2D optical trapping potential for the confinement of heteronuclear molecules 6

Chapter 6

Iterative Correction Algorithms
In this chapter, we will describe the construction of a correction algorithm that improves the
reflectivity pattern of the SLM based on the observed camera output. We compare the performance of two alternative algorithms: one adapted from reference [Liang et al., 2010] and our
newly-constructed algorithm. With the best result attained by the application of the correction
algorithm, we will summarize some further characterization of the flat-top beam such as its
depth of field and temporal noise analysis. Finally, we end this chapter with some suggestion
steps which could be carried out to further improve the profile of the output flat-top beam.

6.1

First Correction Algorithm

In this section, we will discuss the performance of the correction algorithm as described in
reference [Liang et al., 2010]. The main idea of this algorithm is summarized in the following
steps:
1. Take a picture of the output beam and compute the error profile.
2. Define a region of interest centered around the pixel with maximum error.
3. Map the region of interest to the SLM plane.
4. Within the region of interest in the SLM plane, calculate the number of SLM pixels to
change and implement the change.
5. Repeat the procedure with the output beam now produced by the updated SLM reflectivity.
The first step is the same as the beam analysis procedure discussed in the previous chapter,
where the beam error is obtained by subtracting the fitted profile from the observed profile. For
this algorithm however, we take an average over 10 pictures before proceeding with the fitting
routine. The averaging procedure is inserted to reduce the impact of the time-fluctuation of the
output profile which we have seen from the previous chapter. For this algorithm, this averaging
step is especially important since the correction is based around the pixel with maximum error.
For the second step, we first find the location of the output picture pixel containing the
largest error from an ideal flat-top profile. This pixel will be the center of a square-box region

of interest which will be the working area for the current step of the iterations. The square box
is first defined as one pixel in size, and then continually increased as long as doing so increases
the error contained in the box. This procedure allows us to model the size of the error peak, as
we can see from figure 6.1.
Given the information of the error peak location and size in the camera plane, we need to find
the corresponding location at the SLM plane where we can correct for this error by modifying
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the reflectivity pattern. The pixel mapping calculation from the camera plane (camera pixel)
to the SLM plane (SLM pixel) consists of two steps: mapping the position of the central SLM
pixel and mapping the distance between the two planes. The first step is the exact same step
as what have been done in chapter 5, in the context of centering the input beam with respect
to the SLM. By observing the reflection off only the central pixel of the SLM, we map the
position of the center of the SLM in terms of the pixel of the camera. Secondly, we want to map
the distance between two pixels from the SLM plane to the camera plane. To achieve this, we
measure the reflection off one pixel which is located at a certain distance away from the central
SLM pixel. In this regard, we measured the reflection off four of such points in sequence. Each
point is located 50 pixels away from the center of the SLM pixel, one above, one below, one to
the right and one to the left of the center. For each point, we find that the reflection is located
54 pixels away from the pixel associated with the center of the SLM. This finding is consistent
with the magnification of the telescope which is set at 2/3. Since the pixel size of the SLM is
7.637 µm and the pixel size of the camera is 4.65 µm, we expect that one SLM pixel will be
mapped to (7.637 ∗ 2)/(4.65 ∗ 3) ≈ 1.1 camera pixel. The measured ratio of 54/50 = 1.08 is
indeed very close to the expected value. With these two information, we can convert a pixel of
coordinate r in the SLM plane to its corresponding pixel in the camera plane R according to
the equation:
R = Rc + α(r − rc ),

(6.1.1)

where rc and Rc are the coordinates of the central SLM pixel in the camera plane and the SLM
plane respectively.

Figure 6.1: Illustration of the first correction algorithm: (Left) initial error profile and the square
region of interest around the maximum error peak, (Right) error profile after the implementation
of the first correction step.

Once we complete the coordinate mapping process, we need to change the SLM reflectivity
based on the error profile. If the error inside the region of interest is predominantly positive, it
indicates that a certain number of SLM pixels have to be converted from the on state to the off
state. To determine this number, we assume that the observed beam intensity is proportional
to the number of on state pixel. Therefore, the ideal number of on pixels Nid inside the region
of interest should be proportional to the fitted flat-top intensity Iav according to equation 5.4.1
in chapter 5. Similarly, the current number of on pixels inside the region of interest Nc is
proportional to the average beam intensity inside the region of interest:
Nc ∝

I(x, y)|ROI
,
NROI
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(6.1.2)


where NROI indicates the size of the region of interest. Therefore, the number of SLM pixels to
be changed is a certain proportion of the current number of on state pixels Nc :
∆Nc = Nid − Nc =

Iav NROI
I(x, y)|ROI

Nc − Nc =

Iav NROI
− 1 Nc .
I(x, y)|ROI

(6.1.3)

In the above equation, a negative ∆Nc indicates the number of pixels to be converted to the off
state, whereas a positive ∆Nc indicates the number of off state pixels to be turned on. In either
case, the pixels to be flipped are randomly chosen around the center of the region of interest by
a normal probability distribution. The standard deviation of the distribution is chosen of the
order of the size of the region of interest. Once the new reflectivity pattern has been defined,
we upload this pattern to the SLM. Finally, we proceed with a new iteration, this time with the
updated SLM pattern.

Figure 6.2: Evolution of the output profile RMS and maximum error in function of the number
of iterations with the first algorithm.
We apply this correction algorithm to the initial reflectivity pattern while monitoring the
evolution of the RMS error. As we can see in figure 6.2, the RMS error starts to stabilize
after 80-100 iterations. The profile displayed in figure refIter1OptProfile is one representation
of 100 data measurement of the output profile spaced in 1 second time lapse; where we use the
SLM pattern optimized with 100 iterations of algorithm 1. The observed output beam profile
has a better error figures, where the average RMS error is 4.25% (down from 7.23%) and the
maximum error is 18.3% (down from 27%).

Figure 6.3: The RMS and maximum error in of the output profile optimized by algorithm 1,

taken with 1 second time lapse.
Although we have observed a real improvement of the beam profile with the application
of this first correction algorithm, its mechanism becomes increasingly less effective as more
iterations are added. The correction breaks the large error peaks into smaller ones and thus,
the size of the region of interest decreases as the iterations proceed. Consequently, the number
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Figure 6.4: Profile of the output beam optimized by algorithm 1. (Top) 2-dimensional camera
profile, (Bottom Left) cut along X axis, and (Bottom Right) cut along Y axis.

of flipped pixels (which is proportional to the number of pixels inside the region, see equation
6.1.3) also decreases. This fact is reflected in the evolution of the RMS error improvement
rate in function of the number of iterations as illustrated in figure 6.2. There, we observe a
considerable slowing down after a very fast improvement during the first 10 to 20 steps. In the
end, the beam profile is grainy due to the numerous small error peaks as we can see from figure
6.4 which cannot be efficiently corrected by the algorithm as constructed.

6.2

Second Correction Algorithm

The problem with the first algorithm led us to construct an algorithm which considers the whole
beam profile to create a correction, instead of an isolated region of interest. Our approach is to
return to the fundamental equation governing the beam-shaping action:
Eout = Ein rid .

(6.2.1)

Our technique was to replace the exact reflectance pattern rid with a combination of a spatial

filtering and an approximated binary reflectance rSLM created by processing rid with the Error
Diffusion algorithm. Subsequently, we hypothesize that the errors in the output profile Eout are
created due to a mismatch between the input profile and the reflectance pattern. One possible
source of this mismatch is our modelization of the input beam as a perfect Gaussian beam. To
amend the error, one can attempt to better represent the input profile and later modify the
exact reflectance rid .
For this second version of the correction algorithm, we choose to apply the correction based
on the observed output profile. At the beginning of the iterations, we produce an output pattern
0 . Our goal is to convert the output
Iout using an a binary approximation of reflectance pattern rid
profile into the flat-top profile If t whose parameters are obtained by fitting the current output
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Figure 6.5: Definition of Iout and If t in equation 6.2.2.

1 as:
profile. In accordance with equation 6.2.1, we define a new target reflectance pattern rid

1
rid
=

Ef t 0
r =
Eout id

If t 0
r .
Iout id

(6.2.2)

Note that in the above equation, we need to convert the coordinate of the beam intensity ratio
from the camera plane into the SLM plane which follows the same procedure as described in
the first algorithm description. Using the Error Diffusion algorithm, we calculate the binary
1 that will be used as the SLM pattern for the following iteration. The
approximation of rid
process is then repeated until the error figures are optimized. The summary of the this second
algorithm is thus as following:
1. Take a picture of the output beam and fit it to a flat-top profile.
2. Calculate the new target reflectance pattern according to equation 6.2.2.
3. Using the Error Diffusion algorithm, calculate the binary approximation of the new reflectance pattern and use it as the new SLM pattern.
4. Repeat the procedure with the output beam now produced by the updated SLM reflectivity.

Figure 6.6: Evolution of the output profile RMS and maximum error in function of the number
of iterations with the second algorithm.
The evolution of the RMS and maximum error of the beam during the iterations with this
second algorithm is depicted in figure 6.6. We see that this second algorithm converges with
significantly fewer number of steps at 10-20 iterations. Since the amount of time spent in each
step for the two algorithms are similar, the second algorithm is superior in terms of the time
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required to achieve the correction. To compare the final result of this second algorithm, we take
the same statistical sampling of the optimized output beam profile (100 data with 1 second time
lapse) with the error chart displayed in figure 6.7. The error figures achieved by this algorithm
is even better than the first algorithm, with average RMS error down to 3.53% and maximum
error down to 15.7%. Comparing these values with those achieved by all previous measurements,
we conclude that the output produced by this second correction algorithm is the best flat-top

beam realized experimentally.

Figure 6.7: The RMS and maximum error in of the output profile optimized by algorithm 2,
taken with 1 second time lapse.

Figure 6.8: Profile of the output beam optimized by algorithm 2. (Top) 2-dimensional camera
profile, (Bottom Left) cut along X axis, and (Bottom Right) cut along Y axis.
To finish the description of this section, it is interesting to see how compatible is the flattop beam with the envisioned optical lattice setup. Therefore, we would like to measure the
depth of field of the beam by moving the camera along the propagation axis of the beam. This
measurement is realized by moving the CCD camera with a translation stage, which is capable
of 1 cm of displacement in both forward and backward directions. We measure the beam profile
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in 4 chosen positions along the axis: -10 mm, -5 mm, 5 mm, and 10 mm (the reference 0 mm
refers to the focus point of the telescope). To produce a statistical average, 30 shots of the beam
are taken for each position.

Figure 6.9: Profile of the flat-top beam, cut along the X axis for various camera positions along
the Z axis.

Figure 6.10: Profile of the flat-top beam, cut along the Y axis for various camera positions
along the Z axis.

As we expect due to diffraction phenomenon, going out of the focus plane induces a wavy
pattern along the flat intensity area of the beam. This effect is more prominent along the
horizontal axis as can be seen in figure 6.9. Referring to figure 6.11, we can infer that the
diffraction effect increases the maximum error from 15.3% at the focus to the order of 17% in
the plane 5 mm away from the focus and 19% in the plane 1 cm away from the focus. This
shows that the the error increases by 10-15% with 5 mm displacement from the focus plane,

which should be tolerable for a lattice. With an interference range of 10 mm, the lattice could
accommodate up to 20000 pancake layers.
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Figure 6.11: RMS and maximum error, cut along the Y axis for various camera positions.

6.3

Conclusion and Outlook

In figure 6.12 below, we summarize the error figures of the flat-top beam during the different
stages of the experimental testing and during the numerical simulation for comparison’s sake.
Our second correction algorithm succeeded in improving the errors to around half from the
initial output profile. However, we see that the error figures are still a few times higher than
the limit given by the numerical simulation. Looking at the system as designed, we set the
trap depth to be around 100 times the molecule temperature to assure a tight longitudinal confinement (2D molecule geometry). Therefore, a 15% maximum error could potentially create
a trapping/anti-trapping region of 15 times the temperature in depth around the flat-intensity
region of the beam. It is desirable to bring the error figures as close as possible to the numerical
simulation limit of a few percent.

Figure 6.12: Summary of RMS and maximum error figures obtained from numerical simulations
and experimental results.
The path to reduce the beam error could be done by improving the correction algorithm as
we have done in this chapter. However, such algorithm could only work given that the input
beam intensity is not fluctuating in time. Input beam fluctuation could come in the form of intensity and pointing (beam position) fluctuations and both are equally problematic for the trap
system. An optimized SLM pattern will no longer be optimal if the input intensity distribution
changes or if the beam position shifts.
To identify the presence of an intensity noise of the laser, we attempt a series of measurement
using a photodiode. The laser output beam is measured with an AC coupling which detects the

change in the intensity. We then record the FFT of the signal to obtain the spectral component
of the intensity noise. Firstly, we calibrate the response time of the photodiode using an infrared
LED whose current is controlled by a signal generator. In figure 6.13, we present the photodiode
response to a 20 µs periodic square wave signal from the LED. We can see that the rise time
of the photodiode is of the order of 1 to 3 µs, compared to a much faster rise time of the LED.
Therefore, we infer that the maximum detectable frequency of the photodiode is of the order
of 300 kHz. Our primary interest is the frequency range of a few tens of kHz, as this range is
range is usually close to the trap frequency. A trap fluctuation at this frequency is known to
produce heating effect which will remove the molecules from the trap [Savard et al., 1997].
We measured the laser intensity fluctuation in several different setups. To measure the global
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Figure 6.13: Calibration of the response time of the photodiode using a square wave signal from
an infrared LED.
fluctuation in the beam intensity, we directly measure the beam intensity after collimation lens.
Subsequently, we also consider the local intensity fluctuation by first magnifying the beam with
a divergent lens, and select a part of the beam using an iris before the photodiode. In particular,
we select two regions of the beam in this measurement: a region near the center of the beam
and a at the bottom left part of the beam. Finally, we measure the possibility of an induced
fluctuation by the SLM by measuring the beam after being reflected by the SLM. For the SLM
group, we distinguish the two cases where we put all the pixel in the on state and where we use
the flat-top shaping pattern. In all those groups, we adjust the power of the light incident to
the photodiode such that the DC signal strength is homogeneous and far from the saturation
intensity. As a control, we take the reading from the photodiode without any incident light.

Figure 6.14: Intensity fluctuation measurement with a photodiode in the spectral range of 200
kHz.
As we can see from figure 6.14, there is no noise detected in the kHz spectral range for any
data group. A significant amount of noise is only found in the low frequency (less than 50 Hz)

component of the signal for the reflection off the flat-top pattern and the local region of the
laser output. The noise found in the reflection off the flat-top pattern can be explained by a
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Figure 6.15: Intensity fluctuation measurement with a photodiode in the spectral range of 200
Hz.

beam pointing fluctuation. As the beam has a non-uniform intensity distribution at the SLM
plane, a movement of the beam around this plane can change the total intensity incident to the
on state pixels. To quantify the beam pointing fluctuation, we measure the input beam at the
SLM plane. We took 100 pictures of the beam with 1 second time lapse. For each, we fit a
Gaussian beam profile and record the position of the beam center. In figure 6.16, we display
the difference of the position between each data and the mean over 100 data points. As we can
see, the beam displacement is of the order of 10 µm (around 1.5 SLM pixels) for the horizontal
direction and 20 µm (around 3 pixels) for the vertical direction. Such a pointing instability
could also contribute to the high error figures of the optimized flat-top pattern. If one decides
to remove this pointing instability, an acousto-optic modulator or a single-mode fiber could be
installed before the SLM.

Figure 6.16: Input Gaussian beam stability at the SLM plane.
Beside a pointing fluctuation, a local intensity fluctuation may also be the cause of the noise
observed in the low frequency range of the input beam masked by an iris. A more thorough
measurement is needed to confirm the time scale of such fluctuation since the photodiode measurement is limited at very low frequency range due to the limitation from the acquisition time.
If the fluctuation is indeed present, and happens at time scales of the order of a few seconds,
one can opt to optimize the correction algorithm to stabilize the output flat-top before the
intensity fluctuation starts to kick in. Since the molecules usually spend not more than one
second inside the trap, a trap optimized by the algorithm before each trapping sequence is in
theory realizable. Our current version of algorithm however, is not yet time-optimized. Each
iteration usually take at best several seconds to complete and so the presence of an intensity

fluctuation would be harmful. Thus, one could try to get rid of this fluctuation by installing a
single-mode fiber before the SLM.
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Figure 6.17: Intensity captured by the camera without any incident light.
Finally, the high error figures we observe might be an artifact of the CCD camera in the
setup. The camera could introduce errors in the output profile from several mechanisms; such
as the pixel dark count (intensity recorded when no light is incident to the camera), the saturation effect, and the etaloning/interference effect between various interfaces (the chip and the
camera window, or inside the camera chip). In figure 6.17, we can see that the dark count of the
camera is rather significant. We found that the intensity of the dark count fluctuates between
0.04 to 0.07 and is changing with time. Moreover, the spatial variation of the dark pixel count
appears to contain high frequency components, which cannot be corrected by the SLM via our
correction algorithm because of the spatial filtering of the beam. Therefore, we suspect that
our camera might contribute significantly to the large errors observed in our beam profile. To
mitigate this problem, one could try characterizing the camera by taking a picture of a ’clean’
beam (such as the output of a single-mode fiber). Otherwise, a better camera such as those
equipped with a cooled CCD array and windowless chip would reduce the errors from the three
effects described above.
In conclusion, we have demonstrated a correction algorithm capable of significantly improve
the flat-top beam profile. The output profile observed by our test system is apparently still
not smooth enough to be implemented as a trap beam. However, an improvement over the
camera setup has to be done to be able to observe the flat-top beam without the introduction
of artificial reading errors. If the flat-top is still considered not smooth enough by then, one
could try to improve the flat-top output by improving the stability of the input beam (e.g.
by the introduction of a single-mode fiber before the SLM) or to use more SLM pixels to
modulate the beam (e.g. by using the larger version of the DLP mirrors with more pixels, see
[Texas Instruments, 2014b]). As a final note, the SLM has not been tested with a high power
(of the order of W in power) beam which would be necessary to ensure the necessary trap depth
and 2-dimensional confinement of the molecules. When a satisfactory flat-top beam is produced,

one could test the power limit of the DLP mirrors by reducing the power attenuation before the
SLM (see the setup in chapter 5). If the SLM cannot handle the necessary power to produce
the trap, the SLM together with the correction algorithm can be used to design a static binary
reflection mask (such as a chrome-platted fused silica glass) which can replace the SLM when
the high power beam is used.

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