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Numerically focused full-field swept-source opticalcoherence microscopy with low spatial coherence

illuminationAnton Grebenyuk, Antoine Federici, Vladimir Ryabukho, Arnaud Dubois

To cite this version:Anton Grebenyuk, Antoine Federici, Vladimir Ryabukho, Arnaud Dubois. Numerically focused full-field swept-source optical coherence microscopy with low spatial coherence illumination. Applied op-tics, Optical Society of America, 2014, 53 (8), pp.1697-1708. �10.1364/AO.53.001697�. �hal-00956673�

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Numerically focused full-field swept-source opticalcoherence microscopy with low spatial

coherence illumination

Anton Grebenyuk,1,* Antoine Federici,2 Vladimir Ryabukho,1,3 and Arnaud Dubois2

1Department of Optics and Biophotonics, Saratov State University, 83 Astrakhanskaya, Saratov 410012, Russia2Laboratoire Charles Fabry, Institut d’Optique Graduate School, CNRS UMR 8501, Université Paris-Sud,

2 Avenue Augustin Fresnel, 91127 Palaiseau Cedex, France3Institute of Precision Mechanics and Control, Russian Academy of Sciences, 24 Rabochaya,

Saratov 410028, Russia

*Corresponding author: [email protected]

Received 28 May 2013; revised 9 December 2013; accepted 14 January 2014;posted 30 January 2014 (Doc. ID 191145); published 10 March 2014

We propose a 3D imaging technique based on the combination of full-field swept-source optical coherencemicroscopy (FF-SSOCM) with low spatial coherence illumination and a special numerical processing thatallows for numerically focused coherent-noise-free imaging without mechanical scanning in longitudinalor transversal directions. We show, both theoretically and experimentally, that the blurring effects aris-ing in FF-SSOCM due to defocus can be corrected by appropriate numerical processing even when lowspatial coherence illumination is used. A FF-SSOCM system was built for testing the performance of thistechnique. Coherent-noise-free imaging of a sample with longitudinal extent exceeding the optical depthof field is demonstrated without displacement of the sample or any optical element. © 2014 OpticalSociety of AmericaOCIS codes: (110.4500) Optical coherence tomography; (180.3170) Interference microscopy;

(070.7345) Wave propagation; (100.3010) Image reconstruction techniques; (110.6880) Three-dimensional image acquisition; (090.1995) Digital holography.http://dx.doi.org/10.1364/AO.53.001697

1. Introduction

Low-coherence interferometry is a noncontact opticalsensing technology that provides the possibility of 3Dimage reconstruction. In the most recent technique,referred to as optical coherence tomography (OCT),an optical probe directs a broadband light beam onthe sample and sends reflected light back to theinterferometer [1,2]. The interference pattern associ-ated with each single scan point is interpreted as adepth profile (A-scan). In the time domain approach,the length of the interferometer reference arm is

scanned to bring forth the appearance of the interfer-ence signal. In the frequency domain approach, thereference arm length is fixed. The spectrum of theinterference pattern is captured by the detectorand then converted to the time domain using aFourier transformation. Imaging speed is dramati-cally improved in the frequency domain OCT. In bothapproaches, a cross section (B-scan) is obtained byscanning the probe linearly across the sample. 3Dimages can be generated by combining multiple crosssections. Relatively low numerical aperture lensesare used in OCT to preserve a sufficient depth offield, especially in the frequency domain approach,where the depth profile (A-scan) is acquired in par-allel. This limits the transverse resolution, which

1559-128X/14/081697-12$15.00/0© 2014 Optical Society of America

10 March 2014 / Vol. 53, No. 8 / APPLIED OPTICS 1697

http://dx.doi.org/10.1364/AO.53.001697

is a major drawback of OCT. Several solutions havebeen implemented to improve the transverse resolu-tion, including dynamic focusing [3–5], illuminationof the sample with a Bessel beam [6,7], or image post-processing as implemented, for example, in interfero-metric synthetic aperture microscopy (ISAM) [8].

Another way to achieve high transverse resolutionimaging is to acquire en face images rather thancross sectional B-scan images using high numericalaperture optics, as can be accomplished in optical co-herence microscopy (OCM). Two general approachesfor OCM have been reported to date. The first ap-proach of OCM is based on the combination ofscanning confocal microscopy with low-coherenceinterferometry, as initially demonstrated in [9] andrevisited later [10]. A drawback of this approach isthe relatively slow acquisition speed, since three di-mensions have to be scanned to get a 3D image. Thesecond approach of OCM involves full-field illumina-tion and detection using an area camera. En facesections are then acquired without scanning in thetransversal directions. This approach has attracteda lot of attention because of its potentially fasteroperation speed for 3D imaging. It has been realizedin a number of modalities in both the time domainand Fourier domain with spatially coherent illumi-nation [11–13] and low spatial coherence illumina-tion [14–17].

When achieved with plane-wave illumination (i.e.,spatially coherent illumination), OCM correspondsto conventional digital holography or digital holo-graphic microscopy (DHM) utilizing broadband orfrequency tunable (swept) light sources. Like digitalholography, these holographic OCM techniquesoffer the possibility of numerical refocusing withoutadjustment of the optical focus [11–13,18,19]. There-fore, when realized in the Fourier domain using fre-quency swept sources, these techniques can acquire3D images without displacement of the sample oroptical elements in either the transversal or longi-tudinal directions. This is very promising for high-speed 3D imaging. However, the transversal spatialcoherence of plane-wave illumination makes thesetechniques very sensitive to coherent noise (e.g., be-cause of dust in the optical path) and degradeimage quality because of the possible presence ofgranular speckle and the manifestation of cross talkeffects [20].

When achieved with partially spatially coherent(low coherence) illumination, the technique is usu-ally termed as full-field optical coherence microscopy(FF-OCM), or sometimes full-field optical coherencetomography (FF-OCT). FF-OCM produces tomo-graphic images in the en face orientation by thearithmetic combination of interferometric images ac-quired with an area camera and by illuminating thewhole field to be imaged with low-coherence light(temporally and spatially). This technique has beendevelopedmostly in the time domain with broadbandlow-coherence light sources, including light emittingdiodes (LEDs) [14], thermal light lamps [16,17], or

fluorescence-based sources [21]. Full-field OCM canimage at high resolution with highly reduced coher-ent noise and cross talk effects [20]. However, as atime domain technique, it requires scanning of thelongitudinal sample structures to produce a 3Dimage, which is time consuming.

Recently, a full-field OCT/OCM technique withtransversally partially coherent illumination hasbeen demonstrated in the frequency domain usinga frequency swept source [22]. This work demon-strated that full-field swept-source OCM (FF-SSOCM) provides the possibility of parallel detectionof 3D image data with highly reduced coherent noise.Nevertheless, with this kind of technique, as in con-ventional frequency domain OCT, the imaged regionis limited by the original depth of field related to theobjective NA.

In this paper, we propose a 3D imaging technique,based on FF-SSOCM with partial spatial coherenceillumination, in which the imaging depth is extendedby numerical processing. This processing performs anumerical correction of the complex coherencesignal in FF-SSOCM, including both correction ofthe angular spectrum gate (numerical focusing) andcorrection of the temporal spectrum gate (similarlyto conventional Fourier domain OCT). This numeri-cal processing takes into account the specific proper-ties of illumination with an arbitrary degree oftransversal spatial coherence. It thus allows fornumerically focused coherent-noise-free 3D imagingwithout mechanical scanning in longitudinal ortransversal directions. The theory of this numericalprocessing is developed based on a previous theoreti-cal model [23,24]. A FF-SSOCM system was built fortesting the performance of this numerical processingtechnique. Experimental observations confirm thetheoretical predictions and the validity of the pro-posed method.

2. Image Formation in FF-SSOCM

A. Interference Analysis

The theoretical analysis presented in this paper isbased on the theoretical model of image formationin optical microscopes proposed in [23,24]. Thismodel is based on the scalar diffraction theory andcombines an interference analysis similar to theWiener–Khintchin theorem derivation of [25] witha special analysis of the optical microscope transmis-sion functions.

To consider the signal delivered by a two-beaminterference microscope using illumination witharbitrary transversal spatial coherence, we first needto analyze the interference of the optical fields fromboth arms of the interferometer. For this purposesome results from [23,24] are summarized in thissubsection.

The intensity of a quasi-monochromatic opticalfield of circular temporal frequency, ω, in the regis-tration plane �x; y� can be considered proportionalto the power spectral density of the field, I�ω; x; y�:

1698 APPLIED OPTICS / Vol. 53, No. 8 / 10 March 2014

I�ω; x; y� � IS�ω; x; y� � IR�ω; x; y� � Γ�ω; x; y�� Γ��ω; x; y�; (1)

where the spectral densities, IS�ω; x; y� andIR�ω; x; y�, are proportional to the intensities of theoptical fields coming from the sample and the refer-ence arms; the mutual spectral density, Γ�ω; x; y�, isproportional to their mutual coherence function.These functions are determined by the followingexpressions [23,24]:

2πδ�ω − ω0�IS�ω; x; y� � hVS�ω; x; y�V�S�ω0; x; y�i; (2a)

2πδ�ω − ω0�IR�ω; x; y� � hVR�ω; x; y�V�R�ω0; x; y�i; (2b)

2πδ�ω − ω0�Γ�ω; x; y� � hVS�ω; x; y�V�R�ω0; x; y�i; (2c)

where the functions, VS�ω; x; y� and VR�ω; x; y�, areproportional to the complex amplitude distributionsof the sample and reference fields. The angularbrackets indifferently denote time averaging orstatistical averaging, since the optical fields aresupposed to be stationary.

To determine the averages in Eq. (2) we need toanalyze the optical field propagation in the two inter-ferometer arms from some initial planes �x0; y0� tothe registration plane �x; y�, i.e., determine the opti-cal transmission functions of the two arms. Theinitial �x0; y0� planes should be chosen in such away that the optical field distributions in theseplanes are identical for both arms.

In the theoretical model [23,24], the optical fieldpropagation is analyzed using the Fresnel–Kirchhoffintegral and angular spectrum representation. Theimaging and illumination systems are treated undera parabolic approximation and with the assumptionthat the aberrations of the microscope optics are cor-rected for the analyzed part of the field of view insuch a way that the general shape of the equationshold when higher numerical apertures are consid-ered (some aberrations can still be taken into accountby the aperture function, A). Such an assumptioncannot be applied to the optical field propagationthrough the sample and thence it is treated morestrictly for being applicable to high numerical aper-ture imaging (the ultimate limitation being thescalar approximation) [23].

Considering the back focal planes (apertureplanes) of the objectives at the stage of illumination(before reflection from the sample and referencemirror) as the initial planes (Fig. 1), we can writefor the reference field [24]:

VR�ω;x;y�≈μ�ω;x;y�ZZ

dx0dy0V0�ω;x0;y0�Φ2R

×�ω;−k

x0f 0;−k

y0f 0

�rR

�ω;−k

x0f 0;−k

y0f 0

�×A�ω;−x0;−y0�exp�ik�x0x�y0y�∕f 0L�; (3)

μ�ω; x; y� � iλf 0L

expfik�2f 0�1� n20� � L

� f 0L�g exp�iπλf 0L

�1 −

Lf 0L

��x2 � y2�

�; (4)

ΦR�ω; kx; ky� � exp�i�zR − jf j��k2n20 − k2x − k2y�1∕2�; (5)

where f 0 and f 0L are the back focal distances of theobjective and tube lenses, respectively; n0 � nim isthe refractive index of immersion; k � ω∕c is thewavenumber; λ � 2πc∕ω is the wavelength; rR isthe amplitude reflectivity of the reference mirror(which can depend on the reflection angle);A�ω; x0; y0� is the aperture function of the objectives;V0�ω; x0; y0� is proportional to the complex amplitudedistribution of the illuminating optical field immedi-ately after passing the objective aperture plane; L isthe distance between the objective aperture plane(back focal plane) and the tube lens front principalplane; zR is the distance from the reference objectivefront principal plane to the reference mirror surface;f is the front focal distance of the objectives. Thefunction, ΦR�ω; kx; ky�, arises from the angular spec-trum representation and describes the transforma-tion of the angular spectrum due to propagationat a distance zR − jf j. μ�ω; x; y� is a complex coefficientaccounting mostly for the mean optical path ofthe field propagating through the microscope

Fig. 1. Schematic of the FF-SSOCM setup. Imaging system:OP—principal planes of the microscope objectives; OA—apertureplanes of the objectives; TP—principal planes of the tube lens; BS,cube beamsplitter; �x; y�, registration plane, where a matrix photo-detector (CCD) is placed. Illumination system: light from theswept-source is guided via an optical fiber and directed by a lenssystem, L1, onto the rotating diffuser, RD; AD, diaphragm of var-iable radius, located right after the diffuser and determining theillumination aperture size; lenses, L2 and L3, image AD onto theaperture planes, OA; FD, field diaphragm of variable radius, whichis optically conjugated by lens, L3, and the objectives’ lenses withthe front focal planes of the objectives.


(interferometer arm) and its possible curvature dueto the mismatch of the objective back focus and thetube lens front focus.

As a model object we shall consider an N-layeredmedium with layer thicknesses fΔzjg and refractiveindices fnjg. The upper N interfaces have a uniformreflectivity, while the �N � 1�th interface has a trans-versal pattern, rS�ω; xS; yS�, of the amplitude reflec-tion coefficient distribution. Such a model objectallows us to take into account the refractive proper-ties of the immersion medium as well as the outerlayers of the sample (and the respective coherenceeffects related to the refractive index variation[19,26–29]) and at the same time to avoid dealingwith scattering of light by the outer layers (whichis a usual simplification in OCT/OCM analysis ofthe defocus effects [11,12,18,19,30]) when analyzingthe signal from the sample inner structure.

The function, VS�ω; x; y�, can then be written in thefollowing form [24,31]:

VS�ω; x; y�

≈μ�ω; x; y��λf 0�2

ZZdxSdySrS�ω; xS; yS�

×ZZ

A�ω; x3; y3�T2

�ω; k

x3f 0; k

y3f 0

�

exp�−ik

�x3

�xf 0L

� xSf 0

�� y3

�yf 0L

� ySf 0

��dx3dy3

×ZZ

V0�ω; x0; y0�T1

�ω;−k

x0f 0;−k

y0f 0

�× exp�−ik�x0xS � y0yS�∕f 0�dx0dy0; (6a)

T1�ω; kx; ky� �YNj�1

tj−1;j�ω; kx; ky�

YNj�0

exp�iΔzj�k2n2j − k2x − k2y�1∕2�; (6b)

T2�ω; kx; ky� �YNj�1

tj;j−1�ω; kx; ky�

YNj�0

exp�iΔzj�k2n2j − k2x − k2y�1∕2�; (6c)

where Δz0 � zS − jf j, zS being the distance from thesample arm objective front principal plane to thesample surface; tj−1;j and tj;j−1 are the amplitudetransmission coefficients from the �j − 1�th layer tothe jth layer and backward. The complex phasedistributions in Eq. (6) arise from the angular spec-trum representation of the optical field propagationthrough the sample, similarly to ΦR.

It is important to note that the fields arising due tothe reflection from the N upper interfaces are not

taken into account here. VS�ω; x; y� represents onlypart of the actual optical field in the registrationplane, corresponding solely to the reflection fromthe �N � 1�th interface in the sample. This isdone because the interference between optical fields,generated by reflection from different interfaces, isusually considered as a spurious signal (apart fromthe autocorrelation low-coherence interferometry[32]) and is therefore not of interest in this paper.However, if necessary, these signals can as well beconsidered on the basis of this theory, by inclusionof corresponding fields in the VS�ω; x; y� function.B. Analysis of the Complex Coherence Function

Let us assume the field V0�ω; x0; y0� in the illumina-tion aperture plane �x0; y0� to be spatially incoherent,which means that

hV0�ω; x0; y0�V�0�ω0; x00; y

00�i

� 2πδ�ω − ω0�I0�ω; x0; y0�δ�x0 − x00�δ�y0 − y00�: (7)

Substitution of Eqs. (3)–(7) into Eq. (2) allows for theanalysis of the mutual coherence function, Γ�ω; x; y�,and of both the sample and reference intensities,IS�ω; x; y� and IR�ω; x; y�.

Substitution of Eq. (3) into Eq. (2c) taking intoaccount Eq. (7) yields:

Γ�ω; x; y� ≈ μ0�ω�ZZ

dxSdySrS�ω; xS; yS�

×ZZ

A2�ω; x3; y3� exp�−ik

�x3

�xf 0L

� xSf 0

�

� y3

�yf 0L

� ySf 0

��dx3dy3

×ZZ

A1�ω; x0; y0� exp�−ik

�x0

�xf 0L

� xSf 0

�

� y0

�yf 0L

� ySf 0

��dx0dy0; (8)

where

μ0�ω� �jμ�ω; x; y�j2

�λf 0�2 � λ−4f 0−2f 0−2L ; (9)

A1�ω;x0; y0�

� I0�ω;x0; y0�A��ω;−x0;−y0�r�R�ω;−k

x0f 0;−k

y0f 0

�

×T1

�ω;−k

x0f 0;−k

y0f 0

��Φ�

R

�ω;−k

x0f 0;−k

y0f 0

��2; (10a)

A2�ω; x3; y3� � A�ω; x3; y3�T2

�ω; k

x3f 0; k

y3f 0

�: (10b)

If we consider the situation when the objectives inthe sample and reference arms are placed at differentdistances from the beam splitter (i.e., when the


elements in the dashed box in Fig. 1 are shifted alongthe optical axis as a whole), then, strictly speaking,the optical fields in the aperture planes of the objec-tives are no longer identical and we have to select theinitial plane earlier in the optical path. However, ifthis asymmetry,ΔL � LS − LR, of the interferencemi-croscope arms is small compared to the longitudinalcoherence length due to the angular spectrum of theillumination field in the objective aperture plane (adescription of this type of coherence is given in[33,34]) andΔL∕λ ≪ f 02L∕�x2 � y2�max, then themutualcoherence function can be approximated as:

ΓL�ω; x; y� ≈ exp�2ikΔL�Γ�ω; x; y�; (11)

whereΓ�ω; x; y� is determinedbyEq. (8). Finally, it canbe noted that, in the case when the approximationΔL∕λ ≪ f 02L∕�x2 � y2�max is not valid, it can be shownby means of more strict analysis that under some ad-ditional assumptions an additional transversalphase modulation factor, exp�−2πiΔL�x2 � y2�∕�λf 02L ��,appears in the expression of the coherence function,ΓL�ω; x; y�. If present, such transversal phasemodulation of the coherence function can be correctednumerically.

3. Numerical Correction of the Coherence Function

A. Temporal Spectrum Gate and Angular Spectrum Gate

In a swept-source OCT/OCM experiment, the coher-ence function (11) for every particular quasi-monochromatic state of the swept source can be re-constructed from the experimental interference pat-terns in a number of ways, including phase shiftingreconstructions, off-axis reconstructions or Fourierfiltration in the temporal spectrum domain [i.e.,when a sufficient optical delay is introduced betweenthe interferometer arms to produce high frequencymodulation over ω of the ΓL�ω; x; y� signal and toallow for separation from other terms in Eq. (1)].In this section we shall consider the properties ofthe complex coherence function without consideringthe way it has been reconstructed from the raw inter-ferometric data.

It can be clearly seen from Eqs. (8) and (11), thatthe coherence function of the sample and referencefields in an interference microscope with quasi-monochromatic partially spatially coherent illumi-nation has the shape of the convolution of the samplereflection coefficient distribution of interest, rS, witha certain impulse response function. It means thatthe shape of rS can be reconstructed by means ofFourier analysis. Let us consider the transversalspatial spectrum of ΓL:

~ΓL�ω; kx; ky� �ZZ

ΓL�ω; x; y� exp�−i�kxx� kyy��dxdy:

(12)

Then, we can obtain for the spatial spectrum of thecoherence function when imaging the multilayeredmodel sample,

~ΓL�ω; kx; ky� ≈ μ0�ω�M2

"YNj�1

tj−1;j�ω�tj;j−1�ω�#

× ~rS�ω;−Mkx;−Mky�Ξ�ω; kx; ky�; (13a)

Ξ�ω;kx;ky�

� exp�2ikΔL�ZZ

dxSdyS exp�−iM�kxxS�kyyS��

×ZZ

A�ω;x3; y3�exp�ikXNj�0

Δzj

��n2j −

x23� y23f 02

s �

×exp�−i

kf 0�x3xS� y3yS�

�dx3dy3

×ZZ

Ai�ω;x0; y0�exp�ik��zS −2zR�jf j�

��n20 −

x20� y20f 02

s

�XNj�1

Δzj

��n2j −

x20� y20f 02

s ��

×exp�−i

kf 0�x0xS� y0yS�

�dx0dy0; (13b)

Ai�ω;x0; y0�

� I0�ω;x0; y0�A��ω;−x0;−y0�r�R�ω;−k

x0f 0;−k

y0f 0

�;

(13c)

where the tilde sign in ~rS (and everywhere in this pa-per) denotes the transversal spatial spectrum, simi-larly to Eq. (12) and M � f 0L∕f

0 is the absolute valueof transversal image magnification. In Eq. (13), wehave neglected the angular dependence of the trans-mission coefficients of the outer layers, tj−1;j and tj;j−1.It can be seen from Eq. (13) that the Ai�ω; x0; y0� func-tion stands quite similarly to the aperture function,A�ω; x3; y3�. However, Ai is determined by the proper-ties of the illumination rather than the properties ofthe imaging system. Therefore, we shall call the Aifunction the “illumination aperture” function. It is in-teresting to note that, when zR � jf j, the shape of thecoherence signal [Eq. (13)] is very similar to the caseof scanning confocal Fourier domain OCT/OCM [35].However, the illumination aperture in Eq. (13b) is de-termined by the intensity, I0, of the illuminationfield, while in the confocal modality it is determinedby the amplitude of the illumination. In particular,this means that, unlike the confocal modality, theillumination aperture shape of the full-field modalityis not sensitive to the objective aberrations (providedthat the objectives are identical in both arms). Also,it can be noted that the sectioning effect due to theillumination aperture is dependent in Eq. (13b) onthe reference mirror position, zR.


For more detailed analysis, the functionΞ�ω; kx; ky� can be represented as a multiplication:

Ξ�ω; kx; ky� � Ξt�ω�Ξa�ω; kx; ky�; (14)

Ξt�ω� � exp�2ik

�ΔL� n0�zS − zR� �

XNj�1

njΔzj��

;

(15a)

Ξa�ω; kx; ky� � Ξ�ω; kx; ky�Ξ�t �ω�: (15b)

It can be seen from Eqs. (13)–(15) that the Ξ functionchanges the shape of the coherence signal, ~ΓL, mak-ing it differ from the function of interest, ~rS. TheΞt�ω� function affects only the temporal spectrum,ω (and corresponds to the signal of conventionalOCT with low NAs), while Ξa�ω; kx; ky� affects pre-dominantly the spectrum of the transversal spatialfrequencies �kx; ky�, which is often regarded to asthe angular spectrum. The effect of Ξt�ω� corre-sponds to longitudinal coherence gating due to thetemporal spectrum (temporal spectrum gate), whilethe effect of Ξa�ω; kx; ky� corresponds to the blurringdue to defocus and longitudinal coherence gating dueto the angular spectrum (angular spectrum gate).The limitations of this study do not allow us to studyin detail the physical interpretation and terminologyof these coherence effects. Investigation of the coher-ence effects due to the temporal and angular spectraof the optical field for different modalities oflow-coherence interferometers can be found in theliterature [15,24,26,27,29,33,34,36–38].

B. Transversal Frequency Response Analysis andNumerical Processing Technique

Let us consider the transversal spatial frequency re-sponse of the coherence signal, which is determinedby the shape of the Ξa�ω; kx; ky� function. The graphsin Fig. 2 represent the absolute value and the argu-ment of the Ξa�ω; kx; ky� function for the case whenthe sample consists of a single surface, n0 � 1;zR � jf j; Δz0 � Δz; the imaging aperture functionis circular, NA � 0.1. Figures 2(a) and 2(c) presentthe extreme case of point-like illumination aperture,i.e., when Ai�ω; x0; y0� ≈ Ai�ω�δ�x0�δ�y0� and NAi → 0.It corresponds to spatially transversally coherent il-lumination over the sample. Figures 2(b) and 2(d)present the case of relatively wide circular illumina-tion aperture, with NAi � 0.05, which corresponds tolow spatial coherence illumination of the sample.Owing to the revolution symmetry of the illumina-tion and imaging systems, the Ξa�ω; kx; ky� functionhas circular symmetry, and hence Ξa�ω; kx; 0�presented in Fig. 2 has the same shape asΞa�ω; 0; ky�. Computation of theoretical expressionsand numerical processing of experimental data, pre-sented in Figs. 2–5, have been performed using

home-made programs, written inWolframMathema-tica software.

As can be seen in Fig. 2, the modulation ofcoherence signal ~ΓL by Ξa affects both its phaseand amplitude. A straightforward approach tonumerical correction of the experimental data andreconstruction of a sharp image of the interface ofinterest even outside the focus depth would be thefollowing: for every particular frequency ω, divide~ΓL�ω; kx; ky� by appropriate Ξ�ω; kx; ky� function. Thiscan be realized for nonzero Ξ values, if the signal-to-noise ratio (SNR) of the experimental data ~ΓL ishigh enough. Otherwise, such direct division wouldalso lead to an increase of the noise.

To avoid problems with SNR, a more reliableprocessing algorithm can be implemented, whichconsists in multiplication of ~ΓL by a special correctionfunction. To produce sharp, coherence gated image ofan interface, located under N sample layers, the ~ΓLfunction should be multiplied by

Ψ�ω;kx; ky�

��Ξ��ω;kx; ky�∕jΞ�ω;kx; ky�j; if jΞ�ω;kx; ky�j > 0;

0; if jΞ�ω;kx; ky�j � 0:

(16a)

Fig. 2. Absolute value (a), (b) and phase (c), (d) of the Ξa�ω; kx;0�distribution versus the reduced spatial frequency k0x and the defo-cus value Δz; (a), (c), NA � 0.1, NAi → 0; (b), (d), NA � 0.1,NAi � 0.05. A frequency ω � 1.2 × 1015 Hz (λ � 1.57 μm) was con-sidered. Note that we introduced here the normalized spatial fre-quency k0x � M × kx∕kwhereM is the magnification of the imagingsystem.


The resultant function ~ΓL�ω; kx; ky�Ψ�ω; kx; ky�should then be integrated (summed) over ω andFourier transformed over �kx; ky�.

Ψ�ω; kx; ky� is only a phase function with unity ab-solute value; its zero part is necessary just to avoiddivision by zero and suppress some noise. Thismeans that multiplication of ~ΓL by Ψ does not affectthe SNR of ~ΓL data, except for suppression of somehigh-frequency noise, and leads to image sharpeningor “numerical focusing”.

It can be seen in Fig. 2(b), that the amplitude of theΞa function decreases with increase of the defocusdistance. The larger the illumination aperture, thestronger this effect of Ξa amplitude decrease. Itfollows readily from Eq. (13) that if the shape ofillumination intensity distribution I0 becomespoint-like, i.e., when Ai�ω; x0; y0� ≈ Ai�ω�δ�x0�δ�y0�,then the shape of the coherence function (13) be-comes similar to the case of interference microscopywith transversally coherent plane-wave illumination

[19], which does not have a numerical refocusingrange limitation caused by the SNR [Fig. 2(a)]. Itmeans that if the illumination aperture is narrowand the numerical refocusing distance is not toolarge, the numerical processing algorithm forspatially coherent illumination [19] can be appliedto FF-SSOCM with partial spatial coherence ofillumination. The use of this algorithm providescertain advantage in computation speed in compari-son to Eq. (16a). However, for larger illuminationaperture sizes, the shape of phase modula-tionΨ�ω; kx; ky� becomes different, which can be seenfrom comparison of Figs. 2(c) and 2(d). Hence, in gen-eral, the numerical processing according to Eq. (16a)should be used for producing sharp coherence gatedFF-SSOCM images with partially transversally co-herent illumination. For the sake of improvementof the computation speed, a C-mode-like processingalgorithm can be applied. Then the ~ΓL�ω; kx; ky�function should be multiplied by

Fig. 3. (a) and (b) 3D graph represents the absolute value of the ΓPSF�ω; xS;0� distribution over xS versus defocus value Δz; the 2D pro-jected plot corresponds to the evolution of the absolute value of ΓPSF�ω; 0;0� with respect to the defocus value. (a), without numericalcorrection; (b), with numerical correction according to Eq. (16a). (c) and (d) absolute value of the ΓPSF�ω; xS;0� distribution over xS afternormalization to unity maximum for comparison of the widths for three defocus values: Δz � 0 μm (red dash line), Δz � 300 μm (bluepoint-dash line), Δz � 600 μm (black continuous line), (c), without numerical correction; (d), with numerical correction according toEq. (16a). For all images NA � 0.1, NAi � 0.05, ω � 1.2 × 1015 Hz.


Ψa�ω;kx; ky�

��Ξ�

a�ω;kx; ky�∕jΞa�ω;kx; ky�j; if jΞa�ω;kx; ky�j > 0;

0; if jΞa�ω;kx; ky�j � 0;

(16b)

and the resultant function ~ΓL�ω; kx; ky�Ψa�ω; kx; ky�should then be Fourier transformed over the threefrequencies �ω; kx; ky�. Such numerical focus adjust-ment is performed for several depths inside the sam-ple, with a step of the order of the focus depth (not atthe depth sampling step as in Eq. (16a) and theobtained data are combined into a final 3D image.In this case, the required number of numerical focusadjustments is decreased similarly to decrease thenumber of optical focus adjustments in the mechani-cal C-mode scanning approach [39], and provideshigher numerical processing speed.

As can be seen from Eqs. (13)–(15), for everyfrequency ω, a 2D Fourier transform of the Ξa

function over the spatial frequencies �kx; ky� deter-mines the transversal impulse response (point-spread-function, PSF) of the coherence functionΓL�ω; x; y�. Scaled to the object space coordinates, thisfunction can be denoted as ΓPSF�ω; x∕M; y∕M�. Fig. 3illustrates the influence of the numerical correctionaccording to Eq. (16a) on the transversal PSF of thecoherence function in the case of relatively wide illu-mination aperture NAi � 0.05. Owing to the revolu-tion symmetry of the illumination and imagingsystems, the ΓPSF�ω; xS; yS� function has circularsymmetry, so that ΓPSF�ω; xS; 0� presented in Fig. 3has the same shape as ΓPSF�ω; 0; yS�. The processingaccording to Eq. (16a) preserves a sharp PSF in spiteof the defocus, as can be seen from Fig. 3(b). This ef-fect can be even more clearly seen by comparison ofFigs. 3(c) and 3(d), which presents the PSFs for threevalues of defocus (0, 300, and 600 μm), after normali-zation to unity maximum at each defocus position.

Since the illumination aperture is relativelywide, the amplitude is lower in the defocus region.

Fig. 4. Comparison of FF-SSOCM images obtained without the focalization algorithm for two mechanically focused position (a), (b) andwith numerical focalization (c), (d) by using the algorithm for spatially coherent illumination (c) or the proposed algorithm (d). Positions ofcuts (see Fig. 5) along the xS axis are indicated in the zoomed windows. The field of view is 700 μm× 870 μm.


This amplitude decrease can be clearly seen from thetwo graphs in Fig. 3(b). For example, at 600 μm fromthe focus plane the maximum value of the ΓPSF, evenafter numerical correction, is reduced by a factor of 5compared to the maximum value of the focused ΓPSF.

As can be seen in Fig. 2(b), wider illuminationaperture results in wider nonzero regions ofjΞa�ω; kx; ky�j, which enables the detection of largerspatial frequencies. Also, for wider illuminationaperture the decrease of jΞa�ω; kx; ky�j with increaseof kx and ky becomes smoother, which leads to sup-pression of sidelobes in the PSF.

Finally, it should be noted that when the complexcoherence signal reconstruction is based on Fourierfiltration in the temporal frequency domain, thenumerical processing procedure, according toEq. (16), can be applied directly to the interferencesignal without prior reconstruction of the complex co-herence function. This results from the linearity ofthis processing. Therefore, the presence of other in-terference terms [see Eq. (1)] does not affect thecorrection of the complex coherence function.

4. Experimental Verification of the NumericalProcessing Technique

A. Experimental Setup

To verify the applicability of our numerical processingtechnique to focus correction, we have built a dedi-cated FF-SSOCM system. The experimental set-upis based onaLinnik-type interferometer, as in conven-tional full-field OCT/OCM [15]. A schematic of the ex-perimental arrangement is shown in Fig. 1. Twoidentical dry microscope objectives are used (5×, 0.1N.A.). The light source is a swept-source laser (ModelAQ4320d, Ando Electric Co.) emitting light continu-ously from ν1 � 1.97 × 1014 Hz (λ1 � 1.52 μm) to ν2 �1.85 × 1014 Hz (λ2 � 1.62 μm) with an output opticalpower of 5 mW. The light emitted by the laser isdirected into a single-mode fiber and then collimatedby a fiber collimation package (F280 APC-C, Thor-labs). A rotating diffuser, conjugated with the aper-ture plane of the microscope objectives, is used toreduce the spatial coherence of the illuminationand then remove the coherent noise related to it. Un-like conventional full-field OCM, the referencemirroris immobile, since no phase shifting is used here to ex-tract the interferometric signal amplitude and nomechanical scan is required to acquire the depth in-formation in the sample. Furthermore, the transverseinformation is acquired in parallel by an area camera,leading to a full 3D motion-free imaging system.The images are projected onto an Indium GalliumArsenide (InGaAs) camera (OWL SW1.7-CL-HS,RaptorPhotonics, 320 × 256pixels, 14 bits) via an ach-romatic 400 mm lens optimized in the 1.5 μm wave-length region. The field of view is 700 μm × 870 μm.

B. Methods

The source is swept in wavelength to acquire thespectral interference pattern which is then processed

in order to obtain directly en face tomographicimages of the sample owing to the full field illumina-tion. To produce a 3D-image, 256 2D interferencepatterns are acquired and then processed after ap-plying a Gaussian apodization. A sufficient opticaldelay, 2ΔL, was introduced between the interferencemicroscope arms prior to the imaging, to allowsufficient separation of the complex coherence termΓL�ω; x; y� from the other terms of Eq. (1). Since theswept-source laser generates light that is nonli-nearly sampled in wavenumber, a numerical pixel in-terpolation is done to provide equidistant sampling[40]. Each of the 256 interference patterns corre-sponds to a source linewidth of 3.91 × 10−1 nm(47.6 × 109 Hz), which provides a theoretical maxi-mum imaging depth in the sample of 1580 μm[41]. Besides, before applying the numerical focusingalgorithms, for every ω the spatial spectrum of coher-ence function is centered by appropriate linear phasemodulation over �x; y�, correcting the effect of someresidual misadjustment in the setup

The imaging depth of field is governed by the NA ofthe microscope objectives and the mean temporalfrequency. In our experimental conditions, the depthof field is around zmax � 150 μm, and correspondsto the actual imaging depth without focalizationcorrection.

By using the proposed numerical processing ac-cording to Eq. (16a) (which is applied directly tothe interference signal, as described above inSubsection 3.B), this yields numerically focused im-ages and expands the imaging depth which dependsthenmostly on the illumination properties [Fig. 3(b)].

The transverse resolution is also related to thenumerical aperture of the microscope objectives(NA) and of the illumination (NAi), and to the meantemporal frequency (ω). The theoretical full widthat half-maximum of the transversal PSF was calcu-lated to be ∼10 μm (with NA � 0.1, NAi � 0.05 andω � 1.2 × 1015 Hz). In the same conditions, thewidth of the step-response [42] has a theoreticalvalue of ∼6 μm. Experimentally, the width of thestep-response was estimated to be ∼8 μm. Thedeviation of the theoretical and experimental valuesis presumably caused by the objectives, which are notoptimized for these wavelengths.

The longitudinal resolution, depending on thetemporal coherence length of the light source, wasmeasured, after apodization of the spectral interfer-ence pattern, to be 21 μm in air.

The use of such microscope objectives, with rela-tively high numerical aperture (NA � 0.1), yieldsbetter transverse resolution than in conventionalswept-source OCT (SSOCT) at the cost of the depthof field, which is then much smaller than the usualone in SSOCT. However, the numerical focaliza-tion expands the maximum reachable imagingdepth and makes it similar to the one of conventionalSSOCT.

By considering low spatial coherence illumination,the larger the illumination numerical aperture, the


lower the amplitude of the detected signal from de-focused regions in the sample. Assuming that the il-lumination aperture function has a uniform circularshape, we have estimated the illumination numeri-cal aperture (NAi) to be 0.05. Then, the NAi limitsthe imaging depth, since, without enough intensity,no image can be produced, even if they are numeri-cally focused.

Given the evolution of the PSF amplitude alongthe longitudinal axis after numerical correction[Figs. 3(b) and 3(d)], the use of an illumination aper-ture NAi � 0.05 for objectives with NA � 0.1 can beconsidered as a good trade-off between transverseresolution and loss of signal from defocused regions.

C. Results

For testing the performance of our numerical process-ing technique, we have compared tomographic im-ages obtained with our FF-SSOCM techniquewithout and with focalization algorithms. We choosea sample whose longitudinal extent exceeds the opti-cal depth of field. The sample consists of two pieces ofan integrated circuit wafer placed one on the other.The two pieces partially overlap making the samplerepresent two highly contrasted structures locatedin two distinct parallel planes. The difference ofheight between the two planes is ∼330 μm.

The different images are presented in Fig. 4, wherethe upper and lower circuits appear on the left andright parts of the images respectively. Two sets of fig-ures are displayed. Images in Figs. 4(a) and 4(b) cor-respond to FF-SSOCM processing without numericalfocalization. Images in Figs. 4(c) and 4(d) compare thetwo focalization algorithms. As indicated on thefigure, in Fig. 4(b) the front focal plane of the micro-scope objective matches with the lower circuitwhereas all of the three other images correspond toa mechanically focused position on the upper circuit.Figure 4(c) presents the result of processing with thealgorithm proposed in [19] for spatially coherent illu-mination, andFig. 4(d) presents the result after usingour specific algorithm. It can be seen that, without thefocalization algorithm [Figs. 4(a) and 4(b)], only theimages of the mechanically focused surfaces appearsharp. However, application of an appropriate focali-zation numerical processing [Fig. 4(d)] allows us toobtain sharp images of the two circuits irrespectiveof their axial positions. In accordance with thetheoretical analysis presented in Subsection 3.B,the amplitude of the defocused image becomes lowerin comparison to the focused one because of the wideillumination aperture. Figure 4(c) shows that thealgorithm for interference microscopes with plane-wave illumination is not suitable to numerically focuswith wide illumination aperture.

Profiles of the absolute value of the coherence func-tion along a line (see yellow-dashed line in Fig. 4)across a thin scratch on the sample surface are pre-sented in Fig. 5 for quantitative illustration of the ef-fect of numerical focalization. It can be seen fromFig. 5 that the sharp scratch profile visible in the

focused image (black line with filled dots) getsblurred in the defocused image whether the numeri-cal focusing for spatially coherent illumination isapplied or not. However, our specific algorithm forpartially spatially coherent illumination applied tothe defocused OCM data (green line with filledrectangles) is able to restore a sharp profile similarto the one from the optically focused image, except forthe presence of side lobes. We suppose that thisdeviation is caused by some artifacts in the numeri-cal processing, in first place by the ignorance of theactual illumination aperture shape which is sup-posed to have a uniform circular shape. Enhance-ment of the image quality could be achieved bycomplete characterization of the illumination aper-ture shape.

5. Conclusions

We have demonstrated a 3D imaging technique,based on the combination of FF-SSOCM with lowspatial coherence illumination and an originalnumerical processing technique for extension ofthe imaging depth of field. It is shown, theoreticallyand experimentally, that this numerical focusingenhances the sharpness and contrast of the recon-structed images.

This numerical processing includes the processingfor spatially coherent plane-wave illumination [19]as a particular case of extremely narrow illuminationaperture. This effect can be explained using theterminology of transversal coherence of illumination

Fig. 5. Distributions of the absolute value of the reconstructedcoherence function along the xS axis in the lower circuit images(see Fig. 4) with mechanical focalization on the lower circuit (blackline with filled dots) and mechanical focalization on the upper cir-cuit (all other data points): without numerical focusing (orangeline with white dots), with numerical focusing for spatially coher-ent illumination (red line with white rectangles) and with numeri-cal focusing by the proposed technique (green line with filledrectangles). The distributions for the case of focalization on theupper circuit are magnified 4 times in comparison to the distribu-tions for focalization on the lower circuit. Unlike the algorithm forspatially coherent illumination, the proposed algorithm for arbi-trary state of spatial coherence clearly reveals the dark thinscratch, visible in the optically focused image.


field, incident on the sample. The smaller transver-sal coherence length of this field corresponds to widerillumination aperture and vice versa. Hence, thecase of narrow illumination aperture correspondsto transversally almost coherent illumination, simi-larly to plane-wave illumination. The proposednumerical processing is more general and allowsfor numerical focusing even in the case of low spatialcoherence illumination.

As can be seen from the numerical simulations andexperimental results, the use of a wide illuminationaperture leads to a decrease of the coherence signalamplitude with increased defocus. If this decrease ofthe amplitude is tolerable through the whole re-quired image depth, the system is completely motionfree and does not require mechanical displacement ofthe optical elements or the sample. If this amplitudedecrease is too large, a smaller illumination apertureshould be chosen or some longitudinal scanning canstill be performed in the C-mode regime [39]. In thelatter case, the proposed technique provides an ex-tension of the original depth of field and decreaseof the number of required mechanical adjustments.

On the other hand, it should be noted that a veryhigh sensitivity is peculiar to OCT/OCM systems.This sensitivity is necessary to overcome the signalattenuation due to scattering and absorption and de-tect the weak signals from structures inside the im-aged sample. It means that if the optical focus isadjusted at the maximum required image depth,the effects of signal attenuation due to scatteringand absorption inside the sample and attenuationdue to defocus would somewhat compensate eachother, leading to more uniform SNR in the imagedvolume. The proposed FF-SSOCM system could thenbe used as a totally motion free imaging system withextended imaging depth, even with wider numericalapertures. The experimental investigation of thepeculiarities of imaging scattering samples shouldbe a topic of further research.

We have considered here the possibilities of thetechnique for tomographic 3D imaging. The tech-nique can also be considered as a reflection-mode dig-ital holographic microscope (DHM) with partiallyspatially coherent illumination for coherent-noise-free quantitative phase imaging. All the theoreticalresults derived in this paper can be directly appliedto this DHM mode as well as to the discussedFF-SSOCM mode.

The authors are grateful to Nicolas Dubreuil forlending the swept laser source. A. Grebenyuk andV. Ryabukho have been supported in part by RussianFederation state contract 14.B37.21.0728 and grantNSh-1177.2012.2. A. Grebenyuk is also grateful tothe Development Program of National ResearchSaratov State University for supporting his stay atthe Institut d’Optique Graduate School, where theexperimental investigations were carried out.

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