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Optimal spectral reshaping for resolution

improvement in optical coherence tomography

Jianmin Gong, Bo Liu, Young L. Kim, Yang Liu, Xu Li, Vadim Backman

Biomedical Engineering Department, Northwestern University, IL 60208, USA

j-gong@northwestern.edu

Abstract: We analyze the resolution limit that can be achieved by means of

spectral reshaping in optical coherence tomography images and demonstrate

that the resolution can be improved by means of modelessly reshaping the

source spectrum in postprocessing. We show that the optimal spectrum has

a priory surprising “crater-like” shape, providing 0.74 micron axial

resolution in free-space. This represents ~50% improvement compared to

resolution using the original spectrum of a white light lamp.

©2006 Optical Society of America

OCIS codes: (100.2980) Image enhancement; (170.4500) Optical coherence tomography;

(350.5730) Resolution

References and links

1.

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, “Optical coherence tomography,” Science, 254, 1178 (1991).

Brett E. Bouma, Guillermo J. Tearney, Handbook of optical coherence tomograph, (Marcel Dekker, New

York, 2002)

B. Bouma, G. Tearney, S. Boppart, M. Hee, M. Brezinski, and J. Fujimoto, “High-resolution optical

coherence tomographic imaging using a mode-locked Ti:Al2O3 laser source,” Opt. Lett. 20 1486-1488

(1995)

B. Povazay, K. Bizheva, A. Unterhuber, B. Hermann, H. Sattmann, A. F. Fercher, W. Drexler, A.

Apolonski, W. J. Wadsworth, J. C. Knight, P. St. J. Russell, M. Vetterlein and E. Scherzer, “Submicrometer

axial resolution optical coherence tomography,” Opt. Lett. 27 1800-1802 (2002)

Wolfgang Drexler, “Ultrahigh-resolution optical coherence tomography,” J. Biomed. Opt. 9, 47–74 (2004)

A. Dubois, G. Moneron, K. Grieve and A.C. Boccara, “Three-dimensional cellular-level imaging using full-

field optical coherence tomography,” Phys. Med. Biol. 49 1227–1234 (2004)

A. Wax, C.H. Yang, J.A. Izatt, “Fourier-domain low-coherence interferometry for light-scattering

spectroscopy,” Opt. Lett. 28 1230-1232 (2003)

Yan Zhang and Manabu Sato, “Resolution improvement in optical coherence tomography by optimal

synthesis of light-emitting diodes,” Opt. Lett. 26, 205-207 (2001)

Renu Tripathi, Nader Nassif, J. Stuart Nelson, Boris Hyle Park and Johannes F. de Boer, “Spectral shaping

for non-Gaussian source spectra in optical coherence tomography,” Opt. Lett. 27, 406-408 (2002)

10. E. Smith, S. C. Moore, N. Wada, W. Chujo, and D. D. Sampson, “Spectral domain interferometry for

OCDR using non-Gaussian broadband sources,” IEEE Photon. Technol. Lett. 13, 64-66 (2001)

11. A. Ceyhun Akcay, Jannick P. Rolland and Jason M. Eichenholz, “Spectral shaping to improve the point

spread function in optical coherence tomography,” Opt. Lett. 28, 1921-1923 (2003)

12. Daniel Marks, P. Scott Carney, Stephen A. Boppart, “Adaptive spectral apodization for sidelobe reduction

in optical coherence tomography images,” J. Biomed. Opt. 9, 1281-1287 (2004)

13. T.F. Coleman, and Y. Li, “A Reflective Newton Method for Minimizing a Quadratic Function Subject to

Bounds on Some of the Variables,” SIAM J. Optimiz. 6, 1040-1058 (1996)

2.

3.

4.

5.

6.

7.

8.

9.

1. Introduction

Optical coherence tomography is a promising non-invasive in vivo imaging technique, which

has undergone rapid development since its invention in 1991 [1,2]. In an OCT system, the

axial resolution is determined by the temporal coherence length lc of a light source. If the

spectrum of the light source has a Gaussian shape, lc is proportional to λ0

2/Δλ with λ0 the

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central wavelength and Δλ the bandwidth of the emission spectrum, respectively [2].

Depending on the choice of a light source, the axial resolution of a typical OCT system varies

from ~1 to 30 µm. Several techniques have been proposed to improve the resolution of OCT.

One possible strategy is to develop new light sources with shorter coherence lengths. For

example, a Kerr-lens mode-locked Ti:sapphire laser [3], Ti:sapphire pumped super-continuum

[4,5], and thermal light sources [6,7] were used to obtain 3.7 [3], 0.5 [4], ~1 [5], 0.7 [6], and

~1 µm [7] axial resolution in biological tissue, respectively. Alternatively, resolution can be

improved by combining multiple light sources, e.g. after optimizing the power ratios of three

LEDs, resolution in OCT images was improved from 12 µm to 7 µm [8]. Moreover, by

digitally reshaping the source spectra to known modes, such as Gaussian, white, and

Hamming windowed shapes, OCT resolution was enhanced from over 10 to a few microns

[9,10,11]. A modeless spectral reshaping, in which a spectral profile can be changed to any

arbitrary shape, was used to effectively reduce sidelobes of in OCT images [12].

In this Letter we analyze the resolution limit that can be achieved by means of the

modeless spectral reshaping approach, determine the shape of the optimal spectrum, and use

this to improve the resolution of OCT.

2. Theory and simulation

The modeless spectral reshaping we propose here is based on a spectral domain OCT

modality in which the interference signal I(k,r) is given by [2]

)},(])()([ )()(){()(

)](2exp[)()( )](2 exp[)()()(),(

2/ 1

2

0

0

2 / 1

0

2/ 1

rkIBkSkRBkSkRkQkP

dz nzz ikzakSrz ikkRkQkPrkI

d

++=

where k=2π/λ is the wave number; n is the refractive index of the sample; z0 is the path

length from the beam splitter to the top surface of the sample, z is the depth of the sample, r is

the path difference between the reference mirror and the top surface of the sample; a(z) is the

depth-resolved backscattered amplitude of the sample; P(k), Q(k), R(k) and S(k) are the source

spectrum, spectral response of the detector, reflection spectrum of the reference mirror, and

the scattering spectrum in the sample (dispersion within the sample was neglected),

respectively;

∫ ∫

00

the light scattered from the sample;

2),(

rkId

=

resolved OCT signal from the sample a(z), which we aim to obtain. In an experiment, by

blocking the reference or sample arm, respectively, P(k)Q(k)S(k)B and P(k)Q(k)R(k) can be

recorded, and Id(k,r) can then be obtained as

),([),(

PQSBPQRrkIrkId

−−=

The bandwidth of an OCT system is limited by the spectrum of a light source as well as

the spectral range of the detector and other components. Assuming that the minimum and

maximum detectable wave numbers are k1 and k2, respectively, the sum of the interference

signal weighted with a distribution V(k) within the range [k1, k2] can be expressed as follows:

),()()(

1

k

∫

where

−≡

PSF() PSF(

rZ

is the point spread function (PSF) that determines axial resolution and V(k) can be reshaped in

postprocessing. In order to improve the resolution, the width of PSF has to be minimized.

Equation (4) can be rewritten as

cos),()()PSF(

0

∫

where ⊗ denotes convolution operation, rect(k1, k2) is the rectangle function which equals to 1

within the range [k1,

k2]

+++=

∫

∞

(1)

∞∞

a

−≡

' )]'(2exp[) '

z

()(

dzdzzzinkazB

describes the mutual interference of

2 / 1

0

/)(2 cos)(

B dznzrkza

∫

∞

−

decodes the depth-

2/ 1

)]()/[(]

PQSBPQR

×

. (2)

2/ 1

0

/)PSF()(2

2

BdzZza dkrkIkVrI

k

dd

∫

∞

==

(3)

∫

−≡

2

1

)(2 cos)()

k

k

dknzrkkVnz

(4)

)(]2cos)([2

0

21

ZUkZdkkVkZdkkkrectkVZ

⊗==

∫

∞∞

(5)

and zero elsewhere, and

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4/ ]

Z

) sinc[(])cos[()()(

122112

kkZkkkkZU

envelope is a sinc function. If V(k) is a constant (i.e. white light source), the first term in Eq.

(5) would be a delta function, and PSF would equal to U(Z). The peak of this PSF is narrow;

however the sidelobes are large. If V(k) is Gaussian or Hamming windowed, the sidelobes can

be significantly compressed, however the peak of the PSF is broadened. In this paper we

optimized V(k) so that U(Z) functions with different shifts compensate each other to create a

narrower PSF peak than that of white light, at the same time the sidelobes are suppressed to a

lower level than that of white light also.

It is hard to get an analytical solution for the optimal spectrum V(k). We discretize the

optical path difference (OPD) and wave number as Zi (i=1, 2,…M) and kj (j=1, 2 ,…W),

respectively, and rewrite Eq. (5) in matrix notations:

⎡

⎟

⎜

ggf

???

−+−=π

is a high frequency oscillation term whose

⎟⎟

⎠

⎟

⎟

⎟

⎞

⎜⎜

⎝

⎜

⎜

⎜

⎛

⎥

⎦

⎥

⎥

⎥

⎤

⎢

⎣

⎢

⎢

⎢

=

⎟⎟

⎠

⎟

⎟

⎞

⎜⎜

⎝

⎜

⎜

⎛

WMWM

cos

M

g

W

W

M

v

v

v

g

. In order to find the optimal spectrum V,

and

min

V

gg

g

Ggg

f

f

?

?

??

?

?

2

1

21

2 21 21

1 12 11

2

1

or F=GV, (6)

where fi is the PSF for Zi, vj=V(kj), and

we solved the nested optimization problem:

ijij

Zk

2

=

)( min

F

GVFWHM

2

2

F GV

−

, where

FWHM represents the full-width half-maximum. The optimization procedure is shown as Fig.

1. Since the shape of the PSF of a broad-band source is similar to the sinc function, we

selected the sinc function with a variable width T, sinc(Z/T), as the target PSF. The initial

width T was set to 1/k1. In the m-th iteration, the target function Fm was chosen as sinc(Z/Tm),

the half-optimal spectrum Vm was found so that

corresponding PSF, Dm, was calculated by Hilbert transform. In order to optimize F, we

applied a 0.1% perturbation to Tm to obtained another FWHM Dm', and in the following

iteration, Tm was set as

) '(

111

−−−

−+

mmm

DDT

γ

proportional coefficient. Normally a smaller value of γ leads to more stable convergence but

also increases the complexity of computations. In our computations, the convergence criterion

was set as γ(D-D')<10-5Tm, and γ was chosen as 10 to ensure the convergence after 100-1000

iterations. The reflective Newton iterative method [13], which allows setting the range of vj,

was chosen to solve the matrix-form least-squares problem, and set the lower limit of vj to

zero to satisfy the physics of real light sources. The system is spectrally sensitive, and the

signal-to-noise ratio (SNR) values differ significantly at different wave numbers. In order to

limit the use of noisy spectral data, we applied an additional condition

2

2

mm

FGV −

is minimal, and the FWHM of its

to give a PSF with narrower peak. γ is a

α

<

1

∑=

F'm=sinc(Z/ Tm')

W

w

wwSNRv

for

the optimization, where

w

SNR is the signal-to-noise ratio at wave number

w

v which was

obtained by dividing the signal level at at wave number

white noise. α is a threshold controlling the trade-off of the SNR and the FWHM. For a less α

w

v by the amplitude of background

Fm=sinc(Z/Tm)

Tm'=1. 001*Tm

Calculate the PSF of

Vm and its FWHM Dm

2

2

min

Vm

mm

FGV −

2

2

''

'

min

Vm

mm

FGV −

T0=1/k1

Tm=Tm-1+γ (D m-1-D m-1')

Calculate the PSF of Vm'

and its FWHM Dm'

Increase m by 1

Fig. 1. PSF optimization algorithm

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value, the SNR has larger effect while FWHM has lower effect on the optimization procedure.

After a few attempts we selected α as W, the number of the wave numbers.

The obtained optimal spectrum was shown in Fig. 2. Different from the conventional

symmetric bell-like reshaping spectrum, the optimal spectrum has a priory surprising

asymmetric crater-like shape with two peaks close to the lower and upper bounds of the

spectrum, and the peak at shorter wavelength is higher. It can be understood with this way: the

resolution is primarily limited by the shortest and longest detectable wavelengths, especially

the shortest detectable wavelength, so the side-spectrum enhancement, especially the

enhancement at the shorter wavelength side, helps improve the resolution. In Fig. 2 we also

show five other spectra: the modified optimal spectrum which is more symmetric with the

amplitude of the two peaks being the same, Gaussian profile with FWHM = 300 nm, white

light, white light weighted with Hamming window, and the xenon lamp spectrum. To be

consistent with our experimental system, a wavelength band from 421 nm to 895 nm was

used. The envelopes of theoretical PSFs of the six spectra are shown in Fig. 4(a), and their

FWHM values are given in the notation text. Evidently, the optimal spectrum provides the

highest FWHM resolution, 0.44 μm in free space, with sidelobes lower than those of the PSF

generated by the white light spectrum. The PSF peak of the modified optimal spectrum is

slightly broader than that of the optimal spectrum, which proves that an asymmetric spectral

profile may lead to a higher resolution than a symmetric spectral profile. To quantitatively

compare the sidelobes effect of the PSFs, we calculated the root-mean square width (RMSW)

22

2

])()([

∫∫

dzzPSFdzzPSFz

position z=0 [12]. The theoretical FWHM and RMSW values are shown in Table 1. the

optimal spectrum provides the highest FWHM as well as adequate RMSW.

Table 1. Optimal PSF has improved FWHM and RMSW compared to other PSF given by white light,

Gaussian, and Hamming windowed spectra

From Theory

Properties

Spectra

FWHM (μm) RMSW (μm)

Xenon 1.18 1.05

Gaussian 0.70 0.33

Hamming 0.82 0.29

White Light 0.52 0.89

Optimal 0.44 0.60

2/1

which represents the deviation of the PSF from the central

From Experiments

FWHM (μm)

1.51

1.06

1.03

0.92

0.74

RMSW (μm)

1.12

0.37

0.37

0.67

0.50

Fig. 2. Typical light source spectra

500 600700800900

0.0

0.2

0.4

0.6

0.8

1.0

Normalized intensity

Wavelength (nm)

Optimal

Modified optimal

Gaussian

White

Hamming

Xenon

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-2-1012

-2 5

-2 0

-1 5

-1 0

-5

0

-10-50510

-3 0

-2 5

-2 0

-1 5

-1 0

-5

0

Envelope of PSF (dB)

O ptical path difference (μ m )

Envelope of PSF (dB)

O p tic a l p a th d iffe re n c e (μ m )

O p tim al,0.44 μm

M o dified ,0 .4 4 μ m

G a uss ia n,0.70 μm

W hite ,0 .5 2μ m

H am m in g,0.82 μm

X en on,1 .1 8 μ m

F W H M

(a )

-2 -1012

-1 4

-1 2

-1 0

-8

-6

-4

-2

0

(b )

-10-5 0510

- 20

- 15

- 10

-5

0

Envelope of PSF (dB)

O p tica l path differenc e (μ m )

Envelope of PSF (dB)

O ptic al p ath d iffe ren c e (μm )

O p tim al,0.74μm

G a uss ia n,1.03 μm

W hite ,0 .9 2μ m

H am m in g,1.06 μm

X en on,1 .5 1μ m

FW H M

Fig. 3. The optimal PSF provides improved OCT resolution for the source spectra shown in Fig.

3. (a) Envelopes of theoretical PSFs (b) Envelopes of experimentally measured PSFs. PSFs in

the insets are normalized to conserve the L2 norm.

3. Experimental results

The approach reported in section 2 was also validated experimentally. Our spectroscopic OCT

system setup is shown in Figure 4. A 250W xenon lamp (Oriel) was used as the light source.

The output of the xenon lamp was first collimated by two 250-mm-focal-length lenses L1 and

L2 together with a 1 mm pinhole, spatially filtered by an adjustable aperture, and then

separated into the reference and sample beams by a polarization independent beam splitter

BS. We used the largest facet of an uncoated right-angle prism as the reference mirror M1.

Two identical 4x achromatic objective lenses were used to focus the light beams to M1 and

the sample, and collected the light back reflected/scattered by M1 and the sample. The

collected light was analyzed by means of a spectrograph (Acton SP2150i) coupled into a CCD

camera (Princeton instruments PIMAX 1KHQ), and magnification of the sample to the CCD

was 20. The width of the entrance slit of the spectrograph was adjusted to be 10μm, that is,

the vicinity of 10/20=0.5μm of the reference mirror or the sample was regarded as a single dot.

light from the . The detection system was sensitive to wavelengths from 421nm to 895 nm,

with a spectral resolution of 2 nm. When biological tissue was used as a sample, the CCD was

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set to a higher gain and longer integration time, and in order to avoid the CCD saturation, a

neutral density filter with 10 dB single-trip loss was inserted in the reference arm. A glass was

placed in the sample arm to compensate the additional optical path introduced by the neutral

density filter. Reference mirror M1 and lens L3 were moved as one unit to change the OPD

between the reference and sample beams with a step size of 0.1 μm. Interference spectrum

I(λ, r) was recorded at each OPD step. For calibration, I(λ, r) reflected by the reference mirror

and the sample were also recorded at both the beginning and the end of each imaging

experiment. We then followed Eq. (2) to obtain Id(k,r) for different k and r values, and applied

different spectra V(k) to get the corresponding Id(r) values according to Eq. (3).

Fig. 4. Schematic of a spectral domain OCT system: BS, beam splitter; NDF, neutral density

filter; PCG, phase compensation glass; RM, reference mirror

Fig. 5. OCT images of an onion root tip tissue (a) Image obtained with the originally detected

xenon spectrum. (b) Image obtained with the optimal spectrum. (c) Intensity cross section from

the image shown in panel (a) at lateral position 80 μm. (d) Intensity cross section from the

image shown in panel (b) at lateral position 80 μm.

Xenon

Lamp

RM

Spectroscope

Sample

L1

L2

L5

BS

L3

L4

Pinhole

Aperture

CG

NDF

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A piece of microscope cover glass was used as the sample first. The light was focused on

the top surface of the glass and a(z) can be regarded as a delta function. Thus the resulted Id(r)

gave the PSFs for different source spectra V(k). Fig. 3(b) shows the envelope of the

experimental PSFs. There are multi-peaks in the theoretical PSF, with about 100nm between

the adjacent peaks. In the experiments, the un-flatness of components surfaces and the

dispersion induced optical path mismatch, which are in the order of 100nm, smooth the peaks.

So there is no multiple peaks in the experimental PSFs. The corresponding FWHM and

RMSW values are given in Table 1. As shown in Fig. 3(b) and Table 1, the modeless spectral

reshaping indeed provides resolution superior to those obtained with other conventionally

used spectra including the original xenon-lamp spectrum, white light, Gaussian spectrum, and

Hamming windowed profile, with the FWHM reduced by 51%, 20%, 28%, and 30%,

respectively. Although the spectrum was not purposely reshaped to optimize the sidelobes, we

also notice that the optimal spectrum suppress sidelobes compared to the original xenon-lamp

spectrum with RMSW reduced by 54%. The sidelobes are larger, however, than those of

corresponding to the Gaussian and Hamming profiles.

As a further illustration of the improved resolution, some onion root tip tissue was used

the sample. The experiment was repeated by 100 times, and the results were averaged to give

a higher SNR. Figs. 5(a, b) show the B-scan images before and after spectral reshaping,

respectively, and Figs. 5(c, d) show the corresponding A-scan traces taken from the images at

75 μm lateral position. Evidently, the image/trace obtained using the optimal spectrum (Figs.

5(b, d)) has an improved higher resolution compared to those obtained using the original

xenon-lamp spectrum (Figs. 5(a, c)).

4. Conclusions

In summary, we showed that the optimal spectrum has a priory surprising crater-like spectral

shape, which is different from the conventionally adopted bell-shape spectra. Although the

resolution provided by the optimal spectrum is primarily determined by the spectral range, we

further demonstrated, for the first time to our knowledge, the optimal spectral reshaping

results in an extra improvement in resolution. In particular, in our experiments, the resolution

of 0.74 μm in free space was obtained, which is 51% higher than the one provided by the

original xenon-lamp spectrum. Therefore, the modeless spectral reshaping technique provides

a flexible way to improve the resolution of OCT and may help achieve sub-micron resolution

for subcellular OCT imaging.

Acknowledgments

This work is supported by HIH grants R01CA112315, R01EB003682, and a grant from the

Wallace H. Coulter Foundation.

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