Comparison between Hadamard and canonical bases for in-situ wavefront correction and the eect of ordering in compressive sensing Dennis Scheidt and Pedro A. Quinto Su

2025-04-29 0 0 2.2MB 8 页 10玖币

侵权投诉

Comparison between Hadamard and canonical bases for in-situ wavefront correction

and the eﬀect of ordering in compressive sensing

Dennis Scheidt and Pedro A. Quinto Su∗

Instituto de Ciencias Nucleares, Universidad Nacional Aut´onoma de M´exico,

Apartado Postal 70-543, 04510, Cd. Mx., M´exico

(Dated: October 24, 2022)

In this work we compare the Canonical and Hadamard bases for in-situ wavefront correction

of a focused Gaussian beam using a spatial light modulator (SLM). The beam is perturbed with

a transparent optical element (sparse) or a random scatterer (both prevent focusing at a single

spot). The phase corrections are implemented with diﬀerent basis sizes (N= 64,256,1024,4096)

and the phase contribution of each basis element is measured with 3 step interferometry. The

ﬁeld is reconstructed from the complete 3Nmeasurements and the correction is implemented by

projecting the conjugate phase at the SLM. Our experiments show that in general, the Hadamard

basis measurements yield better corrections because every element spans the relevant area of the

SLM, reducing the noise in the interferograms. In contrast, the canonical basis has the fundamental

limitation that the area of the elements is proportional to 1/N, and requires dimensions that are

compatible with the spatial period of the grating. In the case of the random scatterer, we were

only able to get reasonable corrections with the Hadamard basis and the intensity of the corrected

spot increased monotonically with N, which is consistent with fast random changes in phase over

small spatial scales. We also explore compressive sensing with the Hadamard basis and ﬁnd that

the minimum compression ratio needed to achieve corrections with similar quality to those that

use the complete measurements depend on the basis ordering. The best results are reached in the

case of the Hadamard-Walsh and cake cutting orderings. Surprisingly, in the case of the random

scatterer we ﬁnd that moderate compression ratios on the order of 10 −20% (N= 4096) allow to

recover focused spots, although as expected, the maximum intensities increase monotonically with

the number of measurements due to the non sparsity of the signal.

I. INTRODUCTION

Most optical instruments and applications can be very

sensitive to aberrations. However, programmable opti-

cal elements like deformable mirrors and 2 dimensional

arrays like spatial light modulators (SLM) and digital

micromirror arrays (DMD) have enabled corrections of

arbitrary errors that can result in better focusing and

enhancing light transmission through scattering materi-

als [1, 2].

A very simple solution to correct aberrations with a

spatial light modulator is the ‘in-situ’ method that can-

cels the errors at the point where the beam is used (i.e.

focused spot) demonstrated by ˇ

Ciˇzmar and coworkers [3].

That approach uses an orthogonal canonical basis repre-

sentation of the ﬁeld at the surface of an SLM, where the

basis is implemented in a square domain with an area A

that is divided into a grid of Nnon-overlapping elements.

Each element is represented by a unit vector in Ndimen-

sional space ˆei=δij (i, j = 1, ..., N). The amplitude and

phase contribution of each square wavelet at the point of

interest is measured with interferometry. In that way, it

is possible to enhance the intensity of the focused beam

by assigning each mode a phase that maximizes the in-

terference, cancelling the aberrations and enhancing the

intensity. This method has also been implemented with

a DMD that has faster update rates [4] to generate very

∗Pedro.quinto@nucleares.unam.mx

accurate structured beams.

A fundamental limitation of the in-situ method with

the canonical basis is that increasing the spatial resolu-

tion of the measurement by increasing the size of the basis

Ndecreases the area of the individual elements to A/N ,

reducing the amount of light diﬀracted by each element

which can increase noise in the measurement. Further-

more, if the size becomes smaller than the grating period

that is used to diﬀract the light, the diﬀraction eﬃciency

decreases exacerbating the problem.

Other orthogonal bases (i.e. Hadamard) utilize over-

lapping modes that ﬁll the relevant area Aof the SLM

where the basis is deﬁned [5]. For example, the Nor-

thogonal vectors of the Hadamard basis do not have zero

entries but ±1 values; as a result the signal to noise ratio

increases and is similar for all elements.

Furthermore, the spatial overlap of the modes in the

case of the Hadamard basis allows the use of compres-

sive sensing (CS) [6], where the full signal can be recon-

structed measuring only a subset Mof vectors in the

basis (M < N). This can be done because most sig-

nals are sparse with only a few components contributing

signiﬁcantly. An important consideration in the imple-

mentation of compressive sensing is the ordering of the

Hadamard basis [7–9], which can signiﬁcantly decrease

the number of measurements needed to reconstruct a sig-

nal.

Several groups have already implemented compressive

sensing for imaging a complex ﬁeld (phase and ampli-

tude) [5], but to our knowledge there are no studies com-

paring the original ’in-situ’ experiment with a Hadamard

arXiv:2210.11483v1 [eess.IV] 20 Oct 2022

basis to correct a focused beam, which is important for

many applications like micromanipulation and microfab-

rication.

In this work we demonstrate in-situ wavefront correc-

tion for two extreme cases: a sparse perturbation with a

transparent optical element and a random scatterer. In

general, we ﬁnd that the signal to noise ratio degrades

for the largest canonical basis (N=4096), while the case

of the Hadamard basis is independent of N. The case

of the random scatterer cannot be corrected with the

canonical basis, while the Hadamard correction improves

monotonically with N. We also investigate the following

orderings of the Hadamard basis for compressive sensing:

Hadamard, Hadamard-Walsh, cake cutting and random.

The best results are achieved with the Hadamard-Walsh

and cake cutting orderings.

This article is arranged in the following way: The ex-

perimental setup is described in section 2, including the

implementation of the bases at the SLM, the 3 step inter-

ferometry to recover the phase and amplitude contribu-

tion of each element and the description of the perturbing

elements. The results of the full measurements (3N) of

the bases are presented in section 3. Compressive sens-

ing is described in section 4 and the results are in sec-

tion 5 where wavefront corrections are implemented using

subsets of the full measurements (between 2 −90%) and

diﬀerent orderings of the Hadamard basis (Hadamard,

Hadamard-Walsh, cake cutting, random). Section 6 con-

tains the concluding remarks.

II. EXPERIMENT

This section contains details about the experimental

setup, bases, encoding at the SLM, 3 step interferom-

etry, ﬁeld reconstruction and speciﬁc details about the

perturbations and the realization of the experiment.

A. Experimental Setup

The experiment schematic (not to scale) is depicted

in Figure 1a, where a CW laser beam (λ= 1064 nm,

4.95 ±0.01 mW) with linear polarization controlled by a

half wave plate (HWP1) is reﬂected by a spatial light

modulator (SLM - Hamamatsu: LCOS-SLM X10468,

20 µm pixel size) that imprints a 2d prism phase to the

horizontal polarization component to separate it from

the zero order. Then a polarizing beam splitter cube

(PBS) separates the modulated horizontal and unmodu-

lated vertical polarization that makes the reference beam

which propagates through a small aperture (diameter of

∼1.5 mm) and then through a half wave plate (HWP2)

projecting it into the horizontal polarization state. A

lens (f= 40 cm) at a distance of 40 cm from the SLM fo-

cuses the beams into the camera (Thorlabs DCC1645C,

3.6µm pixel size), that can acquire the transverse inten-

sity proﬁle of the uncorrected/corrected beams (with no

FIG. 1. Experimental setup. (a) Experimental setup. (b)

Measurement bases in their matrix representation Φ (16 ×16

elements): (1) CAN: Canonical (2) H: Natural Hadamard (3)

HW: Hadamard-Walsh (4) CC: cake-cutting. The 4th row

is highlighted in all Φ, the orange highlighted components

in HW and CC show the position of the 4th element of H

showing the diﬀerent ordering. (c) 2d vector representation

of the 4th 1d-measurement vector.

reference) or the interferograms with are measured at a

single pixel. We introduce an additional element to the

modulated beam at an arbitrary position to perturb it

preventing a good focus: a transparent element or a ran-

dom scatterer (Fig 1a, D1 and D2).

The experiment is similar to the original in-situ article

[3] and the main diﬀerence is the interferometer. ˇ

Ciˇzmar

and coworkers extracted the reference beam from a sin-

gle basis element located at the center of the grid, while

our reference emerges from a small section (controlled

with the iris) of the unmodulated beam which is equiva-

lent. Both experiments perform the measurements with a

CCD camera, which allows to quickly measure the trans-

verse proﬁles of the beams.

B. Bases and 2d representation at the SLM

We consider the canonical and the Hadamard bases.

The bases are represented by matrices with N×Nele-

ments, and each vector (row in the matrix) is represented

in the SLM as a square grid with √N×√Nthat is re-

sized to the size of the screen area that encodes the basis

at the SLM (512 ×512 px).

1. Canonical basis

The canonical basis is represented by an N×Nidentity

matrix ΦCAN =IN, where the orthogonal vectors are

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

ComparisonbetweenHadamardandcanonicalbasesforin-situwavefrontcorrectionandtheeectoforderingincompressivesensingDennisScheidtandPedroA.QuintoSuInstitutodeCienciasNucleares,UniversidadNacionalAutonomadeMexico,ApartadoPostal70-543,04510,Cd.Mx.,Mexico(Dated:October24,2022)InthisworkwecomparetheCano...

展开>> 收起<<

Comparison between Hadamard and canonical bases for in-situ wavefront correction and the eect of ordering in compressive sensing Dennis Scheidt and Pedro A. Quinto Su.pdf

共8页,预览2页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Comparison between Hadamard and canonical bases for in-situ wavefront correction and the eect of ordering in compressive sensing Dennis Scheidt and Pedro A. Quinto Su

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: