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

... 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... 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 75 Figure 6. 15:... 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

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 perfor-mance 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 reflec-tivity

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

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 rcand Rcare 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 ∝ P I(x, y)|ROI

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 =



IavNROI

P I(x, y)|ROI



Nc− Nc =



IavNROI

P I(x, y)|ROI − 1



Nc (6.1.3)

In the above equation, a negative ∆Ncindicates the number of pixels to be converted to the off state, whereas a positive ∆Ncindicates 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

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 = Einrid (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

Ioutusing an a binary approximation of reflectance pattern r0id Our goal is to convert the output profile into the flat-top profile If t whose parameters are obtained by fitting the current output

Figure 6.5: Definition of Iout and If t in equation 6.2.2.

profile In accordance with equation 6.2.1, we define a new target reflectance pattern rid1 as:

r1id = Ef t

Eout

rid0 =

s

If t

Iout

rid0 (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 approximation of r1id that will be used as the SLM pattern for the following iteration The 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 re-flectance pattern and use it as the new SLM pattern

4 Repeat the procedure with the output beam now produced by the updated SLM reflec-tivity

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

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 flat-top 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

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

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 con-finement (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 in-tensity 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

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

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 mea-surement 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