Optical low-coherence reflectometry with resolution beyond
the Fourier transform limit
Chih-Wei Lu, Meng-Tsan Tsai, Yih-Ming Wang, Su-Feng Chen,
Yean-Woei Kiang, C.C. Yang
*Graduate Institute of Electro-Optical Engineering, Department of Electrical Engineering, National Taiwan University, 1, Roosevelt Road, Section 4, Taipei 106, Taiwan, ROC
Received 12 January 2005; received in revised form 29 August 2005; accepted 30 August 2005
Abstract
We demonstrate experimentally and theoretically a method for enhancing the axial resolution of optical low-coherence reflectometry (OLCR) beyond the Fourier transform limit of light source spectrum. The resolution enhancement originates from the superposition of multiple OLCR scans with different offsets of the center-wavelength light-ray with respect to the rotation axis of the mirror mounted on the galvanometer optical scanner in the used optical phase delay line. After certain software process of the multiple scan fringe patterns of different offsets, the superposition leads to an OLCR axial resolution beyond the transform limit. Experimentally, by using four dif-ferent offsets for superposition, about 1.5 times enhancement of the transform-limit resolution is obtained.
2005 Elsevier B.V. All rights reserved.
1. Introduction
Optical low-coherence reflectometry (OLCR) has been widely used for noninvasive scanning of lm-scale struc-tures, particularly in biological tissues[1–4]. It was success-fully developed to become a useful technology, named optical coherence tomography (OCT), for medical diagno-sis[5]. Basically, this technique uses the interfered signals in a Michelson interferometer for monitoring the backscat-tered light such that the index-variation structures within a sample can be mapped. The axial resolution cell size of such a system is inversely proportional to the spectral width of the used light source. With a light source of an ex-tremely large spectral width, the axial resolution can be in the range of a few microns[6–12]. The used light sources of extremely large spectral widths include super-luminescence sources[6], mode-locked lasers[8,9], super-continuum gen-eration[7,10,12], and thermal light[11]. OLCR or OCT has
caught much attention because of its high-resolution and high-sensitivity scanning capabilities for biomedical diag-nosis. Such a high resolution provides us tools for sub-cellular observations. However, for achieving such a high resolution, costly broadband light sources, such as ultra-short solid-state laser and super-continuum generation pumped with an ultra-short pulse, are needed. The need of high cost and complicated (i.e., unstable) light source systems will eventually hinder the development of compact and inexpensive OLCR or OCT systems for broad applica-tions. Also, in using the light sources of extremely broad bands for achieving high resolution, it becomes difficult to employ the widely used fast-scanning optical phase delay line (OPDL). In this situation, the scanning speed is tre-mendously reduced[6,9–12]. Meanwhile, with such a broad bandwidth, dispersion compensation becomes a challeng-ing problem, particularly in a fiber-based system. With a free-space OLCR setup, it is usually difficult to implement a compact and stable system.
In this paper, we demonstrate a novel approach for reach-ing a high resolution with a light source of a relatively small bandwidth in an OLCR system. We can achieve the axial
0030-4018/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2005.08.058
* Corresponding author. Tel.: +886 2 23657624; fax: +886 2 23652637.
E-mail address:[email protected](C.C. Yang).
resolution beyond the Fourier transform limit of the light source spectrum. The basic idea of implementing such a high OLCR resolution is as follows: we vary the offset, x0, of the
center-wavelength light-ray with respect to the rotation axis of the mirror, which is mounted on the galvanometer optical scanner (GOS) in the OPDL used in our OLCR system. With different x0values, the multiple longitudinal scans result in
different interference fringe patterns. The superposition of the multiple scan images, after certain software process, can lead to a result of a high resolution beyond the Fourier transform limit of the light source spectrum.
This paper is organized as follows: In Section2, we pres-ent the experimpres-ental procedures and results. In Section3, theories behind the experimental implementation are reported. Discussions are given in Section4. Finally, con-clusions are drawn in Section 5.
2. Experimental procedures and results
The OLCR system setup used for implementing the high-resolution result is shown in Fig. 1. Here, we use a super-luminescence diode (SLD) of 950 nm in central wavelength. After passing through an isolator for protecting the SLD, the effective spectral full-width at half-maximum (FWHM) is 45 nm. Therefore, the theoretical axial resolution is 8.9 lm if a Gaussian spectral shape is assumed[13]. In the reference arm, an OPDL is used for group- and phase-delay modula-tions. To match the optical paths between the reference and sample arms, a fiber length difference of about 25 cm is required. By using the OPDL and the extra length of disper-sive fiber, dispersion of the OLCR system leads to a broad-ened interference fringe envelope. For changing the x0
value, the GOS was placed on a stage translating in the direc-tion perpendicular to the incident optical axis.
InFig. 2, we show four interference fringe patterns of mir-ror surface reflection for four x0values, obtained by
translat-ing the GOS 1 mm per step. One can see that the frtranslat-inges are located at different positions with slightly different widths and chirps. The FWHMs of all the fringe patterns are around 20 lm. The four fringe patterns inFig. 2were processed by (1) shifting the fringes to align the envelope maxima, and
(2) multiplying the horizontal axis by a factor for each fringe pattern. The multiplication factors are determined by com-paring the distances of the fringe envelopes between the two interfaces of a given physical separation in the cases of different x0values. In other words, this multiplication
proce-dure represents the correction of the slight variation in fringe pattern width when different x0values are used. The four
multiplication factors for parts (a)–(d) in Fig. 2 are 1, 0.9818, 0.9591, and 0.9321, respectively. Also, the shifts of the envelope peaks in parts (a)–(d) are 0, 36, 61, and 117 lm, respectively. The results after the aforementioned software process are shown inFig. 3. Then, the four fringe patterns inFig. 3are superimposed to obtain a new fringe envelope, as shown inFig. 4. Here, one can see that an OLCR resolution of 5.9 lm (FWHM) is achieved. This value is about two-third the Fourier transform limit.
To further demonstrate the resolution enhancement, we scanned a sample consisting of two contacting glass plates with the aforementioned procedure. Parts (a)–(c) inFig. 5
Isolator
= 45 nm
Optical phase delay line
Sample
Data Processing and Image Construction
A/D Converter
SLD (2mW) = 950 nm
Beam splitter with fiber pigtails
x0
Fig. 1. Experimental setup of the OLCR system.
Fig. 2. Four scans of a mirror face with different x0values. The FWHMs
are all around 20 lm.
Fig. 3. The fringe patterns of the four scanning results inFig. 2after the software process.
shows the 1-D scanning results with three different x0
values. Here, although the nearby interfaces between the two glass plates can be more or less identified in parts (a)–(c), their separation is much better resolved in part (d), which is obtained by combining parts (a)–(c) following the procedures described above. The resolution in part (d) is clearly enhanced.
3. Theories
To understand the theories behind the resolution enhancement, we demonstrate the concept with the follow-ing derivations. For the convenience in derivation, we use the frequency and time domains (x, t), instead of the do-mains of wave number and space. Their correspondence is quite straightforward. The phase evolution, /(x, t), of an optical signal after returning from a typical OPDL can be expressed as[14] /ðx; tÞ ¼8p f Ct k h0 sin 1 k d sin hi þx0 f / xð 0; tÞ þ o/ ox x¼x0 x x0 ð Þ. ð1Þ
Here, x is the angular frequency, k is the wavelength, f is the focal length of the used lens, C is the rotation angular
frequency of the GOS, d is the grating period, h0 is the
incident angle onto the diffractive grating, and h1is the
dif-fracted angle from the grating. In(1), we series-expand the phase /(x, t) and keep only the zeroth and first-order terms. The higher-order terms, which are related to disper-sion, are ignored in our concept demonstration. Here, x0is
the central angular frequency. For expression simplicity, we define a(x0) and b(x0) as
/ xð 0; tÞ ¼ 8pCx0 k0 t¼ a xð Þ t;0 ð2Þ o/ ox x¼x0 ¼ b xð Þ t.0 ð3Þ
Note that a = 0 when x0is zero. In collecting the interfered
signals, we are concerned with the cross-correlation func-tion, Cr(t), between the reference and sample arms as
CrðtÞ ¼ Z h~Esðt0Þ~E rðt 0þ tÞi dt0 ¼E 2 0 2pexp j a þ xð 0Þt þ jx0sgs Z Gðx00Þ exp j sgs 1 þ bð Þt x00 dx00 ¼ E2 0exp j a þ xð 0Þt þ jx0sgs ~G sgs 1 þ bð Þt . ð4Þ Here, ~Esð Þ and ~t0 E
rðt0þ tÞ are the time-domain optical
sig-nals returned from the sample and reference arms, respec-tively, and E0 is their equal amplitude. The symbol h•i
represents the ensemble average. Also, sgs stands for the
group delay difference between the two arms. It is also a function of x0. Meanwhile, G(x) is the light source spectral
intensity and ~GðtÞ is the counterpart in the time domain. From (4), one can see that a nonzero x0(nonzero a) leads
to the shift of the modulation frequency of the interference fringe pattern. Also, the b factor results in the modification of the time axis and the variation of fringe envelope peak position. The x0dependence of sgs also contributes to the
peak position variation of fringe envelope.
To show the resolution enhancement through the superposition of more than one scanning results of differ-ent x0values, we consider an example of two x0values as
follows: Cr1ðtÞ ¼ E20exp j að 1þ x0Þt þ jx0sgs1 ~G sgs1 1 þ bð 1Þt ; Cr2ð Þ ¼ Et 20exp j að 2þ x0Þt þ jx0sgs2 ~G sgs2 1 þ bð 2Þt . The software process described in Section 2 implies the modification of(6) as C0r2ð Þ ¼ Et 2 0exp j að 2þ x0Þ 1þ b1 ð Þ 1þ b2 ð Þtþ jx0sgs1 ~G sgs1 1 þ bð 2Þ 1þ b1 ð Þ 1þ b2 ð Þt ¼ E2 0exp ja að 1þ x0Þt þ jx0sgs1 ~G sgs1 1 þ bð 1Þt . ð5Þ
Fig. 5. (a)–(c) 1-D scanning results of a sample consisting of two contacting glass plates with different x0values. (d) Superposition result of
parts (a)–(c) after the software process.
Fig. 4. The fringe envelope after the superposition of the four fringe patterns inFig. 3. The OLCR axial resolution becomes 5.9 lm.
In other words, the time scale is modified by a factor of (1 + b1)/(1 + b2). Also, the peak position of the fringe
envelope is shifted by sgs1– sgs2. Here,
a¼ða2þ x0Þ 1 þ bð 1Þ a1þ x0
ð Þ 1 þ bð 2Þ
. ð6Þ
Therefore, the superposition leads to Cð Þr2ð Þ ¼ Ct r1ð Þ þ Ct 0r2ð Þt ¼ E2 0exp jx0sgs1 exp½j að 1þ x0Þt f þ exp ja a½ ð 1þ x0Þtg ~G sgs 1 þ bð 1Þt . ð7Þ The two terms in the curved brackets produce beating to form a sinusoidal modulation for the fringe envelope, as schematically shown inFig. 6. Here, side-lobes can be seen in the final fringe envelope (after taking the absolute value). Such a theoretical result may explain the side-lobes in
Fig. 4. The side-lobes can be reduced if more scans of different x0values are considered. Theoretically, if the x0
values can be in multiple, the superposition of these scan components is similar to the case of laser mode locking, in which coherent frequency components are superimposed to form a short pulse. Therefore, a narrow hump can be produced for modifying the fringe envelope, leading to an OLCR resolution beyond the Fourier transform limit of the source spectrum.
4. Discussions
Resolution beyond the transform limit in an OLCR system is useful for building high-resolution systems with low-cost light sources. However, the proposed approach requires multiple scans of a sample that may require a longer imaging time. When the GOS is operated at 1 kHz, the required imaging time for a four-scan experiment includes:
(1) <0.4 s for the four scans of a 1 mm· 1 mm frame (100 A-mode scans per frame); (2) 3 s for changing the x0value;
and (3) <0.8 s for data process. Hence, the four-scan exper-iment requires <4.2 s, which is about 40 times that of sin-gle-scan imaging. Nevertheless, the requirement of such a multiple-scan operation with a light source of a relatively narrower bandwidth does not necessarily represent a draw-back in an OLCR or OCT system when compared with the case of an extremely broad bandwidth. With a light source of an extremely broad bandwidth, the fast-scanning OPDL can no longer be used in such a system due to the limited width of the mirror mounted on the GOS. The use of other axial scanning mechanisms, such as a constant-speed trans-lation stage, makes the scanning speed thousand times slower than that using an OPDL [6,9–12]. For instance, with 1 Hz operation frequency of a translation stage, a 1 mm· 1 mm image requires 100 s, which is longer than that of the aforementioned four-scan experiment. There-fore, if the multiple-scan method with a reasonably narrow source bandwidth can lead to an OLCR or OCT resolution comparable to that of a system with an extremely large spectral width, the multiple-scan does not necessarily repre-sent a drawback.
Regarding the imaging time, another important point deserves further discussions as follows. In operating an OLCR or OCT system with various offsets, x0, we do not
need to adjust the x0values or search for the matching
con-ditions by calculating a, b, and sgsfor an individual sample.
We simply need to choose one set of x0values and optimize
the process parameters in advance. Once those parameters are optimized, they can be used for scanning various sam-ples. Therefore, the key problem in imaging time elonga-tion in the proposed method is the change of the x0
value. The variation of the x0value for multiple scans is
in-deed impractical. By using a phase modulator for multiple-frequency scanning will be practical. In this situation, only a few milliseconds is needed for changing the modulation frequency. Recently, the development of spectral-domain OCT has led to much higher scanning speed, up to tens frames per second, which is beyond the video rate [15]. With this development, the required longer imaging time due to the multiple-scan procedure should not be a major problem.
The compensation of the system dispersion mismatch is a crucial issue in the operation of an OLCR or OCT sys-tem. In our derivations, we considered only up to the term of the first-order derivative with respect to frequency in(1)
for concept demonstration. If we considered the term of the second-order derivative (dispersion effect), Eq. (4) would become more complicated. In this situation, the data pro-cess procedures for the superposition of multiple-scan results might become difficult. Also, the implementation of a resolution beyond the transform limit is ineffective. A significant dispersion effect will broaden the fringe enve-lope and hence cancel the resolution improvement. There-fore, our idea for improving the resolution is designed for an OLCR or OCT system with a relatively smaller spectral
Fig. 6. Schematic demonstration of the OLCR resolution modification to reach a level beyond the Fourier-transform limit.
width. In such a system, the dispersion can be easily compensated.
Sample material dispersion is a key issue in evaluating the function of the proposed method even the system spec-tral width is not broad. In this situation, the conditions for resolution improvement can still be optimized through a careful process of the data, i.e., adjusting the required parameters. After such an optimization procedure, the OLCR or OCT system can be used for scanning all sam-ples of the similar material dispersion property. In other words, the system parameters do not need frequent adjust-ments. Finally, the major effect of scattering in tissues is to attenuate the returned signal intensity. Scattering actually does not have the direct impact on the proposed method.
5. Conclusions
In summary, we have demonstrated experimentally and theoretically a method for enhancing the axial reso-lution of an OLCR or OCT system beyond the Fourier transform limit of light source spectrum. The resolution enhancement originated from the superposition of multi-ple longitudinal scans with different offsets of the center-wavelength light-ray with respect to the rotation axis of the mirror mounted on the GOS in the used OPDL. After certain software process of the multiply scanned fringe patterns of different offsets, the superposition led to a resolution beyond the Fourier transform limit. Experimentally, by using four different offsets for super-position, about 1.5 times enhancement of the trans-form-limit value was obtained.
Acknowledgment
This research was supported by National Health Re-search Institute, The Republic of China, under the Grant of NHRI-EX94-9220EI.
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