Ab Initio Spatial Phase Retrieval via Intensity Triple Correlations NOLAN PEARD 12KARTIK AYYER 34AND_2

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Ab Initio Spatial Phase Retrieval via Intensity

Triple Correlations

NOLAN PEARD,1,2 KARTIK AYYER,3,4 AND

HENRY N. CHAPMAN2,4,5,6,*

1Department of Applied Physics, Stanford University, Stanford, CA, USA

2Center for Free-Electron Laser Science CFEL, Deutsches Elektronen-Synchrotron DESY, Notkestr. 85,

22607 Hamburg, Germany

3Max Planck Institute for the Structure and Dynamics of Matter, 22761 Hamburg, Germany

The Hamburg Center for Ultrafast Imaging, Universität Hamburg, Luruper Chausee 149, 22761 Hamburg,

Germany

5Department of Physics, Universität Hamburg, Luruper Chausee 149, 22761 Hamburg, Germany

6Department of Physics and Astronomy, Uppsala University, Box 516, Uppsala SE-75120, Sweden

*henry.chapman@cfel.de

Abstract: Second-order intensity correlations from incoherent emitters can reveal the Fourier

transform modulus of their spatial distribution, but retrieving the phase to enable completely

general Fourier inversion to real space remains challenging. Phase retrieval via the third-order

intensity correlations has relied on special emitter conﬁgurations which simpliﬁed an unaddressed

sign problem in the computation. Without a complete treatment of this sign problem, the general

case of retrieving the Fourier phase from a truly arbitrary conﬁguration of emitters is not possible.

In this paper, a general method for ab initio phase retrieval via the intensity triple correlations is

described. Simulations demonstrate accurate phase retrieval for clusters of incoherent emitters

which could be applied to imaging stars or ﬂuorescent atoms and molecules. With this work, it is

now ﬁnally tractable to perform Fourier inversion directly and reconstruct images of arbitrary

arrays of independent emitters via far-ﬁeld intensity correlations alone.

1. Introduction

Coherent diﬀractive imaging uses the stationary far-ﬁeld interference of elastically-scattered light

to infer the geometry of a scattering potential via Fourier analysis. Since most photodetectors

perform an intensity measurement, information about the relative phases

𝜙( ®𝑚)

of the scattered

waves at pixels

®𝑚

is lost and Fourier inversion to real space is incomplete [1]. This “phase

problem” is shared across a variety of imaging modalities, including x-ray crystallography and

optical microscopy, and research in each ﬁeld has arrived at a variety of techniques to obtain the

phase information.

Non-stationary or incoherent scattering processes are known to provide more information, as

much as twice the information cut-oﬀ in an optical microscope utilising incoherent illumination

or ﬂuorescence as compared with plane-wave illumination [2]. The far-ﬁeld intensity distribution

of such a process is featureless, but the measurement of intensity-intensity correlations can

nevertheless be used to extract the Fourier amplitude of the object’s structure as ﬁrst demonstrated

by Hanbury Brown and Twiss on the radio emission of bright stars [3]. This approach is attractive

in situations where lenses of high enough angular or spatial resolution do not exist. This is

certainly the case in the X-ray regime where recent work has examined the possibility of using

photon pair correlations to retrieve the Fourier spectrum of x-ray ﬂuorescence emission [4

–

9].

One is still left with the phase problem, which can be solved using iterative phase retrieval [9]

when the correlations are adequately and extensively sampled by detectors with large numbers

of pixels. However, it has been known since the 1960s that intensity triple correlations can

arXiv:2210.03793v3 [physics.optics] 6 Jul 2023

reveal partial phase information directly in the form of the so-called closure phase [10, 11]. This

has been heavily investigated in the ﬁeld of radio astronomy [12

–

16] with the aim to develop

the means to reconstruct an image of an arbitrary arrangement of emitters without the use of

additional constraints.

Retrieval of the Fourier phase,

𝜙( ®𝑚)

, begins by ﬁrst computing the absolute value of the closure

phase,

|Φ( ®𝑚, ®𝑛)|=|𝜙( ®𝑚+ ®𝑛) − 𝜙( ®𝑚) − 𝜙(®𝑛)|

, from the triple correlations. Unfortunately, we

need the signed value of Φto recover completely 𝜙( ®𝑚)for the following reason: At the start of

phase extraction, an estimate value is chosen for the ﬁrst pixel. But for the next pixel, the sign

ambiguity of

|Φ|

returns two possible values of

𝜙( ®𝑚+1)

and an additional two possible values

for every subsequent pixel (except for special arrays where

sgn(Φ)

may be assumed constant).

To avoid this exponential expansion of the solution space, we show how redundant information

contained in

|Φ|

may be used to constrain the possible values of

sgn(Φ)

. Multiple publications

have described the concept but, to the best of our knowledge, no one has yet provided complete or

useful details on calculating

sgn(Φ( ®𝑚, ®𝑛))

[17

–

30]. Ab-initio phase retrieval from the third-order

intensity correlations has thus remained incomplete for decades. With our method, it is now

possible to solve for the phase of an arbitrary array of incoherent emitters from the third-order

intensity correlations alone. Combined with the second-order intensity correlations, we have a

completely general method for reconstructing images of arrays of incoherent emitters.

In this paper, we describe our solution to the sign problem of the closure phase, with the help of

a simple 1D example using round numbers in section 3.1.1, and show a numerical implementation

of our method with simulated data from classical independent light sources. This same method

may be used to reconstruct images of star clusters or, with some corrections to account for the

use of a quantum light source, arrays of ﬂuorescent molecules or atoms.

2. Theory

The diagram in Fig. 1 depicts and contrasts structure determination via coherent scattering to

that obtained from incoherent emission. When illuminated with a plane wave with a wave-vector

𝐾, the elastically scattered ﬁeld has stationary intensity given by

𝐼(®𝑞)=

𝜈



𝑖

𝑓𝑖𝑒𝑖®𝑞·®𝑟𝑖

=

𝑖 𝑗

𝑓𝑖𝑓∗

𝑗𝑒𝑖®𝑞·(®𝑟𝑖−®𝑟𝑗)(1)

for a number

𝜈

of point scatterers with scattering factors

𝑓𝑖

and positions

®𝑟𝑖

relative to an arbitrary

real-space origin. The photon momentum transfer

®𝑞

is equal to the diﬀerence

𝑘−®

𝐾

. The phase of

each scattered wave,

®𝑞· ®𝑟𝑖

, is derived from the diﬀerence in the optical path along the directions

of the incoming and outgoing waves as compared to a scatterer at the origin. The intensity pattern

is thus proportional to the square modulus of the Fourier transform

𝐹(®𝑞)

of the distribution of

scatterers in terms of spatial frequencies equated with

®𝑞

. The origin of the pattern,

®𝑞=0

, is

located in the direction of the incident beam. In this forward direction all scattered waves are in

phase and there is strong constructive interference, with intensity generally falling with scattering

angle. The pattern consists of speckles whose width is inversely proportional to the extent of

the object. The recovery of the object’s scattering potential is obtained by an inverse Fourier

transform of 𝐹(®𝑞), but only after the corresponding phases are obtained.

If, instead, the object consists of a collection of incoherent point emitters, then there is no

dependence on any incident beam and the phase of the emission, relative to that of an emitter

at an arbitrary real-space origin, is

𝑘· ®𝑟𝑖+𝜙𝑖

. We assume that the emission phases

𝜙𝑖

are

random and uncorrelated on timescales greater than the relevant system coherence time,

𝜏𝑐

, due

to independent, spontaneous emission at random times. The total light ﬁeld in this scenario is

Fig. 1. Schematic sketch of coherent diﬀraction in the forward detection plane,

intersecting with the incident beam

𝐾

. Fluorescence speckle is emitted by the atoms

isotropically and the position of the second detector is not dependent on the incident

beam. Coherent diﬀraction data is collected as a function of the scattering vector

®𝑞=®

𝑘−®

𝐾

. Correlations between triples of ﬂuorescence photons (intensities) at pixels

separated by

®𝑞1

®𝑞2

, and

®𝑞3

reveal the spatial phase information lost in the coherent

diﬀraction experiment.

often referred to as pseudo-thermal or chaotic and has intensity

𝐼(®

𝑘)=

𝜈



𝑖

𝑠𝑖𝑒𝑖(®

𝑘·®𝑟𝑖+𝜙𝑖(𝑡>𝜏𝑐))

=

𝑖 𝑗

𝑠𝑖𝑠∗

𝑗𝑒𝑖(𝜙𝑖(𝑡>𝜏𝑐)−𝜙𝑗(𝑡>𝜏𝑐))𝑒𝑖®

𝑘·(®𝑟𝑖−®𝑟𝑗)(2)

with 𝑠𝑖the amplitude of electric ﬁeld emission of the 𝑖th emitter. The intensity pattern depends

on the orientation of the object, and, given the complete independence of emission, at an instant

of time this pattern has a uniform intensity modulated by speckles of the same size as the

case for coherent scattering. When rapid exposures are measured with a photodetector, we

can consider the phases

𝜙𝑖

are reset shot-to-shot, changing the instantaneous speckle pattern.

From the right-hand side of Eqn. 2, we observe that the structure (sum of

®𝑟𝑖

for all

𝑖

) would be

diﬃcult to discern by averaging intensities over many shots—the random phase resets would

drive the interference speckle visibility to zero. However, it remains possible to obtain structural

information via intensity correlations [4].

In the following, we use the word atom to refer to any member of a collection of point

ﬂuorescent (atoms and molecules) or thermal (stars) light sources. We assume these atoms to

undergo spontaneous emission independently, i.e. that each atom emits a ﬁeld with a phase or

time delay that is uncorrelated to the ﬁelds emitted by the other atoms.

2.1. Intensity Correlations

We consider photon emission vectors in reciprocal space,

𝑘1

𝑘2

, and

𝑘3

, and their vector

diﬀerences

®𝑞1=®

𝑘1−®

𝑘2(3)

®𝑞2=®

𝑘2−®

𝑘3(4)

®𝑞3=®

𝑘3−®

𝑘1=−®𝑞1− ®𝑞2(5)

as depicted in Fig. 1. The ensemble average of third-order intensity correlations of the light ﬁeld,

Eqn. 2, over all shots

n𝑔(3)(®

𝑘1,®

𝑘2,®

𝑘3)o=(⟨𝐼(®

𝑘1)𝐼(®

𝑘2)𝐼(®

𝑘3)⟩

⟨𝐼(®

𝑘1)⟩⟨𝐼(®

𝑘2)⟩⟨𝐼(®

𝑘3)⟩)(6)

can be expressed as

n𝑔(3)(®𝑞1,®𝑞2)o≈1−3

𝜈+4

𝜈2+1−2

𝜈|𝑔(1)(®𝑞1)|2+ |𝑔(1)(®𝑞2)|2+ |𝑔(1)(−®𝑞1− ®𝑞2)|2(7a)

+2Re 𝑔(1)(®𝑞1)𝑔(1)(®𝑞2)𝑔(1)(−®𝑞1− ®𝑞2)(7b)

and is called the bispectrum. Similarly, the mean second-order intensity correlation function

n𝑔(2)(®

𝑘1,®

𝑘2)o=(⟨𝐼(®

𝑘1)𝐼(®

𝑘2)⟩

⟨𝐼(®

𝑘1)⟩⟨𝐼(®

𝑘2)⟩)(8)

may be written as n𝑔(2)(®𝑞1)o≈1−1

𝜈+𝑔(1)(®𝑞1)2

(9)

This equation is often referred to as the Siegert Relation in quantum optics [31]. For a full

derivation of Equations 7 and 9 please review the Supplement.

In Eq. 7, we have an expression for

𝑔(3)

in terms of constants, the square modulus of

𝑔(1)

, and

the real part of a product of complex-valued

𝑔(1)

. Since

𝑔(1)

may be acquired from

𝑔(2)

Eq. 9, it is possible to extract the last term 7b alone. This term is referred to as the closure in the

astronomy literature. We can rewrite the closure as

2Re 𝑔(1)(®𝑞1)𝑔(1)(®𝑞2)𝑔(1)(−®𝑞1− ®𝑞2)=

2𝑔(1)(®𝑞1)𝑔(1)(®𝑞2)𝑔(1)(−®𝑞1− ®𝑞2)cos 𝜙(®𝑞1) + 𝜙(®𝑞2) + 𝜙(−®𝑞1− ®𝑞2)(10)

where we have expressed

𝑔(1)

in polar coordinates in the complex plane. As the radial component

(𝑔(1)) is easily obtained from 𝑔(2), the phase information, 𝜙(®𝑞), can be isolated as follows.

Suppose we set

®𝑞1=®𝑚

and

®𝑞2=®𝑛

where

®𝑚

®𝑛

map to discrete pixels on a detector. The

symmetry of 𝑔(1)(®𝑞)and anti-symmetry of 𝜙(®𝑞)allow us to rearrange the closure into

cos 𝜙( ®𝑚+ ®𝑛) − 𝜙( ®𝑚) − 𝜙(®𝑛)≈

𝑔(3)( ®𝑚, ®𝑛)−(1−3

𝜈+4

𝜈2)−(1−2

𝜈)(|𝑔(1)( ®𝑚)|2+ |𝑔(1)(®𝑛)|2+ |𝑔(1)( ®𝑚+ ®𝑛)|2)

2𝑔(1)( ®𝑚)𝑔(1)(®𝑛)𝑔(1)( ®𝑚+ ®𝑛)(11)

The inverse cosine of this expression is known as the closure phase, which we represent via the

symbol

Φ( ®𝑚, ®𝑛)=±[𝜙( ®𝑚+ ®𝑛) − 𝜙( ®𝑚) − 𝜙(®𝑛)](12)

Just as in the Siegert Relation, the third-order correlation function encodes the phase

𝜙(®𝑞)

pixels in

𝑘

-space beyond the physical spatial extent of the detector (

|®𝑞max |=2®

𝑘max

) as depicted

in Fig. 1. Together, the double and triple correlations allow retrieval of the equivalent of a

coherent diﬀraction pattern and its phase across an area of

𝑘

-space four times larger than the area

of detector coverage [4, 6].

3. Phase Retrieval

Equation 12 for the closure phase

Φ( ®𝑚, ®𝑛)

is a diﬀerence equation which can be used like

a discrete derivative to estimate the slope of

𝜙( ®𝑚)

between pixels separated by

®𝑛

. The anti-

symmetry of the phase pins

𝜙(®𝑞=®

0)=0

. Since overall translation in real-space results in phase

ramps in reciprocal space, we can estimate the value of the phase at a nearest-neighbor pixel

𝜙(®𝑞=®

0)=0

without loss of generality. This estimate can be reﬁned later by seeking to

minimize the total error (see Section 3.2) of all pixels and treating the initial value as a parameter.

The diﬀerence equation and calculated

Φ( ®𝑚, ®𝑛)

from experimental data reveals the value of the

phase at the next-nearest-neighbor pixels and so forth until the phase on the entire pixel array has

been calculated.

Once the second pixel of

𝜙

(next-nearest neighbor) in any direction is calculated, the interval

of the diﬀerence equation (

®𝑛

) may be increased to ﬁnd the slope between every other (instead of

every) pixel in the same direction. Essentially, the phase values calculated for pixels near the

origin constrain the possible phase values of pixels far from the origin.

3.1. Determining the sign of the closure phase sgn(Φ)

Due to the sign ambiguity of the inverse cosine, every datum from

Φ( ®𝑚, ®𝑛)

points to two possible

values of the phase 𝜙( ®𝑚+ ®𝑛)

𝜙( ®𝑚+ ®𝑛)=+Φ( ®𝑚, ®𝑛) + 𝜙( ®𝑚) + 𝜙(®𝑛) ≡ 𝜃+(13)

𝜙( ®𝑚+ ®𝑛)=−Φ( ®𝑚, ®𝑛) + 𝜙( ®𝑚) + 𝜙(®𝑛) ≡ 𝜃−(14)

for any

®𝑚

and

®𝑛

. Assuming a global sign often leads to an incorrect slope for

𝜙

, as shown in

Fig. 2. The fact that multiple values of

Φ( ®𝑚, ®𝑛)

relate to the value of the phase at a single pixel

allows us to determine the proper sign of Φ( ®𝑚, ®𝑛)for each ®𝑚and ®𝑛.

Suppose for a given pixel at

®𝑢

there exist

𝑁

sets

( ®𝑚

®𝑛)

Φ( ®𝑚, ®𝑛)

for which

®𝑚+ ®𝑛=®𝑢

. Each

set oﬀers a pair of possible values for

𝜙(®𝑢)

, giving

2𝑁

possible values for

𝜙(®𝑢)

altogether. We

know that each and every one of the

𝑁

pairs contains the correct value, so comparing the

𝑁

pairs

should reveal it. Ideally, the correct value is included

𝑁

times between the

𝑁

pairs and is found

simply by taking the intersection of all pairs. Next, we show a simple 1D example to illustrate

the principle.

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AbInitioSpatialPhaseRetrievalviaIntensityTripleCorrelationsNOLANPEARD,1,2KARTIKAYYER,3,4ANDHENRYN.CHAPMAN2,4,5,6,*1DepartmentofAppliedPhysics,StanfordUniversity,Stanford,CA,USA2CenterforFree-ElectronLaserScienceCFEL,DeutschesElektronen-SynchrotronDESY,Notkestr.85,22607Hamburg,Germany3MaxPlanckInstit...

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Ab Initio Spatial Phase Retrieval via Intensity Triple Correlations NOLAN PEARD 12KARTIK AYYER 34AND_2

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