Interferometric Synthetic Aperture Microscopy€¦ · Interferometric Synthetic Aperture Microscopy...

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Interferometric Synthetic Aperture Microscopy Inverse Scattering for Optical Coherence Tomography P Scott Carney Electrical and Computer Engineering The Beckman Institute for Advanced Science and Technology The University of Illinois Urbana-Champaign [email protected] optics.beckman.illinois.edu

Transcript of Interferometric Synthetic Aperture Microscopy€¦ · Interferometric Synthetic Aperture Microscopy...

Interferometric Synthetic Aperture Microscopy Inverse Scattering for Optical Coherence Tomography

P Scott CarneyElectrical and Computer Engineering

The Beckman Institute for Advanced Science and TechnologyThe University of Illinois Urbana-Champaign

[email protected]

The National Science Foundation, NASA, USAF, Beckman Foundation, NIH NCI

AcknowledgementsP. Scott Carney

Post-Docs Brynmor Davis

Dan Marks

Graduate Students Jin Sun Yang Xu

Lang Wang

Stephen A. Boppart Post-Doctoral Fellows

Dan Marks Stephen Adie

Graduate Students Tyler Ralston Adeel Ahmad

Nathan Shemonski

Diagnostic Photonics, Inc Andrew Cittadine

Kathryn Hyer Amanda Hirsch

Jason Smid Don Darga

Kiran Yemul Adam Zysk

Johns Hopkins and Ann Arundel Dr. Lisa Jacobson, MD Dr. Loraine Tatra, MD Dr. Ed Garielson, MD

Optical Coherence Tomography:range finding with low-coherence

interferometry

Source

!L

Detector

Sample

Reference Mirror

Beam Splitter

A

!L

Lc

Signal processing: speckle reduction, numerical dispersion correction

Transverse ScanningBackscatter Intensity

Axi

al S

cann

ing

(Dep

th)

Tissue Specimen

Transverse scanning in OCT imaging

Tissue SpecimenTissue Specimen

Commercial success

http://www.meditec.zeiss.com/cirrus

The problem

Beam Focusing

Low-NAOCT Column High-NA

=≈

r0

S(r0, k) =

!z=0

d2r U(r, r0, k)g(r − r0, k)

S(r0, k)Ui(r, r0, k) = A(k)g(r − r0)

Ui(r, r0, k)

UiηG

S(r0, k) = A(k)

!z=0

d2r

!d3r′G(r

′, r, k)g(r′− r0, k)g(r − r0, k)η(r

′).

Us(r, r0, k)

Us(r, r0, k) =

!d3r′ G(r

′, r, k)η(r′)Ui(r

′, r0, k)

The forward problem

g(r, k) =

1

2πW 20(k)

e−r2/2W

2

0(k)

g(q, k) = e−q2W

2

0/2

= e−q2α2

/(2k2)

W0(k) = α/k

α = π/NA

G(r′, r, k) =

eik|r−r′|

|r − r′|

=

i

!d2q eiq·(r−r

′)e−ikz(q)(z−z

′)

kz(q)

kz(q) =

!k2

− q2

Details

q

kz(q) k

S(r0, k) = A(k)

!z=0

d2r

!d3r′G(r

′, r, k)g(r′− r0, k)g(r − r0, k)η(r

′).

˜S(Q, k) = i2πA(k)

!d2q

!dz′

1

kz(q)

eikz(q)(z′−z0) eikz(q−Q)(z

′−z0)

× e−α2

Q2

4k2 e

−α2|q−Q/2|2

k2 η(Q, z′)

˜S(Q, k) =

k2

α2i2π2A(k)

e−2ikz(Q/2)z0

kz(Q/2)

e−α2Q2

4k2 ˜η [Q,−2kz(Q/2)]

Asymptotics

˜S(Q, k) = K(Q, k)˜η [Q,−2kz(Q/2)]

. T S Ralston, D L Marks, S A Boppart, and P S Carney, “Inverse scattering for high-resolution interferometric microscopy,” Opt. Lett, 31, 3585-3587 (2006). T S Ralston, D L Marks, P S Carney, and S A Boppart, “Inverse scattering for optical coherence tomography,” Journ. Opt. Soc. Am. A, 23, 1027-1037, (2006).

Interferometric synthetic aperture microscopy

˜⌘(q, �) = K�1(q,�12

pq2 + �2)S(q,�1

2

pq2 + �2)

Radon, MRI,CT Interferometric SyntheticAperture Microscopy

. B J Davis, S C Schlachter, D L Marks, T S Ralston, S A Boppart and P S Carney, “Non-paraxial vector-field modeling of optical coherence tomography and interferometric synthetic aperture microscopy,” Journ. Opt. Soc. Am A, 24,2527-2542, (2007).

3D Rendered3D Rendered

3D Rendered3D Rendered

ISAM vs Histology

T S Ralston, D L Marks, P S Carney and S A Boppart, “Interferometric synthetic aperture microscopy,” Nature Physics, 3, 129-134, (2007).

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. A Ahmad, N D Shemonski, S G Adie, H Kim, W W Hwu, P S Carney, and S A Boppart, “Real-time in vivo computed optical interferometric tomography,” Nat. Phot. 7 444-448 (2013).

Impact

Fixed-focus, high resolution hand-held probe for surgical application.

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Histology: Negative Lumpectomy MarginPatient: Invasive Ductal Carcinoma

1 mm

Diagnostic Photonics: Pilot Clinical Trial - First In Vivo Images

adipose

parenchyma

Clinical results: breast cancer

Tumor (invasive ductal carcinoma)

Normal (adipose)

Positive margin

Bad news

Good news

. A M Zysk, et al, Ann. Surg. Oncol., DOI: 10.1245/s10434-015-4665-2 (2015).

Outlook, open problems, and opportunities

• This is the age of computed imaging because this is the age of computers.

• Data and image processing must be coupled to an understanding of the physics.

• Spectra and structure, sample and instrument, are tied together in the data in nontrivial but understandable ways.

• In the lens vs algorithms battle, algorithms will win.

•There are a number of important open problems in optical computed imaging and inverse scattering.

Full Field OCT and Multiple Scattering

Spatially

incoherent

source

Collimation

Lens

Source

Field

Plane

Focal plane array

Partially coherent

illumination

Iris Relay

Lens

Sample

plane

PupilObjective

lens

Sample Relay

Telescope

Magnification M

Reference

delay

mirror

Reference Relay

Telescope (without a

pupil, magnification M)

Fig. 1. Diagram of full-field optical coherence tomography instrument with a source of

adjustable partial coherence.

31

. D L Marks, T S Ralston, S A Boppart, and P S Carney, “Inverse scattering for frequency-scanned full-field optical coherence tomography,” J. Opt. Soc. Am A, 24, 1034-1041 (2007).

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Polarization

. F A South, Y-Z, Liu, P S Carney, and S A Boppart, “Polarization-sensitive interferometric synthetic aperture microscopy,” Appl. Phys. Lett., doi: 10.1063/1.4936236, 107, 211106 (2015).

in the transverse polarization. This supports the use of theJones vector representation at low-NA as in the standard PS-OCT model.

In general, ISAM imaging systems are identical to thoseused in OCT, with the exception that a higher NA may beused without sacrificing depth-of-field. Experimental datafor this paper were acquired with a custom built spectral do-main PS-OCT system using a traditional free-space PS-OCTdesign.8,9 The optical source was a super luminescent diodecentered at 1300 nm with 100 nm 3 dB bandwidth(Thorlabs). Polarization-maintaining (PM) fiber was used todeliver light to and from the free-space PS-OCT interferome-ter through an in-line linear polarizer (ThorlabsILP13010PM-APC), optical circulator (AFW TechnologiesPMP-13-R-C3N-45-22), and 45 m of PM fiber, which wasincluded to displace the ghost images out of the imagingrange.14 The collected interference signal returned throughthe circulator to a PM fiber polarization beam splitter (ACPhotonics PBS-13-P-2-2-1-1) for polarization diverse detec-tion. The two polarization components of the Jones vectorwere measured with spectrometers using 2048 pixel linescan cameras (Sensors Unlimited GL2048L). Both the axialand transverse resolution of the system were approximately7.65 lm full-width-half-maximum (FWHM), or 13 lm ( 1

e2),giving an NA of 0.065, slightly greater than the 0.05 NA ofthe initial ISAM demonstrations.5 Three-dimensional data-sets were acquired by scanning 512! 256 transverse pointswith an isotropic transverse sampling of 3.4 lm.

The ISAM reconstruction was applied to each componentof the measured Jones vector through a Fourier domain coor-dinate resampling of the data according to the relationship

k ¼ 1

2Q2

x þ Q2y þ Q2

z

! "1=2; (1)

for wavenumbers k ¼ 2pn=k and spatial frequencies Q, wheren is refractive index and k is wavelength. Following this step,the sample reflectivity R and phase retardation d were calcu-lated for each position ðx; y; zÞ in the 3-D volume as

Rðx; y; zÞ / jHðx; y; zÞj2 þ jVðx; y; zÞj2; (2)

d x; y; zð Þ ¼ arctanjH x; y; zð ÞjjV x; y; zð Þj

!; (3)

where H and V are the horizontal and vertical components ofthe measured Jones vector, respectively. A two-dimensional

median filter of approximately two resolution elements wasapplied to the phase retardation data to remove random fluc-tuations for improved visualization. The standard OCT struc-tural data were given by R, while a change in the phaseretardation indicated a change in the polarization state.

To demonstrate the PS-ISAM reconstruction, a scatter-ing phantom consisting of TiO2 particles (<5 lm) suspendedin a silicone gel was imaged. The results are shown in Fig. 2.An en face plane taken from far above focus shows strongblurring in the OCT image (Fig. 2(a)) caused by the limiteddepth-of-field. The corresponding ISAM reconstructionshown in Fig. 2(b) shows clear improvement in the trans-verse resolution. PS-OCT and PS-ISAM phase retardationimages are shown in Figs. 2(c) and 2(d). The valid polariza-tion information is localized to areas with sufficient signalcorresponding to the scattering particles in the intensityimages, which have phase retardation values near theextremes of the scale corresponding to right- and left-handedpolarization states.15 Comparison of the PS-OCT and PS-ISAM images reveals improved localization of the phase re-tardation information in the PS-ISAM reconstruction due tothe improved transverse resolution. The traces for a singleparticle shown in Figs. 2(e) and 2(f) highlight the improve-ment of the ISAM and PS-ISAM reconstruction over thestandard techniques.

Figure 3 demonstrates PS-ISAM imaging in a birefrin-gent material. The phantom consisted of small molded plasticpieces suspended in an agarose gel. In dense scattering sam-ples such as this, the improvement from OCT to ISAM maynot be dramatic due to the relatively uniform scattering struc-ture (Figs. 3(a) and 3(d)). However, the birefringence of thesample causes change in the polarization state seen as varia-tions in the phase retardation images (Figs. 3(c) and 3(d)).

FIG. 1. Simulation of the longitudinal (z) and transverse (xy) beam powervs increasing numerical aperture for light circularly polarized prior tofocusing.

FIG. 2. Imaging of a silicone phantom consisting of sub-resolution micro-particles. En face planes are taken from 1502 lm optical distance abovefocus (10.5 Rayleigh ranges). (a) OCT intensity image. (b) ISAM intensityimage. (c) PS-OCT phase retardation image. (d) PS-ISAM phase retardationimage. (e) Trace of the OCT and ISAM intensities for a single particle(white arrow) showing the FWHM resolution. (f) Trace of the PS-OCT andPS-ISAM phase retardation for the same particle as in (e). Shaded cyan andmagenta areas indicate regions of valid signal from OCT and ISAM intensitymeasurements respectively, determined from the FWHM measurements in(e). Scale bar indicates 200 lm.

211106-2 South et al. Appl. Phys. Lett. 107, 211106 (2015)

Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. Download to IP: 130.126.255.133 On: Sun, 10 Jul 201609:24:12

needed to setup a invertible linear system for each grid point�kk, �

�,

2

666666666666664

S(1)

S(2)

S(3)

S(4)

...

S(n)

3

777777777777775

| {z }S

=

2

66666664

H 0(1)xx 2H 0(1)

xy 2H 0(1)xz H 0(1)

yy 2H 0(1)yz H 0(1)

zz

......

......

......

......

......

......

H 0(n)xx 2H 0(n)

xy 2H 0(n)xz H 0(n)

yy 2H 0(n)yz H 0(n)

zz

3

77777775

| {z }H

2

666666666666664

˜⌘0xx

˜⌘0xy

˜⌘0xz

˜⌘0yy

˜⌘0yz

˜⌘0zz

3

777777777777775

| {z }⌘

, (4.2)

where the n in the superscript represents the nth measurement, and the�kk, �

�dependence of S(n), H 0(n)

↵� ,

and ˜⌘0↵� is omitted here for brevity. For convenience, as is indicated in the equation, the vectors will be

referred to as vector S and ⌘, and the matrix will be referred to as matrix H in the following discussion.

To achieve this, we first need to make sure the weights H↵� are large enough for all combinations of ↵�.

If a certain polarization component of the light is too weak, the problem will easily become ill-conditioned,

and the information in ˜⌘0↵� becomes di�cult to recover. We decide to use high NA Gaussian beam which

brings an adequate amount of axial polarized light. Fig. 4.1 (a) - (c) shows the transverse field amplitude at

focal spot of a 0.6 NA focused x polarized Gaussian beam. Fig. 4.1 (d) shows the optical power distribution

between three polarization components, as a function of system NA. It can be seen that in low NA Gaussian

beam (NA < 0.2), the axial polarization component is negligible. When the system NA goes beyond 0.2,

axial polarization increase significantly with higher NA. In addition to the high NA focusing, we use circularly

polarized beam, which ensures the optical power of x and y polarization components are on a similar level.

This can be generated by passing a horizontally linearly polarized collimated beam through a quarter-wave

plate (QWP) at 45� angle with respect to the horizontal direction. Fig. 4.2 (a) - (c) shows the transverse

field amplitude at focal spot of a 0.6 NA circularly polarized Gaussian beam. Fig. 4.2 (d) shows the optical

power distribution between three polarization components, as a function of system NA. It can be observed

that an NA larger than 0.6 will make the power of three polarization components on the same order of

magnitude, resulting in a reasonably well-conditioned inverse problem.

In order to vary H↵� , we propose to include two rotatable polarizers before and after the objective lens,

as is shown in Fig. 4.3. In the experiment, datasets are recorded multiple times on the same sample area

with di↵erent polarizer angle combinations in each scan.

To solve the linear system in Eqn. 4.2, we need to calculate H(n)↵� for each scan. In our proposed system,

33

Opportunity in high-NA PS-ISAM

Doppler

the validity and performance of both methods are demonstrated.While the “phase tracker method” requires more computationper pixel, we have shown that the process can be significantlyshortened by implementation in a parallel processing architec-ture, making semi-real-time processing possible. Because thisLetter focuses on the denoising aspect, a simple one-dimensionalunwrapping algorithm is used. However, if combined with moreadvanced two-dimensional unwrapping algorithms, such asquality-guided unwrapping, it should be capable of unwrappingeven more challenging DOCT velocity maps. It should be notedthat both algorithms, while effective in denoising, also inevitablycause the loss of details in the images. The filter parameters orthe window size should be carefully chosen to balance the trade-off to suit the intended applications.

Funding. National Cancer Institute, National Institutesof Health, Department of Health and Human Services(HHSN261201400044C); National Institutes of Health(NIH) (1 R01 CA166309); National Science Foundation (NSF)(CBET 14-45111); Howard Hughes Medical InstituteInternational Student Research Fellowship.

Acknowledgment. P. Scott Carney and Stephen A.Boppart are co-founders of Diagnostic Photonics, Inc., whichmakes the system used in this work.

REFERENCES

1. J. A. Izatt, M. D. Kulkarni, S. Yazdanfar, J. K. Barton, and A. J. Welch,Opt. Lett. 22, 1439 (1997).

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4. R. A. Leitgeb, R. M.Werkmeister, C. Blatter, and L. Schmetterer, Prog.Retinal Eye Res. 41, 26 (2014).

5. D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W.Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G.Fujimoto, Science 254, 1178 (1991).

6. A. F. Fercher, J. Biomed. Opt. 1, 157 (1996).7. C. Sun, F. Nolte, K. H. Cheng, B. Vuong, K. K. Lee, B. A. Standish, B.

Courtney, T. R. Marotta, A. Mariampillai, and V. X. Yang, Biomed. Opt.Express 3, 2600 (2012).

8. V. Westphal, S. Yazdanfar, A. M. Rollins, and J. A. Izatt, Opt. Lett. 27,34 (2002).

9. V. Yang, M. Gordon, B. Qi, J. Pekar, S. Lo, E. Seng-Yue, A. Mok, B.Wilson, and I. Vitkin, Opt. Express 11, 794 (2003).

10. A.Mariampillai,B.A.Standish,N.R.Munce,C.Randall,G.Liu,J.Y.Jiang,A. E. Cable, I. A. Vitkin, and V. X. Yang, Opt. Express 15, 1627 (2007).

11. Y. Wang, B. A. Bower, J. A. Izatt, O. Tan, and D. Huang, J. Biomed.Opt. 12, 041215 (2007).

12. Y. Wang, B. A. Bower, J. A. Izatt, O. Tan, and D. Huang, J. Biomed.Opt. 13, 064003 (2008).

13. A. Davis, J. Izatt, and F. Rothenberg, Anat. Rec. 292, 311 (2009).14. D. C. Ghiglia, G. A. Mastin, and L. A. Romero, J. Opt. Soc. Am. A 4,

267 (1987).15. D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping:

Theory, Algorithms, and Software (Wiley, 1998), Vol. 4.16. H. C. Hendargo, M. Zhao, N. Shepherd, and J. A. Izatt, Opt. Express

17, 5039 (2009).17. H. C. Hendargo, R. P. McNabb, A.-H. Dhalla, N. Shepherd, and J. A.

Izatt, Biomed. Opt. Express 2, 2175 (2011).18. J. Tokayer, Y. Jia, A.-H. Dhalla, and D. Huang, Biomed. Opt. Express

4, 1909 (2013).19. M. Servin, F. J. Cuevas, D. Malacara, J. L. Marroquin, and R.

Rodriguez-Vera, Appl. Opt. 38, 1934 (1999).20. S. Waldner and N. Goudemand, in Interferometry in Speckle Light:

Theory and Applications, P. Jacquot and J.-M. Fournier, eds.(Springer, 2000), Chap. 4, pp. 319–326.

21. K. J. Chalut, W. J. Brown, and A. Wax, Opt. Express 15, 3047 (2007).22. Y. Cengel and J. Cimbala, Fluid Mechanics Fundamentals and

Applications (McGraw-Hill, 2013).

Fig. 4. Array of images shows a comparison of the simulation and experimental results using the “phase tracker method.” The average velocities ofthe datasets are (a)–(f ) 4.81 cm/s, (g)–(l) 8.62 cm/s, and (m)–(r) 10.42 cm/s. The peak velocities range from 8.32 to 16.50 cm/s (4.8V max–9.4V max).The velocity profiles were taken at the depth labeled by green arrows. It can be seen that this method generates smooth and nearly noise-free velocitymaps that agree well with the physical model.

Letter Vol. 41, No. 17 / September 1 2016 / Optics Letters 4027

. Y Xu, D Darga, J Smid, A M Zysk, D Teh, S A Boppart, and P S Carney, “Filtering and Unwrapping Noisy Wrapped Doppler Optical Coherence Tomography Images for Extended Microscopic Fluid Velocity Measurement Range,” Opt. Lett., 41, 4024-4027 (2016).

N-specieswhere κ (with jκj ! κ) and κ∥ are wave-vector componentsdefined on a sampling grid, rather than continuous parameters,and D"k∥; k0# is a sampling function. In the simplest case, wecan choose D ! δ"k0 − κ#δ"k∥ − κ∥#, representing the processof sampling instantaneously on a discrete, evenly spaced lattice.The summation over κ in Eq. (7) is over the range $κmin; κmax%,which is, in turn, dictated by the position resolution of thereference mirror.

This model works under the first Born approximation anduses scalar Gaussian fields. It, thus, fails in cases of high samplecontrast or high numerical aperture. We also assume that theautocorrelation terms in the interference measurement can beneglected. For a discussion of the effect of these terms and theirremoval, see [30,31].

A. Single Species, 1D Heterogeneous Object

First, we consider the case of a sample heterogeneous objectonly in the principal direction of light propagation, such thatjη1d i ! δ"k∥#jηi, where δ is the Dirac delta function. Then,our estimate of the object is given by

jηi ! "D1dK1dW1d #&jSi; (8)

where the + superscript denotes the Moore–Penrose pseudoin-verse, and the operators above reduce in the 1D case to

W1d !Z

dk0 dzX

zF

jk0; z; zF iW 1d "k0; z; zF #hk0; zj; (9)

K1d !Z

dk0 dz; ∶X

zF

jk0; zF ie−2ik0"z−zF #hk0; z; zF j; (10)

and

D1d !Z

dk0X

κ;zF

jκ; zF iD"κ#hk0; zF j; (11)

where

W 1d "k0; z; zF # !4π3U 0"k0#jz − zF j& zR

: (12)

Here, we have used the fact that kz jk∥!0 ! k0.In OCT/interferometric synthetic aperture microscopy

(ISAM), the susceptibility is implicitly taken as separable, suchthat jηi ! jρijf i and hr; k0iη ! hriρhk0if ! ρ"r#f "k0#.That is, the sample consists of a single species with spectral re-sponse f as a function of wavenumber k0, and spatial densityvariation ρ as a function of spatial coordinate r. The spectralresponse is typically estimated from a known sharp feature ofthe sample (e.g., a planar interface). Figures 2(a) and 2(b) aresimulated demonstrations of OCT under these conditions. In(a), the z-dependent density of a single-species object is plottedas a function of depth. A measurement is produced usingthe matrix D1dW1dK1d assuming a numerical aperture (NA)of 0.5 and three foci, equally spaced throughout the depthof the object. In (b), the density of the sample is retrieved.Here, and, in subsequent results, Tikhonov regularization isused, such that the pseudoinverse of an operator is given byG& ! "G'G& ϵI#G', where ! I is the identity operatorand ϵ is a constant. A discrete fast Fourier transform is usedto compute the Fourier transform.

B. Multispecies, 1D Heterogeneous Object

We now consider a special class of objects known to be com-posed of a limited number of species with different absorptionspectra, so that the susceptibility can be represented ashr; k0iη !

PNj!1 ρj"r#f j"k0#. In Fig. 2(c), the simulated sam-

ple is composed of a mixture of two species with complexLorentzian spectra, given by

f n"k0# ∝k2n − k20

"k2n − k20#2 & k20γ2&

iγk0"k2n − k20#2 & k20γ2

; (13)

where k is the center wavenumber and γ is a damping constant.Moreover, we assume that the f j’s are linearly independent.The N ×M matrix formed by the spectral functions (M beingthe number of spectral points) needs to be full rank, so it isimplicitly assumed that M > N , i.e., the number of spectralpoints collected is at least as many as the number of species.

The real parts of these spectra for two species are plotted inFig. 2(d). Figure 2(e) shows the result of applying an OCT-typeinversion scheme in which only the spectrum of species 1 isknown. The total density of the object is recovered but not theindividual densities of each species. Since we have no way to

Fig. 2. Comparison of OCT with and without compositional priorinformation. (a) Density profile of a single-species object. (b) OCT-type regularized inversion for single-species object, in which it is as-sumed that we have knowledge of its spectral response. (c) Densityprofile of a two-species object. (d) Real parts of the spectral responsesof species 1 and 2. (e) OCT-type regularized inversion of two-speciesobject, in which we are missing information about the spectral re-sponse of species 2. The inversion reconstructs the total density profilebut assigns equal weight to each species at each depth. (f) Retrievedregularized density with full spectral information, as provided by theN -species constraint. The individual densities of species 1 and 2 areretrieved as well as the total density.

1128 Vol. 32, No. 6 / June 2015 / Journal of the Optical Society of America A Research Article

Models for ÷

Simplifications

I OCT/ISAM: ÷(x, y, z, k0) = p(x, y, z)f(k0)I FTIR (Bulk): ÷(x, y, z, k0) = f(k0)I FTIR Microscopy: ÷(x, y, z, k0) = ÷(x, y, k0)I Proposed

÷(x, y, z, k0) =Nsÿ

j=1pj(x, y, z)fj(k0)

8

. B Deutsch, R Reddy, D Mayerich, R Bhargava, and P S Carney, “Compositional prior information in computed infrared spectroscopic imaging,” Journ. Opt. Soc. Am. A 32(6), 1126-1131 (2015).

Scanning microscopy Holography

Synthetic holography Exotic reference fields

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x

y

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Fig. 2. Simulations of SOH with linear- and sinusoidal-phase reference waves. The real

and imaginary part of the original complex-valued image are shown in panels (a) and (b)

respectively. The mirror position in radians as a function of position in the scanned image is

shown in (c) and the resulting hologram is shown in (d) with a 41× zoom inset. The Fourier

transform of the hologram is shown in (e) demonstrating the separation of the direct and

conjugate images with the filter used indicated by the red box. The recovered images are

shown in panels (f) and (g). The mirror position for sinusoidal-phase SOH is shown in panel

(h) with resulting hologram and 41× zoom showing the fringe pattern in (i). The Fourier

transform of the hologram is shown in (j) with the two filters used to recover the real and

imaginary parts of the field indicated by red boxes. The recovered images are shown in (k)

and (l).

Aberration correction

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ratic phase term of Z11. Fig. S1 shows a plot of image metrics (asa function of the frame number in Movie S1), based on sensorlessAO techniques (40–42) that provided feedback on the optimi-zation of the aberration correction. Although the dominantaberration was astigmatism, the correction of quartic sphericalaberration produced an appreciable improvement of the image

metrics over the correction of only astigmatism and defocus, evenat this relatively low N.A. of 0.1.

Aberrations are usually viewed as detrimental in optical micro-scopy. However, CAO provides the ability to exploit potentialadvantages of imaging with aberrated beams. An example ofsuch a benefit is shown in Fig. 3B, which compares ISAM recon-structions with and without the astigmatism-producing cylindricallens added to the sample arm optics. ISAM reconstructions afterCAO demonstrate the advantage of a more uniform depth-dependent signal, thereby reducing the dynamic range requiredto represent the signal. At a given depth deep within the sample(e.g., z ¼ 1.4mm), this more uniform signal is also seen to in-crease SNR. The change in depth-dependent SNR can be attrib-uted to the presence of two axially separated line foci, effectivelyforming two (partial) confocal gates within the sample. Thisresult is of even greater benefit for high N.A. OCT, where thesteep roll-off in signal strength with distance from focus demandsdata acquisition with a larger dynamic range.

Fig. 4 demonstrates CAO in a biological sample, ex vivo ratlung tissue. Three-dimensional data were obtained using thesame astigmatic setup used to acquire the phantom data in Fig. 1and processed using the same computational aberration correc-tion. The standard OCT tomogram shows astigmatic blurringof tissue structures, particularly evident in highly scattering re-gions. The application of ISAM without aberration correctioncorrects some defocus, but does not resolve fine sample structure,indicating that when aberrations are present, they must beaccounted for in the reconstruction. (Note that this uncorrectedISAM processing is different to the aberration-corrected OCTprocessing in Figs. 1 and 2.) However, CAO followed by ISAMis seen to reconstruct the fine structure of the sample. In parti-cular, correction of both astigmatism and defocus is clearlydemonstrated at highly scattering regions of the sample.

Computational AO is based on the ability to correct aberra-tions of a virtual (or computed) pupil rather than the physicalpupil of the objective lens (see SI Text for a detailed treatmentof the following description). From Fourier optics, the (trans-verse) optical field distribution at the beam focus, gðx; y; z ¼0; kÞ, at optical wavenumber k, is related to the objective lenspupil function by the Fourier transform (43). Therefore, froma measurement of the optical field at the nominal focus, suchas gðx; y; 0; kÞ, a corresponding (virtual) pupil function can becomputed via the transverse Fourier transform. The complexsystem PSF can be measured using a sparse phantom consistingof subresolution scatterers, such as the one in Figs. 1 and 2. How-ever, due to the double-pass imaging geometry, this system PSF,hðx; y; z; kÞ ∝ g2ðx; y; z; kÞ, is a product of the (identical) illumina-tion and collection beams (44, 45).

According to the convolution theorem, the transverse Fouriertransform of the complex system PSF is the convolution of thesevirtual pupil functions. As with hardware-based AO, we expresspupil aberrations using Zernike polynomials, but compute anaberration-correction filter as the (transverse) convolution ofthese aberrated (virtual) pupil functions. This aberration-correc-tion filter, applied to the 3D Fourier transform of the OCTtomogram, “rephases” components in the transverse frequencydomain of the focal-plane PSF to restore constructive interfer-ence across the band. This constructive interference results inrecovery of diffraction-limited resolution at the nominal focus,accompanied by increased SNR (Figs. 1 and 2).

The 3D aberration correction filter implemented here correctsfor phase deviations from the ideal transverse-frequency re-sponse of the system PSF. Although these results show that itworks well in practice, we note that, in general, the convolutionin Eq. S5 can manifest (depth-dependent) amplitude structure inthe transverse frequency domain. This depth-dependent amplitudestructure is relevant when combining CAO with ISAM, in order toreconstruct object structure away from the nominal aberration-free

Fig. 2. Complex signals from the silicone phantom data, showing the impactof computational correction of astigmatism on both the amplitude andphase. Images are arranged in columns according to the type of processingapplied. The en face (x-y) planes shown are from the 3D silicone phantomdataset near (A) the upper line focus (z ¼ 300 μm), (B) the plane of leastconfusion (z ¼ 0 μm), and (C) the lower line foci (z ¼ −300 μm), where theunits of the z axis denote optical path length. Dimensions of all images are256 × 256 μm.

Fig. 3. Depth-dependent resolution and signal-to-noise ratio in the siliconetissue phantom. (A) Resolution (along the x axis) vs. depth for the aberration-corrected OCT and ISAM, and (B) signal-to-noise ratio after ISAM reconstruc-tion, comparing the cylindrical lens setup producing two axially separatedline foci to a standard single-focus setup. The optical focus appears atz ≈ 1 mm because depth is plotted relative to zero optical path delay in theinterferometer.

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A feature, not a bug?

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ratic phase term of Z11. Fig. S1 shows a plot of image metrics (asa function of the frame number in Movie S1), based on sensorlessAO techniques (40–42) that provided feedback on the optimi-zation of the aberration correction. Although the dominantaberration was astigmatism, the correction of quartic sphericalaberration produced an appreciable improvement of the image

metrics over the correction of only astigmatism and defocus, evenat this relatively low N.A. of 0.1.

Aberrations are usually viewed as detrimental in optical micro-scopy. However, CAO provides the ability to exploit potentialadvantages of imaging with aberrated beams. An example ofsuch a benefit is shown in Fig. 3B, which compares ISAM recon-structions with and without the astigmatism-producing cylindricallens added to the sample arm optics. ISAM reconstructions afterCAO demonstrate the advantage of a more uniform depth-dependent signal, thereby reducing the dynamic range requiredto represent the signal. At a given depth deep within the sample(e.g., z ¼ 1.4mm), this more uniform signal is also seen to in-crease SNR. The change in depth-dependent SNR can be attrib-uted to the presence of two axially separated line foci, effectivelyforming two (partial) confocal gates within the sample. Thisresult is of even greater benefit for high N.A. OCT, where thesteep roll-off in signal strength with distance from focus demandsdata acquisition with a larger dynamic range.

Fig. 4 demonstrates CAO in a biological sample, ex vivo ratlung tissue. Three-dimensional data were obtained using thesame astigmatic setup used to acquire the phantom data in Fig. 1and processed using the same computational aberration correc-tion. The standard OCT tomogram shows astigmatic blurringof tissue structures, particularly evident in highly scattering re-gions. The application of ISAM without aberration correctioncorrects some defocus, but does not resolve fine sample structure,indicating that when aberrations are present, they must beaccounted for in the reconstruction. (Note that this uncorrectedISAM processing is different to the aberration-corrected OCTprocessing in Figs. 1 and 2.) However, CAO followed by ISAMis seen to reconstruct the fine structure of the sample. In parti-cular, correction of both astigmatism and defocus is clearlydemonstrated at highly scattering regions of the sample.

Computational AO is based on the ability to correct aberra-tions of a virtual (or computed) pupil rather than the physicalpupil of the objective lens (see SI Text for a detailed treatmentof the following description). From Fourier optics, the (trans-verse) optical field distribution at the beam focus, gðx; y; z ¼0; kÞ, at optical wavenumber k, is related to the objective lenspupil function by the Fourier transform (43). Therefore, froma measurement of the optical field at the nominal focus, suchas gðx; y; 0; kÞ, a corresponding (virtual) pupil function can becomputed via the transverse Fourier transform. The complexsystem PSF can be measured using a sparse phantom consistingof subresolution scatterers, such as the one in Figs. 1 and 2. How-ever, due to the double-pass imaging geometry, this system PSF,hðx; y; z; kÞ ∝ g2ðx; y; z; kÞ, is a product of the (identical) illumina-tion and collection beams (44, 45).

According to the convolution theorem, the transverse Fouriertransform of the complex system PSF is the convolution of thesevirtual pupil functions. As with hardware-based AO, we expresspupil aberrations using Zernike polynomials, but compute anaberration-correction filter as the (transverse) convolution ofthese aberrated (virtual) pupil functions. This aberration-correc-tion filter, applied to the 3D Fourier transform of the OCTtomogram, “rephases” components in the transverse frequencydomain of the focal-plane PSF to restore constructive interfer-ence across the band. This constructive interference results inrecovery of diffraction-limited resolution at the nominal focus,accompanied by increased SNR (Figs. 1 and 2).

The 3D aberration correction filter implemented here correctsfor phase deviations from the ideal transverse-frequency re-sponse of the system PSF. Although these results show that itworks well in practice, we note that, in general, the convolutionin Eq. S5 can manifest (depth-dependent) amplitude structure inthe transverse frequency domain. This depth-dependent amplitudestructure is relevant when combining CAO with ISAM, in order toreconstruct object structure away from the nominal aberration-free

Fig. 2. Complex signals from the silicone phantom data, showing the impactof computational correction of astigmatism on both the amplitude andphase. Images are arranged in columns according to the type of processingapplied. The en face (x-y) planes shown are from the 3D silicone phantomdataset near (A) the upper line focus (z ¼ 300 μm), (B) the plane of leastconfusion (z ¼ 0 μm), and (C) the lower line foci (z ¼ −300 μm), where theunits of the z axis denote optical path length. Dimensions of all images are256 × 256 μm.

Fig. 3. Depth-dependent resolution and signal-to-noise ratio in the siliconetissue phantom. (A) Resolution (along the x axis) vs. depth for the aberration-corrected OCT and ISAM, and (B) signal-to-noise ratio after ISAM reconstruc-tion, comparing the cylindrical lens setup producing two axially separatedline foci to a standard single-focus setup. The optical focus appears atz ≈ 1 mm because depth is plotted relative to zero optical path delay in theinterferometer.

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single singledual dual