerical calculation results. (a) Part of modal frequency
spectrum assumed in the numerical calculations. (b-e) Calculated
speckle contrast as a function of exposure time for laser cavity with (b)
M=100 longitudinal modes and N=1 transverse modes, (c) M=1 and
N=100, (d) M=100 and N=10, and (e) M=10 and N=100.
with one another, and generate a new speckle pattern with speckle
contrast of unity.
Figure 3(d) shows the calculated speckle contrast as a function of
exposure time for a laser with N=100 transverse modes and M=10
longitudinal modes. Three distinct temporal regimes are clearly seen, in
agreement with the experimental data in Fig. 2(c). For long exposure
times Δτ≫Δ𝜈𝜀−1 , the total intensity distribution is an incoherent
sum of all transverse modes, whereby the speckle contrast is
−1
C=N-1/2=0.1. For instantaneous exposure times Δτ≪Δ𝜈𝐹𝑆𝑅
, all
transverse modes interfere with one another and the speckle contrast
reaches unity. In the regime of short exposure times,
−1
Δ𝜈𝐹𝑆𝑅
≪Δτ≪Δ𝜈𝜀−1 , the speckle contrast is C=M-1/2=0.32. We attribute
this surprising result to the following spatio-temporal mechanism: all
the longitudinal modes that lie in the vicinity of a given frequency 𝑞𝑐/8𝑓
can be considered as a single super-mode. The transverse modes in each
super-mode interfere coherently at the regime of short exposure times
and generate a single speckle pattern with speckle contrast of unity.
However, the transverse modes in each super-mode have random
phases, so each super-mode generates an independent speckle
realization that is uncorrelated with those of other super-modes. The
frequency spacing between the M super-modes is at least ΔνFSR , and
−1
therefore for Δτ≫Δ𝜈𝐹𝑆𝑅
, they are mutually incoherent and yield a
speckle contrast of C=M-1/2.
Finally, Fig. 3(e) shows the calculated speckle contrast as a function
of exposure time for a laser with N=10 transverse modes and M=100
longitudinal modes (𝑁≪𝑀). As evident, the speckle contrast is only
suppressed to a level of C=0.32=N-1/2. The reason for this is that spatial
diversity is a necessary condition for speckle suppression, and the
speckle contrast cannot be reduced by more than the number of
transverse modes. Thus, for lasers with 𝑁≪𝑀 speckle contrast cannot
be reduced below C=N-1/2.

Funding. This work was supported in part by the Israel Science
Foundation and the Israel-US Binational Science foundation.
Acknowledgment. We thank Prof. Hui Cao (Yale University) for many
fruitful discussions, and Ofir Korech and Alexander Cheplev for their
help in spectral measurements.
Fig. 4. Experimental full field imaging results of an Air Force resolution
target. (a) Detected image of a static object using a spatially coherent
source and long exposure time. (b) Detected image of a static object
using a spatially incoherent source and long exposure time. (c) Detected
image of a fast moving object and short exposure time.

IMAGING OF FAST MOVING OBJECTS IN THE SHORT
EXPOSURE TIME REGIME
To exemplify the usefulness of rapid speckle suppression we performed
full field imaging experiments for an object that was a U.S. Air Force
resolution target. We performed the experiments with a static object
and long exposure time, and a fast moving object and short exposure
time. In the experiment of fast moving object in short exposure time, the
object was placed after a diffuser, and rotated such that it had a linear
velocity of 15m/s. To achieve high resolution, the laser was Q-switched,
resulting in short exposure, limited by the 100ns pulse duration of the
laser, whereby the object could only move 1.5µm in every pulse.
Figure 4 presents the experimentally detected images. Figure 4(a)
shows the detected image of a static object using a coherent source. As
evident, the image suffers from high speckle contrast. Figure 4(b) shows
the detected image of a static object that is illuminated with spatially
incoherent light from the DCL and long exposure time. As evident, the
speckle contrast is suppressed, and was measured to be C=0.015. Figure
4(c) shows the detected image of the fast moving object with short
exposure time. Here again the speckle contrast is suppressed, and was
measured to be C=0.045, while the spatial resolution remains high due
to the short exposure time. These results demonstrate that speckle
suppression can indeed be obtained at short nanosecond time scales.

CONCLUDING REMARKS
We presented and characterized a new spatio-temporal mechanism for
suppressing speckle noise that arises when using laser sources in short
exposure times. Although it is generally believed that speckle noise is
determined only by the number transverse modes, we showed that in
the regime of short exposure times speckle noise is also influenced by
the number of longitudinal modes. Specifically, for a degenerate cavity
with 𝑁≫M≫1, we identified, both in experiments and in simulation,
three distinct temporal regimes of exposure times. In the regime of long
exposure time Δτ≫𝛥𝜈𝜀−1 , the speckle contrast is C=N-1/2. In the regime
−1
of short exposure time Δ𝜈𝐹𝑆𝑅
≪ Δτ≪𝛥𝜈𝜀−1 , longitudinal modes in the
vicinity of the same frequency form independent super-modes, and the
speckle contrast is governed by the number of longitudinal modes
supported by the laser as C=M-1/2. In the instantaneous exposure time
−1
regime Δτ≪Δ𝜈𝐹𝑆𝑅
, the speckle contrast is trivially equal to unity, C=1.
The mechanism described here is general, and can be applied to
various different types of lasers. For example, highly multimode fiber
lasers are expected to suppress speckles even in the sub-picosecond
time scale. Furthermore, the mechanism enables tuning of the speckle
contrast in short exposure time scales by changing the number of
longitudinal modes in the cavity. Such speckle suppression and control
in short time scales can be advantageously exploited in ultra-fast
imaging and detection applications, as was exemplified in a full-field
imaging of a fast moving object.

See Supplement 1 for supporting content.

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