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Phase-drift suppression using harmonics in heterodyne detection and its application to optical coherence tomography
1 October 2000
Optics Communications 184 (2000) 95±104
www.elsevier.com/locate/optcom
Phase-drift suppression using harmonics in heterodyne detection and its application to optical coherence tomography M. Sato a,*, K. Seino a, K. Onodera a, N. Tanno a,b a
Graduate Program of Human Sensing and Functional Sensor Engineering, Graduate School of Science and Engineering, Yamagata University, 4-3-16 Johnan, Yonezawa 992-8510, Japan b Regional Joint Research Project of Yamagata Prefecture, Yamagata Technopolis Foundation, Yamagata 990-2473, Japan Received 3 April 2000; received in revised form 26 July 2000; accepted 2 August 2000
Abstract We demonstrate phase-drift suppression (PDS) to enhance stabilization of the computed signal intensity (CSI) in heterodyne detection without optical and/or electronic feedback systems. CSI is de®ned as the square root of the sum of the squares of each harmonic intensity in the heterodyne beat signal. The PDS theoretically and experimentally demonstrates stabilizing conditions for modulation indexes and stability enhancements; for example, the CSI stability was enhanced 49 times through simulations and by 6.3 times through experiments for a modulation index of 2.62 rad. Optical coherence tomography images using a ®ber Mach±Zehnder interferometer and a superluminescent diode verify the validity of PDS as an application. Ó 2000 Elsevier Science B.V. All rights reserved. PACS: 07.60; 42.30.W; 87.62; 42.30.W Keywords: Heterodyne beat signal; Stabilization; Harmonics; Phase drift; Optical coherence tomography
1. Introduction Optical interferometric measurements are usually combined with heterodyne detections to achieve optimal sensitivity. This measurement technique has a wide range of applications, from detection of minute optical waves and precise distance measurement to optical tomographies for biological tissues. Several imaging methods for high-scattering materials and biological tissues have recently been reported [1±6]. Optical coherence tomography
*
Corresponding author. Fax: +81-238-26-3187. E-mail address: [email protected] (M. Sato).
(OCT) [7,8], in particular, has attracted interest, because the spatial resolution of OCT is about 10 lm, up to one order higher than clinically available diagnostic imaging. OCT systems utilize lowcoherence interferometry and low-coherence light sources, such as a superluminescent diode (SLD). Fluctuations of an optical wavelength, an optical phase in the light source, and the optical distance decrease the output signal stability of interferometric measurement systems. Novel systems using optical and/or electronic feedback have been reported to overcome these problems [9,10]. In particular, the ®ber components were not thermally or acoustically isolated, so random environmental changes caused ¯uctuations in the relative path lengths of the interferometers.
0030-4018/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 ( 0 0 ) 0 0 9 3 0 - 5
96
M. Sato et al. / Optics Communications 184 (2000) 95±104
A technique to control the modulation index and to use the signals of the fundamental and second harmonic has been reported in the ®eld of optical coherence domain re¯ectometry [11] to eliminate these slowly varying ¯uctuations in the detected interferometric signal. The recorded data were calculated to obtain the sum of the powers at the fundamental and second harmonics. The modulation index was adjusted so that the two Bessel functions, J1 /0 and J2 /0 , where /0 is the modulation index, had the same value, making the sum independent of the phase drift. Axial scanning in OCT does not utilize a phase modulator (PM) because the heterodyne beat signal can be detected using the Doppler shift by a moving mirror. However, a PM must generally be used in transversal scanning to detect the heterodyne beat signal as the Doppler shift is not available. The above technique using the fundamental and second harmonics has been used for OCT and for optical coherence microscopy of transversal scanning [12,13]. It has been mathematically and experimentally con®rmed in OCT that the sum of the two powers of the fundamental and second harmonics was independent of the phase drift at a modulation index of 2.63 rad [14]. In this paper, expanding this technique by increasing the number of harmonics, we describe our simulated and experimental results of phasedrift suppression (PDS) [15] to enhance the stability of an output signal in heterodyne detection using higher order harmonics. Assuming a sinusoidal phase modulation such as a mirror/piezo system in the interferometer, the heterodyne beat signal from a detector contains many harmonics that change with random optical phase drifts. Slowly varying phase drifts occur in the interferometer due to thermal and acoustical ¯uctuations in the environment. The computed signal intensity (CSI) is de®ned in PDS as the square root of the sum of the squares of each harmonic intensity in the heterodyne beat signal. It is possible to use the optimum modulation index to increase the stability of the CSI, using higher order harmonics without any optical and/or electronic feedback systems. According to Parseval's theorem, the sum of squaring each of the in®nite harmonics is inde-
pendent of the optical phase dierence. Section 2 describes the practical conditions, such as the phase modulation index and number of harmonics for ®nite harmonics.
2. Phase-drift suppression 2.1. Theory In a conventional interferometer, the phase dierences / between two arms vary with the phase drifts of the wavelength, the optical phase of the optical source, and the path length in the interferometer. Thus, the total phase drift /d t is represented by /d t ÿ
2p 2p L Dkd Dud DLd ; 2 k k
1
where k is the wavelength, L, the optical path dierence, Dkd , the wavelength drift, Dud , the phase drift in the light source, and DLd , the drift of the optical path dierences. The drift in the optical path dierence is due to thermal drift and/or mechanical vibrations. Wavelength and phase drifts are due to stochastic factors in the light source. We assume that the overall phase drifts change randomly. Assuming that the time delay between two arms is within the coherence time, and sinusoidal phase modulation is used, the heterodyne beat signal IHB t is represented as IHB t I0 cos /0 sin xP t /d t;
2
where I0 is the constant intensity that is the square root of the product of the signal and reference intensity, /0 , the modulation index, and xP , the angular frequency of the phase modulation. Using the Bessel function, the heterodyne beat signal is expanded as IHB t; /0 ; /d
1 X
C2iÿ1 sin 2i ÿ 1xP t
i1
1 X j1
C2j cos 2jxP t;
3
M. Sato et al. / Optics Communications 184 (2000) 95±104
where C2iÿ1 ÿ2I0 sin /d J2iÿ1 /0 ; C2j 2I0 cos /d J2j /0 :
4
This indicates that the heterodyne beat signal consists of an in®nite number of harmonics and that the balance between odd and even order harmonics depends on the phase drift. Here, using harmonics from the ®rst to the nth order, we de®ne the CSI as ICS /0 ; /d v uINTf n1=2g INT n=2 X u X 2I t sin2 / J 2 / cos2 / J 2 / ; 0
i1
d
2iÿ1
0
d
j1
2j
0
5
where INT x is the function that gives the nearest integer below x. The signal was light backscattered or transmitted from a biological sample in our measurements. Thus, it is apparent that the scattered light intensity can be detected through the CSI. For a stabilizing condition represented by INTf n1=2g X
INT n=2 X
i1
j1
2 J2iÿ1 /0
J2j2 /0 ;
the CSI is given by v uINTf n1=2g u X 2 J2iÿ1 /0 ICS /0 2I0 t i1
v uINT n=2 u X J2j2 /0 : 2I0 t
6
or
97
between 0 and 2p were used as phase drift /d . The phase drift was generated 100 times for one modulation index /0 . The modulation index range was 0±10 rad in steps of 0.1 rad. The power spectrum of the harmonics changes due to the phase drift, and thus, CSI also change. The standard deviation and average of the CSI were calculated using randomly generated values for the phase drift. The relative standard deviation (RSD) was then de®ned as the ratio of the CSI standard deviation to the CSI average. It is important to evaluate the RSD practically since it represents the instability of the CSI due to phase drifts. The RSD is shown in Fig. 1 as a function of the modulation index. C1 denotes that the CSI is the signal intensity of the fundamentals. C1±j signi®es that the CSI was computed using harmonics from the ®rst to the jth order as given by Eq. (5). The RSD for C1 was almost constant at around 49% because variations of the standard deviation of CSI synchronously correspond to the CSI averages. Supposing that the phase drift is represented by random variables distributed uniformly from 0 to 2p, the RSD for C1 is obtained at 48.5% according to the theory of stochastic processes. 48.5% is the value corresponding to the simulated result of 49%. This analytically con®rms our simulated results.
7
j1
This indicates that the CSI as an output signal is independent of the phase drift, and its stability is enhanced. Considering just the ®rst and second harmonics, the stabilizing condition is approximately given by J1 /0 J2 /0 . The CSI is then given by 2I0 jJ1 /0 j or 2I0 jJ2 /0 j, as shown in Ref. [14]. 2.2. Simulated results and discussion The CSI and its standard deviation were simulated using Eq. (5). Randomly generated values
Fig. 1. Simulated dependence of the RSD of the CSI on the modulation index. The RSD is the ratio of the standard deviation of the CSI to the CSI average.
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M. Sato et al. / Optics Communications 184 (2000) 95±104
Table 1 Modulation indexes at the minimum relative standard deviation of CSI (simulations) Computing conditions
Modulation index (rad) 1st
2nd
3rd
4th
5th
C1±2 C1±2 a C1±4 C1±6 C1±8
2.6 2.62 2.8 2.8 2.8
4.5 4.47 4.1 4.0 4.0
6.1 6.09 5.6 5.9 5.9
7.7 7.71 7.4 7.6 7.3
9.3 9.29 9.0 8.7 9.0
a
Modulation indexes given numerically.
The RSDs for C1±n with n above 2 changed periodically and were smaller than the RSDs of C1 . The modulation indexes that give the minimum RSDs in Fig. 1 are shown in Table 1, using harmonics up to the eighth order. The stability of the CSI for phase drift is enhanced at these modulation indexes. The modulation indexes given numerically by the stabilizing condition of J1 /0 J2 /0 for C1±2 are also shown in the same table, and those values agree with the simulated ones. The dependence of the RSD on the number of harmonics is shown in Fig. 2. The minimum RSD was 1% on C1±2 for a modulation index of 2.6 rad. This indicates that the stability of the CSI was enhanced 49 times, compared with the RSD of 49% on C1 . The minimum RSD was 0.2% on C1±4
at a modulation index of 2.8 rad. This indicates that the stability of the CSI was enhanced 245 times. Assuming sinusoidal phase modulation in interferometers, the conventional feedback system usually attempts to maximize the output signal at the fundamental beat frequency; in short, at the ®rst peak of the Bessel function J1 /. Comparing the CSI with the signal intensity using a feedback system, the ratio of the CSI of C1±2 to the maximum beat signal intensity at the modulation frequency is given by J1 /0 2:62 rad= J1 /0m 1:90 rad 81%, where /0m is the modulation index used to maximize the beat signal intensity at the modulation frequency. This indicates that the stability was enhanced using PDS, but that the CSI was 81% for the maximum beat signal intensity using the conventional feedback system. 3. Experiments 3.1. Experimental setup The optical measurement system mainly consists of a ®ber-optic Mach±Zehnder interferometer that is fundamentally based on Ref. [16], as shown in Fig. 3. This system was utilized to verify simulated results and to measure OCT images using
Fig. 2. Simulated dependence of the RSD of CSI on the number of harmonics.
M. Sato et al. / Optics Communications 184 (2000) 95±104
99
Fig. 3. Experimental setup for an OCT system based on a ®ber Mach±Zehnder interferometer. SLD: superluminescent diode, NDC: ND coupler, CL: coupling lens, RM: re¯ector mirror, PM: phase modulator, HMC: half-mirror coupler, Probe: GRIN lens probe, FC: ®ber coupler, and APD: avalanche photodiode.
PDS. The light source was an SLD with a broad bandwidth of 25 nm (FWHM) and center wavelength of 840 nm. The temperature of the only light source was stabilized at 25°C. The illumination power to the sample was 20 lW. The light beam is launched into two arms by an ND coupler (NDC). The reference arm is comprise, an optical delay and a PM. The optical delay consists of a coupling lens (CL), a re¯ector mirror (RM), and a moving stage. The PM consists of a piezoelectric ®ber stretcher that can control the modulation index by changing the applied sinusoidal voltage of 15 kHz. We ®rst calibrated the peak-to-peak sine voltage applied to the PM. The relation between the modulation index and peak-to-peak voltage of a triangular wave was measured as shown in Fig. 4. As mentioned above, the PM is not usually utilized for axial scanning in OCT because the Doppler shifting is possible by moving a mirror. However, PM was used here in the axial scanning to con®rm the principle of PDS.
The sample arm is comprised of a half mirror coupler (HMC) and an optical probe with a gradient-index (GRIN) lens (1 mm /, 2.6 mm length, 0.25 pitch) for both illumination and collection of backscattered light. The signal light is combined with the reference light by a ®ber coupler (FC) to generate a heterodyne beat signal. The heterodyne beat signal is detected by a Si-avalanche photodiode and is fed into an RF spectrum analyzer only when the total length of the sample arm matches that of reference within the coherence length. A personal computer controls the sample stage in three dimensions and synchronously acquires data from the RF spectrum analyzer. Individual intensities of higher order harmonics in the heterodyne beat signal can also be fed into the computer. The time variations of the heterodyne beat signal from the ®rst to the sixth-order harmonics with time were acquired simultaneously for one modulation index and the CSIs were calculated.
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Fig. 4. Dependence of the modulation index on the applied voltage in the phase modulator.
3.2. Results and discussion An Al coated mirror with 100% re¯ection was used as a sample to verify the simulated results. Two typical power spectra of the heterodyne beat signal are shown in Fig. 5. Odd order harmonics are dominant in Fig. 5(a), and even order harmonics in Fig. 5(b). These two states were randomly exchanged due to phase drifts. The higher order harmonics were below the sixth harmonic of 90 kHz. Therefore, in the following experiments,
we only had to measure the beat signals below the sixth harmonic. The CSIs of C1 , C1±2 , C1±4 , and C1±6 are shown in Fig. 6 as functions of the modulation index. The experimental results almost agree with the simulated ones calculated by Eq. (5). The measured RSDs are shown in Fig. 7 as functions of the modulation indexes for C1 , C1±2 , C1±4 , and C1±6 . The RSD of C1 was almost constant, but the RSDs for C1±2 , C1±4 , and C1±6 changed periodically, as shown in Fig. 7(a). The RSDs are also shown in Fig. 7(b) as functions of the modulation index from 2.0 to 3.2 rad in steps of 0.1 rad for C1 , C1±2 , and C1±6 . It was experimentally veri®ed that the stability of C1±2 at a modulation index of 2.7 rad was enhanced by 6.3 times and the stability of C1±6 at 2.8 rad was enhanced 10 times. The dierences between the experimental and simulated results shown in Fig. 7 are mainly due to the unstable intensities of the light sources. The RSD theoretically indicates the instability of only light intensities with the stable conditions of Eq. (6). The RSD of the light intensity of the SLD was measured at 2.8%. Typical variations of C1 , C2 , C1±2 , and C1±6 over time for modulation indexes of 1.0, 2.5, and 4.0 rad are shown in Fig. 8(a)±(c), respectively. The measurement times and data acquisition intervals were 10 min and 20 s. The variations of the signals were compensated and decreased for C1±2 and C1±6
Fig. 5. Typical power spectra of a heterodyne beat signal: (a) when odd order harmonics are dominant and (b) when even order harmonics are dominant. Modulation index: 2.5 rad, and frequency scale: 10 kHz/div.
M. Sato et al. / Optics Communications 184 (2000) 95±104
101
Fig. 6. Dependence of CSI on the modulation index: the modulation frequency was 15 kHz.
at /0 2:5 rad, and at /0 4:0 rad only for C1±6 . These results correspond to those in Fig. 7.
4. Application to optical coherence tomography 4.1. Basic characteristics of an optical coherence tomography system A coherence length lc of 25 lm was obtained from the equation lc 4 ln 2k2 =p Dk [17], assuming a Gaussian spectrum. Thus, the axial resolution of OCT, given by lc =2, was 13 lm. Using the Al-coated mirror as a sample, the axial resolution was obtained at 17 lm (FWHM) from the measured autocorrelation function, as shown in Fig. 9(b). The dierence between the two axial resolutions of 13 and 17 lm may exist primarily because the interferogram was broadened by a dispersion mismatch between the two arms of the interferometer [18], and because the SLD spectrum was not completely Gaussian. We measured the dependence of the beam spot size on the axial position, and it corresponded to theoretical result given by applying the ABCD law [19]. The transverse resolution from the spot size at the beam waist was 7.5 lm (FWHM).
Fig. 7. Dependence of the RSD of CSI on the modulation index: (a) modulation index region from 1.0 to 4.0 rad and (b) modulation index region from 2.0 to 3.2 rad.
4.2. Optical coherence tomography images with phase drift suppression We measured the OCT images as follows: The measured point within a coherent gate was positioned at the beam waist by the optical delay. (i) The sample was next axially scanned to measure the axial pro®le of backscatters. (ii) The sample
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M. Sato et al. / Optics Communications 184 (2000) 95±104
shown in Fig. 9(a) and (b). It is apparent that the stability in C1±2 is enhanced over that in C1 . We next measured cross-sectional images and signal pro®les using a sample of cover glass (145 lm thick) on the slide glasses in C1 and C1±2 , as shown in Fig. 10(a) and (b). The dimensions of the OCT images were 0:3 axial 0:3 transverse mm2 , which corresponded to 60 axial 60 transverse pixels. It took 80 min to acquire the image due to the slow response of the stages. The boundaries between the air and the cover glass and between the cover glass and the slide glass were very noisy, as shown in Fig. 10(a), but this noise was suppressed for Fig. 10(b). The boundaries of the cover glass had a thickness that exceeded the resolution levels, due to a logarithmic scale display. We con®rmed that the PDS technique is valuable for enhancing the stability of output signals in heterodyne detection. Furthermore, it is easy to apply this technique since it requires only a few lock-in ampli®ers or a few band-pass ®lters and a conventional data acquisition system.
5. Conclusion
Fig. 8. Variations of C1 , C2 , C1±2 , and C1±6 with time: modulation indexes of (a) 1.0 rad, (b) 2.5 rad, and (c) 4.0 rad. The measurement time was 10 min.
stage was then moved transversely. Two-dimensional cross-sectional images were created in computers by alternately executing (i) and (ii). Two autocorrelations of SLD with C1 and C1±2 were measured at a modulation index of 2.5 rad, as
We have demonstrated a PDS technique with an increased number of harmonics for enhancing the stability of CSI in a heterodyne beat signal. The stabilizing conditions in PDS were introduced analytically. The modulation indexes that minimize the standard deviations of CSI were determined through simulations for higher harmonics up to the eighth order. The dependences of CSI and its RSD on the modulation indexes were veri®ed experimentally. For example, the stability of CSI was enhanced 49 times for C1±2 at a modulation index of 2.62 rad through simulations, and the stability was experimentally enhanced 6.3 times at a modulation index of 2.7 rad. This PDS was also con®rmed to be useful for applications in OCT. Complicated and expensive feedback systems must be used to stabilize heterodyne beat signals to an order of 10ÿ6 . However, the PDS is simple to realize and is widely applicable to conventional heterodyne detection in a variety of systems
M. Sato et al. / Optics Communications 184 (2000) 95±104
103
Fig. 9. Autocorrelations of light sources (a) C1 and (b) C1±2 .
Fig. 10. OCT images and pro®les of cover glass and slide glass: (a) C1 and (b) C1±2 . 1 pixel 5 5 lm2 , and 60 transverse 60 axial pixels.
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ranging from optical systems to electromagnetic systems. Acknowledgements Part of this research was supported by the Japan Science and Technology Corporation (JST). References [1] M. Takeda, H. Yamamoto, Appl. Opt. 33 (1994) 7829. [2] N. Tan-no, T. Ichimura, T. Funaba, N. Anndo, Y. Odagiri, Opt. Lett. 19 (1994) 587. [3] B. Devaraji, M. Usa, K.P. Chan, T. Akatsuka, H. Inaba, IEEE J. Sel. Topic Quant. Electron. 2 (1996) 1008. [4] T. Funaba, N. Tan-no, H. Ito, Appl. Opt. 36 (1997) 8919. [5] G. Le Tolguenec, E. Lantz, F. Devaux, Appl. Opt. 36 (1997) 8292. [6] A.C. Millard, P.W. Wiseman, D.N. Fittingho, K.R. Wilson, J.A. Squier, M. Muller, Appl. Opt. 38 (1999) 7393. [7] D. Huang, E.A. Swanson, C.P. Lin, J.S. Schumam, W.G. Pulia®to, J.G. Fujimoto, Science 254 (1991) 1178.
[8] S. Tiba, N. Tanno, 14th Japanese Laser Sensing Symposium, Abstracts of Papers, Laser Radar Society of Japan, and Optoelectric Industry and Technology Development Association, Sendai, Japan, 1991, paper C3. [9] C. Grener, B. Boggs, T. Wang, T.W. Mossberg, Opt. Lett. 23 (1998) 1280. [10] A.A. Freschi, J. Frejlich, Opt. Lett. 20 (1995) 635. [11] S.R. Chinn, E.A. Swanson, Electron. Lett. 29 (1993) 2025. [12] J.A. Izatt, M.D. Kulkarrni, H.W. Wang, K. Kobayashi, M.V. Sivak, IEEE J. Sel. Topics Quant. Electron. 2 (1996) 1017. [13] A.G. Podoleanu, M. Seeger, G.M. Dobre, D.J. Webb, D.A. Jackson, F.W. Fitzke, J. Biomed. Opt. 3 (1998) 12. [14] B.M. Hoeling, A.D. Fernandez, R.C. Haskell, E. Huang, W.R. Myers, D.C. Petersent, S.E. Ungersma, R. Wang, M.E. Williams, Opt. Express 6 (2000) 136. [15] M. Sato, K. Seino, K. Onodera, N. Tanno, IEICE Trans. J83-C (2000) 1. [16] J.A. Izatt, M.R. Hee, G.M. Owen, E.A. Swanson, J.G. Fujimoto, Opt. Lett. 19 (1994) 590. [17] A.F. Fercher, J. Biomed. Opt. 1 (1996) 157. [18] C.K. Hitzenberger, A. Baumgartner, W. Drexler, A.F. Fercher, J. Biomed. Opt. 4 (1999) 144. [19] B.E.A. Saleh, M.C. Teich, Fundamentals of Photonics, Wiley, New York, 1991.
Optics Communications 184 (2000) 95±104
www.elsevier.com/locate/optcom
Phase-drift suppression using harmonics in heterodyne detection and its application to optical coherence tomography M. Sato a,*, K. Seino a, K. Onodera a, N. Tanno a,b a
Graduate Program of Human Sensing and Functional Sensor Engineering, Graduate School of Science and Engineering, Yamagata University, 4-3-16 Johnan, Yonezawa 992-8510, Japan b Regional Joint Research Project of Yamagata Prefecture, Yamagata Technopolis Foundation, Yamagata 990-2473, Japan Received 3 April 2000; received in revised form 26 July 2000; accepted 2 August 2000
Abstract We demonstrate phase-drift suppression (PDS) to enhance stabilization of the computed signal intensity (CSI) in heterodyne detection without optical and/or electronic feedback systems. CSI is de®ned as the square root of the sum of the squares of each harmonic intensity in the heterodyne beat signal. The PDS theoretically and experimentally demonstrates stabilizing conditions for modulation indexes and stability enhancements; for example, the CSI stability was enhanced 49 times through simulations and by 6.3 times through experiments for a modulation index of 2.62 rad. Optical coherence tomography images using a ®ber Mach±Zehnder interferometer and a superluminescent diode verify the validity of PDS as an application. Ó 2000 Elsevier Science B.V. All rights reserved. PACS: 07.60; 42.30.W; 87.62; 42.30.W Keywords: Heterodyne beat signal; Stabilization; Harmonics; Phase drift; Optical coherence tomography
1. Introduction Optical interferometric measurements are usually combined with heterodyne detections to achieve optimal sensitivity. This measurement technique has a wide range of applications, from detection of minute optical waves and precise distance measurement to optical tomographies for biological tissues. Several imaging methods for high-scattering materials and biological tissues have recently been reported [1±6]. Optical coherence tomography
*
Corresponding author. Fax: +81-238-26-3187. E-mail address: [email protected] (M. Sato).
(OCT) [7,8], in particular, has attracted interest, because the spatial resolution of OCT is about 10 lm, up to one order higher than clinically available diagnostic imaging. OCT systems utilize lowcoherence interferometry and low-coherence light sources, such as a superluminescent diode (SLD). Fluctuations of an optical wavelength, an optical phase in the light source, and the optical distance decrease the output signal stability of interferometric measurement systems. Novel systems using optical and/or electronic feedback have been reported to overcome these problems [9,10]. In particular, the ®ber components were not thermally or acoustically isolated, so random environmental changes caused ¯uctuations in the relative path lengths of the interferometers.
0030-4018/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 ( 0 0 ) 0 0 9 3 0 - 5
96
M. Sato et al. / Optics Communications 184 (2000) 95±104
A technique to control the modulation index and to use the signals of the fundamental and second harmonic has been reported in the ®eld of optical coherence domain re¯ectometry [11] to eliminate these slowly varying ¯uctuations in the detected interferometric signal. The recorded data were calculated to obtain the sum of the powers at the fundamental and second harmonics. The modulation index was adjusted so that the two Bessel functions, J1 /0 and J2 /0 , where /0 is the modulation index, had the same value, making the sum independent of the phase drift. Axial scanning in OCT does not utilize a phase modulator (PM) because the heterodyne beat signal can be detected using the Doppler shift by a moving mirror. However, a PM must generally be used in transversal scanning to detect the heterodyne beat signal as the Doppler shift is not available. The above technique using the fundamental and second harmonics has been used for OCT and for optical coherence microscopy of transversal scanning [12,13]. It has been mathematically and experimentally con®rmed in OCT that the sum of the two powers of the fundamental and second harmonics was independent of the phase drift at a modulation index of 2.63 rad [14]. In this paper, expanding this technique by increasing the number of harmonics, we describe our simulated and experimental results of phasedrift suppression (PDS) [15] to enhance the stability of an output signal in heterodyne detection using higher order harmonics. Assuming a sinusoidal phase modulation such as a mirror/piezo system in the interferometer, the heterodyne beat signal from a detector contains many harmonics that change with random optical phase drifts. Slowly varying phase drifts occur in the interferometer due to thermal and acoustical ¯uctuations in the environment. The computed signal intensity (CSI) is de®ned in PDS as the square root of the sum of the squares of each harmonic intensity in the heterodyne beat signal. It is possible to use the optimum modulation index to increase the stability of the CSI, using higher order harmonics without any optical and/or electronic feedback systems. According to Parseval's theorem, the sum of squaring each of the in®nite harmonics is inde-
pendent of the optical phase dierence. Section 2 describes the practical conditions, such as the phase modulation index and number of harmonics for ®nite harmonics.
2. Phase-drift suppression 2.1. Theory In a conventional interferometer, the phase dierences / between two arms vary with the phase drifts of the wavelength, the optical phase of the optical source, and the path length in the interferometer. Thus, the total phase drift /d t is represented by /d t ÿ
2p 2p L Dkd Dud DLd ; 2 k k
1
where k is the wavelength, L, the optical path dierence, Dkd , the wavelength drift, Dud , the phase drift in the light source, and DLd , the drift of the optical path dierences. The drift in the optical path dierence is due to thermal drift and/or mechanical vibrations. Wavelength and phase drifts are due to stochastic factors in the light source. We assume that the overall phase drifts change randomly. Assuming that the time delay between two arms is within the coherence time, and sinusoidal phase modulation is used, the heterodyne beat signal IHB t is represented as IHB t I0 cos /0 sin xP t /d t;
2
where I0 is the constant intensity that is the square root of the product of the signal and reference intensity, /0 , the modulation index, and xP , the angular frequency of the phase modulation. Using the Bessel function, the heterodyne beat signal is expanded as IHB t; /0 ; /d
1 X
C2iÿ1 sin 2i ÿ 1xP t
i1
1 X j1
C2j cos 2jxP t;
3
M. Sato et al. / Optics Communications 184 (2000) 95±104
where C2iÿ1 ÿ2I0 sin /d J2iÿ1 /0 ; C2j 2I0 cos /d J2j /0 :
4
This indicates that the heterodyne beat signal consists of an in®nite number of harmonics and that the balance between odd and even order harmonics depends on the phase drift. Here, using harmonics from the ®rst to the nth order, we de®ne the CSI as ICS /0 ; /d v uINTf n1=2g INT n=2 X u X 2I t sin2 / J 2 / cos2 / J 2 / ; 0
i1
d
2iÿ1
0
d
j1
2j
0
5
where INT x is the function that gives the nearest integer below x. The signal was light backscattered or transmitted from a biological sample in our measurements. Thus, it is apparent that the scattered light intensity can be detected through the CSI. For a stabilizing condition represented by INTf n1=2g X
INT n=2 X
i1
j1
2 J2iÿ1 /0
J2j2 /0 ;
the CSI is given by v uINTf n1=2g u X 2 J2iÿ1 /0 ICS /0 2I0 t i1
v uINT n=2 u X J2j2 /0 : 2I0 t
6
or
97
between 0 and 2p were used as phase drift /d . The phase drift was generated 100 times for one modulation index /0 . The modulation index range was 0±10 rad in steps of 0.1 rad. The power spectrum of the harmonics changes due to the phase drift, and thus, CSI also change. The standard deviation and average of the CSI were calculated using randomly generated values for the phase drift. The relative standard deviation (RSD) was then de®ned as the ratio of the CSI standard deviation to the CSI average. It is important to evaluate the RSD practically since it represents the instability of the CSI due to phase drifts. The RSD is shown in Fig. 1 as a function of the modulation index. C1 denotes that the CSI is the signal intensity of the fundamentals. C1±j signi®es that the CSI was computed using harmonics from the ®rst to the jth order as given by Eq. (5). The RSD for C1 was almost constant at around 49% because variations of the standard deviation of CSI synchronously correspond to the CSI averages. Supposing that the phase drift is represented by random variables distributed uniformly from 0 to 2p, the RSD for C1 is obtained at 48.5% according to the theory of stochastic processes. 48.5% is the value corresponding to the simulated result of 49%. This analytically con®rms our simulated results.
7
j1
This indicates that the CSI as an output signal is independent of the phase drift, and its stability is enhanced. Considering just the ®rst and second harmonics, the stabilizing condition is approximately given by J1 /0 J2 /0 . The CSI is then given by 2I0 jJ1 /0 j or 2I0 jJ2 /0 j, as shown in Ref. [14]. 2.2. Simulated results and discussion The CSI and its standard deviation were simulated using Eq. (5). Randomly generated values
Fig. 1. Simulated dependence of the RSD of the CSI on the modulation index. The RSD is the ratio of the standard deviation of the CSI to the CSI average.
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Table 1 Modulation indexes at the minimum relative standard deviation of CSI (simulations) Computing conditions
Modulation index (rad) 1st
2nd
3rd
4th
5th
C1±2 C1±2 a C1±4 C1±6 C1±8
2.6 2.62 2.8 2.8 2.8
4.5 4.47 4.1 4.0 4.0
6.1 6.09 5.6 5.9 5.9
7.7 7.71 7.4 7.6 7.3
9.3 9.29 9.0 8.7 9.0
a
Modulation indexes given numerically.
The RSDs for C1±n with n above 2 changed periodically and were smaller than the RSDs of C1 . The modulation indexes that give the minimum RSDs in Fig. 1 are shown in Table 1, using harmonics up to the eighth order. The stability of the CSI for phase drift is enhanced at these modulation indexes. The modulation indexes given numerically by the stabilizing condition of J1 /0 J2 /0 for C1±2 are also shown in the same table, and those values agree with the simulated ones. The dependence of the RSD on the number of harmonics is shown in Fig. 2. The minimum RSD was 1% on C1±2 for a modulation index of 2.6 rad. This indicates that the stability of the CSI was enhanced 49 times, compared with the RSD of 49% on C1 . The minimum RSD was 0.2% on C1±4
at a modulation index of 2.8 rad. This indicates that the stability of the CSI was enhanced 245 times. Assuming sinusoidal phase modulation in interferometers, the conventional feedback system usually attempts to maximize the output signal at the fundamental beat frequency; in short, at the ®rst peak of the Bessel function J1 /. Comparing the CSI with the signal intensity using a feedback system, the ratio of the CSI of C1±2 to the maximum beat signal intensity at the modulation frequency is given by J1 /0 2:62 rad= J1 /0m 1:90 rad 81%, where /0m is the modulation index used to maximize the beat signal intensity at the modulation frequency. This indicates that the stability was enhanced using PDS, but that the CSI was 81% for the maximum beat signal intensity using the conventional feedback system. 3. Experiments 3.1. Experimental setup The optical measurement system mainly consists of a ®ber-optic Mach±Zehnder interferometer that is fundamentally based on Ref. [16], as shown in Fig. 3. This system was utilized to verify simulated results and to measure OCT images using
Fig. 2. Simulated dependence of the RSD of CSI on the number of harmonics.
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Fig. 3. Experimental setup for an OCT system based on a ®ber Mach±Zehnder interferometer. SLD: superluminescent diode, NDC: ND coupler, CL: coupling lens, RM: re¯ector mirror, PM: phase modulator, HMC: half-mirror coupler, Probe: GRIN lens probe, FC: ®ber coupler, and APD: avalanche photodiode.
PDS. The light source was an SLD with a broad bandwidth of 25 nm (FWHM) and center wavelength of 840 nm. The temperature of the only light source was stabilized at 25°C. The illumination power to the sample was 20 lW. The light beam is launched into two arms by an ND coupler (NDC). The reference arm is comprise, an optical delay and a PM. The optical delay consists of a coupling lens (CL), a re¯ector mirror (RM), and a moving stage. The PM consists of a piezoelectric ®ber stretcher that can control the modulation index by changing the applied sinusoidal voltage of 15 kHz. We ®rst calibrated the peak-to-peak sine voltage applied to the PM. The relation between the modulation index and peak-to-peak voltage of a triangular wave was measured as shown in Fig. 4. As mentioned above, the PM is not usually utilized for axial scanning in OCT because the Doppler shifting is possible by moving a mirror. However, PM was used here in the axial scanning to con®rm the principle of PDS.
The sample arm is comprised of a half mirror coupler (HMC) and an optical probe with a gradient-index (GRIN) lens (1 mm /, 2.6 mm length, 0.25 pitch) for both illumination and collection of backscattered light. The signal light is combined with the reference light by a ®ber coupler (FC) to generate a heterodyne beat signal. The heterodyne beat signal is detected by a Si-avalanche photodiode and is fed into an RF spectrum analyzer only when the total length of the sample arm matches that of reference within the coherence length. A personal computer controls the sample stage in three dimensions and synchronously acquires data from the RF spectrum analyzer. Individual intensities of higher order harmonics in the heterodyne beat signal can also be fed into the computer. The time variations of the heterodyne beat signal from the ®rst to the sixth-order harmonics with time were acquired simultaneously for one modulation index and the CSIs were calculated.
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Fig. 4. Dependence of the modulation index on the applied voltage in the phase modulator.
3.2. Results and discussion An Al coated mirror with 100% re¯ection was used as a sample to verify the simulated results. Two typical power spectra of the heterodyne beat signal are shown in Fig. 5. Odd order harmonics are dominant in Fig. 5(a), and even order harmonics in Fig. 5(b). These two states were randomly exchanged due to phase drifts. The higher order harmonics were below the sixth harmonic of 90 kHz. Therefore, in the following experiments,
we only had to measure the beat signals below the sixth harmonic. The CSIs of C1 , C1±2 , C1±4 , and C1±6 are shown in Fig. 6 as functions of the modulation index. The experimental results almost agree with the simulated ones calculated by Eq. (5). The measured RSDs are shown in Fig. 7 as functions of the modulation indexes for C1 , C1±2 , C1±4 , and C1±6 . The RSD of C1 was almost constant, but the RSDs for C1±2 , C1±4 , and C1±6 changed periodically, as shown in Fig. 7(a). The RSDs are also shown in Fig. 7(b) as functions of the modulation index from 2.0 to 3.2 rad in steps of 0.1 rad for C1 , C1±2 , and C1±6 . It was experimentally veri®ed that the stability of C1±2 at a modulation index of 2.7 rad was enhanced by 6.3 times and the stability of C1±6 at 2.8 rad was enhanced 10 times. The dierences between the experimental and simulated results shown in Fig. 7 are mainly due to the unstable intensities of the light sources. The RSD theoretically indicates the instability of only light intensities with the stable conditions of Eq. (6). The RSD of the light intensity of the SLD was measured at 2.8%. Typical variations of C1 , C2 , C1±2 , and C1±6 over time for modulation indexes of 1.0, 2.5, and 4.0 rad are shown in Fig. 8(a)±(c), respectively. The measurement times and data acquisition intervals were 10 min and 20 s. The variations of the signals were compensated and decreased for C1±2 and C1±6
Fig. 5. Typical power spectra of a heterodyne beat signal: (a) when odd order harmonics are dominant and (b) when even order harmonics are dominant. Modulation index: 2.5 rad, and frequency scale: 10 kHz/div.
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Fig. 6. Dependence of CSI on the modulation index: the modulation frequency was 15 kHz.
at /0 2:5 rad, and at /0 4:0 rad only for C1±6 . These results correspond to those in Fig. 7.
4. Application to optical coherence tomography 4.1. Basic characteristics of an optical coherence tomography system A coherence length lc of 25 lm was obtained from the equation lc 4 ln 2k2 =p Dk [17], assuming a Gaussian spectrum. Thus, the axial resolution of OCT, given by lc =2, was 13 lm. Using the Al-coated mirror as a sample, the axial resolution was obtained at 17 lm (FWHM) from the measured autocorrelation function, as shown in Fig. 9(b). The dierence between the two axial resolutions of 13 and 17 lm may exist primarily because the interferogram was broadened by a dispersion mismatch between the two arms of the interferometer [18], and because the SLD spectrum was not completely Gaussian. We measured the dependence of the beam spot size on the axial position, and it corresponded to theoretical result given by applying the ABCD law [19]. The transverse resolution from the spot size at the beam waist was 7.5 lm (FWHM).
Fig. 7. Dependence of the RSD of CSI on the modulation index: (a) modulation index region from 1.0 to 4.0 rad and (b) modulation index region from 2.0 to 3.2 rad.
4.2. Optical coherence tomography images with phase drift suppression We measured the OCT images as follows: The measured point within a coherent gate was positioned at the beam waist by the optical delay. (i) The sample was next axially scanned to measure the axial pro®le of backscatters. (ii) The sample
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shown in Fig. 9(a) and (b). It is apparent that the stability in C1±2 is enhanced over that in C1 . We next measured cross-sectional images and signal pro®les using a sample of cover glass (145 lm thick) on the slide glasses in C1 and C1±2 , as shown in Fig. 10(a) and (b). The dimensions of the OCT images were 0:3 axial 0:3 transverse mm2 , which corresponded to 60 axial 60 transverse pixels. It took 80 min to acquire the image due to the slow response of the stages. The boundaries between the air and the cover glass and between the cover glass and the slide glass were very noisy, as shown in Fig. 10(a), but this noise was suppressed for Fig. 10(b). The boundaries of the cover glass had a thickness that exceeded the resolution levels, due to a logarithmic scale display. We con®rmed that the PDS technique is valuable for enhancing the stability of output signals in heterodyne detection. Furthermore, it is easy to apply this technique since it requires only a few lock-in ampli®ers or a few band-pass ®lters and a conventional data acquisition system.
5. Conclusion
Fig. 8. Variations of C1 , C2 , C1±2 , and C1±6 with time: modulation indexes of (a) 1.0 rad, (b) 2.5 rad, and (c) 4.0 rad. The measurement time was 10 min.
stage was then moved transversely. Two-dimensional cross-sectional images were created in computers by alternately executing (i) and (ii). Two autocorrelations of SLD with C1 and C1±2 were measured at a modulation index of 2.5 rad, as
We have demonstrated a PDS technique with an increased number of harmonics for enhancing the stability of CSI in a heterodyne beat signal. The stabilizing conditions in PDS were introduced analytically. The modulation indexes that minimize the standard deviations of CSI were determined through simulations for higher harmonics up to the eighth order. The dependences of CSI and its RSD on the modulation indexes were veri®ed experimentally. For example, the stability of CSI was enhanced 49 times for C1±2 at a modulation index of 2.62 rad through simulations, and the stability was experimentally enhanced 6.3 times at a modulation index of 2.7 rad. This PDS was also con®rmed to be useful for applications in OCT. Complicated and expensive feedback systems must be used to stabilize heterodyne beat signals to an order of 10ÿ6 . However, the PDS is simple to realize and is widely applicable to conventional heterodyne detection in a variety of systems
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Fig. 9. Autocorrelations of light sources (a) C1 and (b) C1±2 .
Fig. 10. OCT images and pro®les of cover glass and slide glass: (a) C1 and (b) C1±2 . 1 pixel 5 5 lm2 , and 60 transverse 60 axial pixels.
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