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Experimental verification of compressive reconstruction of correlation functions in

Ambiguity space

Lei Tian,1,∗Justin Lee,1Se Baek Oh,1and George Barbastathis1,2

1Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

2Singapore-MIT Alliance for Research and Technology (SMART) Centre, Singapore 117543, Singapore∗

(Dated: November 8, 2011)

Phase space tomography estimates correlation functions entirely from multiple snapshots in the

evolution of the system, rather than traditional interferometric methods requiring measurement

of multiple two–point correlations. However, as in every tomographic formulation, undersampling

poses a severe limitation. In the context of quantum correlation function estimation, a new theory

utilizing compressed sensing was recently established [D. Gross et al. Phys. Rev. Lett. 105, 150401

(2010)] whereby both measurement and post–processing dimensionality are reduced without giving

up reconstruction fidelity. Here we present the first, to our knowledge, experimental demonstration

of compressive reconstruction of the classical optical correlation function, i.e. the mutual intensity

function which is of course the analogue to a conservative quantum state. Our compressive algo-

rithm makes explicit use of the physically justifiable assumption of a low–entropy source (or state.)

Since the source was directly accessible in our classical experiment, we were able to compare the

compressive estimate of the mutual intensity to an independent ground–truth estimate from the van

Cittert–Zernike theorem and verify substantial quantitative improvements in the reconstruction.

Correlation functions provide complete characteriza-

tion of wave fields in several branches of physics, e.g.

the mutual intensity of stationary quasi-monochromatic

partially coherent classical light [1], and the density ma-

trix of conservative quantum systems (i.e., with time–

independent Hamiltonian) [2]. The classical mutual in-

tensity expresses the joint statistics between two points

on a wavefront, and it is traditionally measured us-

ing interferometry: two sheared versions of a field are

overlapped in a Young, Mach–Zehnder or rotational

shear [3, 4] arrangement and two–point ensemble statis-

tics are estimated as time averages by a slow detector

under the assumption of ergodicity [1, 5].

As an alternative to interferometry, phase space to-

mography (PST) is an elegant method to measure corre-

lation functions. In classical optics, PST involves mea-

suring the intensity under spatial propagation [6–8] or

time evolution [9].In quantum mechanics, analogous

techniques apply, commonly known as quantum state to-

mography (QST) [10–13]. However, a fundamental dif-

ficulty in performing tomography in both cases is the

large dimensionality of the unknown state. For example,

to recover a n × n correlation matrix, a standard imple-

mentation would require at least n2data points.

A theory based on low-rank assumption for density ma-

trix recovery in QST was recently developed [14]. It was

shown that a nearly pure quantum state of dimension n

and rank r requires only O(rnlogn) [15] data points.

This treatment is directly applicable to classical PST

thanks to the analogy between correlation functions; in

this Letter, we present the first, to our knowledge, exper-

imental measurement and verification of the correlation

function of a classical partially coherent field using low-

rank matrix recovery (LRMR).

Compressed sensing [16–18] exploits sparsity priors to

recover missing data with high confidence from a few

linear measurements. Here, sparsity means that the un-

known vector only contains a small number of nonzero

entries in some specified basis. LRMR [19, 20] is a gener-

alization of compressed sensing from vectors to matrices:

one attempts from very few linear measurements to still

reconstruct a high-fidelity and low-rank description of the

unknown matrix.

The low-rank assumption for classical partially coher-

ent light anticipates a source composed of a small num-

ber of mutually incoherent effective sources, i.e.

herent modes, to describe measurements.

sentially equivalent to the low entropy assumption [14],

i.e. a nearly pure quantum state in the quantum ana-

logue. This assumption is valid for lasers, synchrotron

and table-top X-ray sources [21], and K¨ ohler illumination

in optical microscopes [1]. An additional requirement for

LRMR to succeed is that measurements are “incoherent”

with respect to the eigenvectors of the matrix, i.e. the

measured energy is approximately evenly spread out be-

tween modes [15, 22]. Diffraction certainly mixes the co-

herent modes of the source rapidly, so we expect LRMR

to perform well for classical PST. The same expectation

for QST has already been established [14].

co-

This is es-

The two–point correlation function of a classical sta-

tionary quasi–monochromatic partially spatially coher-

ent field is the mutual intensity function [1]

J(x1,x2) = ?g∗(x1)g(x2)?, (1)

where ?·? denotes ensemble average.

The measurable quantity of the classical field, i.e. the

arXiv:1109.1322v2 [physics.optics] 4 Nov 2011

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intensity, after propagation by distance z is [1],

??

exp

λz(x2

I(xo;z) =

?

dx1dx2J(x1,x2)

?

−iπ

1− x2

2) exp

?

i2πx1− x2

λz

xo

?

, (2)

which can be expressed in operator form as

I = tr(PxoJ), (3)

where P denotes the operator that combines both the

quadratic phase and Fourier transform operations in Eq.

2, tr(·) computes the trace, and xodenotes the lateral co-

ordinate at the observation plane. (For the coherent case,

see [22, 23].) By changing variables x = (x1+ x2)/2,

x?= x1−x2and Fourier transforming the mutual inten-

sity with respect to x we obtain the Ambiguity Function

(AF) [24, 25]

?

Eq. 2 can be written as [6–8, 24],

A(u?,x?)=J

?

x +x?

2,x −x?

2

?

exp(−i2πu?x)dx.(4)

˜I(u?;z) = A(u?,λzu?),(5)

where˜I is the Fourier transform of the vector of mea-

sured intensities with respect to xo. Thus, radial slices

of the AF may be obtained from Fourier transforming the

vectors of intensities measured at corresponding propaga-

tion distances, and from the AF the mutual intensity can

be recovered by an additional inverse Fourier transform,

subject to sufficient sampling.

To formulate a linear model for compressive PST, the

measured intensity data is first arranged in Ambiguity

space. The mutual intensity is defined as the “sparse”

unknown to solve for. To relate the unknowns (mutual in-

tensity) to the measurements (AF), the center–difference

coordinate–transform is first applied, expressed as a lin-

ear transformation T upon the mutual intensity J, fol-

lowed by Fourier transform F, and adding measurement

noise e as

A = F · T · J + e.(6)

The mutual intensity propagation operator is unitary

and Hermitian, since it preserves energy. We use eigen-

value decomposition to determine the basis where the

measurement is sparse. The resulting basis, i.e. the set

of eigenvectors, is also known as coherent modes in classi-

cal optical coherence theory, whereas the whole process is

known as coherent mode decomposition [26]. The goal of

the LRMR method is to minimize the number of coherent

modes to describe measurements. By doing LRMR, we

impose two physically meaningful priors: (1) existence of

the coherent modes [26], and (2) sparse representation of

the partially coherent field in terms of coherent modes.

FIG. 1: Experimental arrangement

z (mm)

x (µm)

19 26 34 41 50 58 67 77 87 99 111 125 141 160 181 207 239 280 334 409

?1500

?1000

?500

0

500

1000

1500

0

0.005

0.01

0.015

0.02

0.025

0.03

FIG. 2: Intensity measurements at several unequally

spaced propagation distances.

Mathematically, if we define all the eigenvalues λiand

the estimated mutual intensity asˆJ, the method can be

written as

minimize rank(ˆJ)

subject to A = F · T ·ˆJ,

λi≥ 0,and

?

i

λi= 1. (7)

Direct rank minimization is NP–hard; however, it can be

accomplished by solving instead a proxy problem: con-

vex minimization of the “nuclear norm” (?1norm) of the

matrix J [19, 27]. In our implementation, we used the

singular value thresholding (SVT) method [28]. Numer-

ical simulations for Gaussian-Schell model sources with

varying degree of coherence showed that SVT yields re-

constructions with mean-square-error less than 3% given

the same number of measurements as in our experiment

in the noiseless case. The reconstruction degrades grace-

fully as the signal-to-noise ratio decreases (see Supple-

mental Material).

The experimental arrangement is illustrated in Fig-

ure 1. The illumination is generated by an LED with

620nm central wavelength and 20nm bandwidth. To gen-

erate partially coherent illumination, a single slit of width

355.6µm (0.014??) is placed immediately after the LED

and one focal length (75 mm) to the left of a cylindrical

lens. One focal length to the right of the lens, we place

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x1 (µm)

x2 (µm)

?100001000

?1500

?1000

?500

0

500

1000

1500

?0.01

0

0.01

0.02

0.03

(a)

x1 (µm)

x2 (µm)

?100001000

?1500

?1000

?500

0

500

1000

1500

?0.005

0

0.005

0.01

0.015

0.02

0.025

(b)

FIG. 3: Real part of the reconstructed mutual intensity

from (a) FBP; (b) SVT. For imaginary parts, see

Supplemental Material.

010 2030

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Modes

Eigenvalue

FIG. 4: SVT Estimated eigenvalues.

the second single slit of width 457.2µm (0.018??), which

is used as a one–dimensional (1D) object.

The goal is to retrieve the mutual intensity immedi-

ately to the right of the object from a sequence of in-

tensity measurements at varying z-distances downstream

from the object, as described in the theory. We measured

the intensities at 20 z-distances, ranging from 18.2mm to

467.2mm, to the right of the object. The data are given in

Figure 2. Each 1D intensity measurement consists of 512

samples, captured by a CMOS sensor with 12µm pixel

size. The dimension of the unknown mutual intensity

matrix to be recovered is 512 × 512. Since only intensi-

ties at positive z, i.e. downstream from the object, are

accessible, we can only fill up the top right and bottom

left quadrants of Ambiguity space. The other two quad-

rants are filled symmetrically, i.e. assuming that if the

field propagating to the right of the object were phase

conjugated with respect to the axial variable z, it would

yield the correct field to the left of the object, i.e. nega-

tive z [8, 13]. Under this assumption, a total of 40 radial

slices are sampled in Ambiguity space. This is only 7.8%

of the total number of entries in the unknown mutual

intensity matrix.

The reconstructions from the filtered–backprojection

(FBP) and SVT methods are compared in Figure 3. The

FBP reconstruction suffers from three types of artifacts.

First, it yields an unphysical result with lower values

along the diagonal of the matrix. However, a correla-

tion function should always have maximum value at zero

separation. Second, the estimated degree of coherence is

lower than the theoretical prediction. The third is the

high frequency noise due to undersampling between the

radial slices. All these artifacts are greatly suppressed or

completely removed in the SVT reconstruction. In the

real part of the reconstruction, the width of the square

at the center is approximately 456µm (38 pixels), which

agrees with the actual width of the slit. The imaginary

part is orders of magnitude smaller than the real part (see

Supplemental Material). The SVT estimated eigenvalues

are shown in Figure 4.

We further validated our compressive estimates by

measuring the field intensity immediately to the right

of the illumination slit (Figure 5a). Assuming that the

illumination is spatially incoherent (a good assumption

in the LED case), the mutual intensity of the field imme-

diately to the left of the object is the Fourier transform

of the measured intensity, according to the Van Cittert–

Zernike theorem [1]. This calculated mutual intensity,

based on the measurement of Figure 5(a) and screened by

the object slit, is shown in Figure 5(b). The eigenvalues

computed by coherent mode decomposition are shown in

Figure 5(c) and are in good agreement with the SVT es-

timates, as compared in Figure 5(d). It is seen that 99%

of the energy is contained in the first 13 modes, which

confirms our low-rank assumption. The compressive re-

construction may also be compared quantitatively to the

FBP reconstruction in terms of the global degree of co-

√?

imentally found as 0.46 and 0.12, respectively; whereas

the estimate yielded by the Van Cittert–Zernike theo-

rem is 0.49. The eigenvectors of each individual mode

are shown and compared in the Supplemental Material.

Small errors in the compressive estimate are because the

missing cone is still not perfectly compensated by the

compressive approach, and other experimental imperfec-

tions.

In this classical experiment, we have the benefit that

direct observation of the one–dimensional object is avail-

able; thus, we were able to carry out quantitative analysis

of the accuracy of the compressive estimate. In the quan-

tum analogue of measuring a complete quantum state,

direct observation would have of course not been possi-

ble, but the accuracy attained through the compressive

estimate should be comparable, provided the low entropy

assumption holds [14].

Here we followed a much simplified version of the ap-

proach described in [22] which showed that the com-

plex operators describing the measurements should be

herence parameters ¯ µ =

iλ2

i|λi|[29], which were exper-

i

?

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4

?1500 ?1000?5000500 10001500

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

x(µm)

Intensity

(a)

x1 (µm)

x2 (µm)

?100001000

?1500

?1000

?500

0

500

1000

1500

0

0.005

0.01

0.015

0.02

0.025

(b)

05 10 1520 25 30

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Modes

Eigenvalue

(c)

0510 1520 2530

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Modes

Absolute Error

(d)

FIG. 5: (a) Intensity measured immediately to the right

of the illumination slit; (b) the real part of the Van

Cittert–Zernike theorem estimated mutual intensity

immediately to the right of the object slit (for imaginary

part, see Supplemental Material); (c) eigenvalues of the

mutual intensity in (b); (d) the absolute error between

the eigenvalues in Figure 4 and 5(c) versus mode index.

uniformly distributed in the n–dimensional unit sphere,

whereas we simply utilized free space propagation, the

classical analogue of rotated quadrature phase opera-

tors [10, 13]. The phase masks described in [22] to im-

plement optimal sampling are outside the scope of the

present work.

The authors thank Baile Zhang, Jonathan Petruccelli,

and Yi Liu for helpful discussions and one anonymous

reviewer for constructive criticism.

was provided by Singapore’s National Research Founda-

tion through the Centre for Environmental Sensing and

Modeling (CENSAM) and BioSystems and bioMechanics

(BioSyM) independent research groups of the Singapore-

MIT Alliance for Research and Technology (SMART),

the US National Institutes of Health and by the Chevron–

MIT University Partnership Program.

Financial support

∗lei tian@mit.edu

[1] L. Mandel and E. Wolf, Optical coherence and quantum

optics (Cambridge University Press, 1995).

[2] K.Blum,Densitymatrix

(Plenum Press, 1981).

[3] K. Itoh and Y. Ohtsuka, J. Opt. Soc. Am. A 3, 94 (1986).

[4] D. L. Marks, R. A. Stack, and D. J. Brady, Appl. Opt.

38, 1332 (1999).

[5] J. W. Goodman, Statistical Optics (Wiley-Interscience,

2000).

[6] K. A. Nugent, Phys. Rev. Lett. 68, 2261 (1992).

[7] M. G. Raymer, M. Beck, and D. McAlister, Phys. Rev.

Lett. 72, 1137 (1994).

[8] C. Q. Tran, A. G. Peele, A. Roberts, K. A. Nugent, D. Pa-

terson,and I. McNulty, J. Opt. Soc. Am. A 22, 1691

(2005).

[9] M. Beck, M. G. Raymer, I. A. Walmsley, and V. Wong,

Opt. Lett. 18, 2041 (1993).

[10] K. Vogel and H. Risken, Phys. Rev. A 40, 2847 (1989).

[11] D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani,

Phys. Rev. Lett. 70, 1244 (1993).

[12] U. Leonhardt, Phys. Rev. Lett. 74, 4101 (1995).

[13] C. Kurtsiefer, T. Pfau, and J. Mlynek, Nature (London)

386, 150 (1997).

[14] D. Gross, Y.-K. Liu, S. T. Flammia, S. Becker,

J. Eisert, Phys. Rev. Lett. 105, 150401 (2010).

[15] D. Gross, IEEE Trans. Inf. Theory 57, 1548 (2011).

[16] E. Cand` es, J. Romberg, and T. Tao, IEEE Trans. In-

form. Theory 52, 489 (2006).

[17] E. Cand` es, J. Romberg, and T. Tao, Comm. Pure Appl.

Math. 59, 1207 (2006).

[18] D. L. Donoho, IEEE Trans. Inform. Theory 52, 1289

(2006).

[19] E. J. Cand` es and B. Recht, Found. Comput. Math. 9,

717 (2009).

[20] E. J. Cand` es and T. Tao, IEEE Trans. Inform. Theory

56, 2053 (2010).

[21] D. Pelliccia, A. Y. Nikulin, H. O. Moser,

Nugent, Opt. Express 19, 8073 (2011).

[22] E. J. Cand` es, T. Strohmer, and V. Voroninski, ArXiv

e-prints (2011), arXiv: 1109.4499v1.

[23] E. J. Cand` es, Y. Eldar, T. Strohmer, and V. Voroninski,

ArXiv e-prints (2011), arXiv:1109.0573.

[24] K.-H. Brenner, A. Lohmann, and J. Ojeda-Casta˜ neda,

Opt. Commun. 44, 323 (1983).

[25] K.-H. Brenner and J. Ojeda-Casta˜ neda, Opt. Acta. 31,

213 (1984).

[26] E. Wolf, J. Opt. Soc. Am. 72, 343 (1982).

[27] E. J. Cand` es and Y. Plan, ArXiv e-prints

arXiv:0903.3131.

[28] J.-F. Cai, E. J. Cand` es,

(2008), arXiv:0810.3286.

[29] A. Starikov, J. Opt. Soc. Am. 72, 1538 (1982).

theoryandapplications

and

and K. A.

(2009),

and Z. Shen, ArXiv e-prints