Autocorrelation low coherence interferometry

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Optics Communications 281 (2008) 1991–1996 www.elsevier.com/locate/optcom

Autocorrelation low coherence interferometry Mark D. Modell a, Vladimir Ryabukho b,*, Dmitry Lyakin b, Vladislav Lychagov b, Edward Vitkin a, Irving Itzkan a, Lev T. Perelman a a

Biomedical Imaging and Spectroscopy Laboratory, Beth Israel Deaconess Medical Center, Harvard University, Boston, Massachusetts 02215, USA b Institute of Precision Mechanics, Control of the Russian Academy of Sciences and Department of Optics, Saratov State University, 155 Moskovskaya Ulitsa, Saratov 410012, Russia Received 9 August 2007; received in revised form 22 November 2007; accepted 12 December 2007

Abstract This paper describes the development of a new modality of optical low coherence interferometry (LCI) that is called autocorrelation LCI (ALCI). The ALCI system employs a Michelson interferometer to measure longitudinal autocorrelation properties of the sample optical field and does not require a reference beam. As the result, there is no restrictions applied on the distance between the sample and the ALCI system, moreover, this distance can even change during the measurements. We report experiments using a proof-ofprinciple ALCI system on a multilayer phantom consisting of three surfaces defining two regions of different refractive indices. The experimental data are in excellent agreement with the predictions of the theoretical model. Ó 2007 Elsevier B.V. All rights reserved.

1. Introduction Low coherence interferometry (LCI) or optical coherence tomography (OCT) relies on the interference of two optical fields, one backscattered from a turbid sample and the other reflected from a reference mirror [1–6]. It provides two-dimensional cross-sectional images of biological tissue with near-micron resolution and up to a millimeter penetration depth. In the standard LCI and OCT setups the sample is located in one of the arms of the interferometer and the area of interest should be in the coherent focus of the instrument within the range of the scanning of the reference mirror. Therefore the distance to the sample in the LCI instrument is limited by that scanning range. However, in conducting in vivo measurements the target may move rapidly during the scan, even beyond the scanning range of the instrument. To compensate for motion artifacts the optical paths in the arms of the interferometer and the depth of

*

Corresponding author. E-mail address: [email protected] (V. Ryabukho).

0030-4018/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2007.12.043

focus of the objective need to be adjusted in real time, an arrangement that is often difficult to implement. This problem is typical to any LCI measurements which employs a reference arm, including time domain setups, such as recently developed for path-length-resolved measurements of multiply scattered light in dynamic turbid media phase modulated LCI technique [7–9], and spectral domain OCT measurements [6,10,11], where constraints on the optical path difference in the sample and reference arms of the interferometer come from the finite spectral resolution of the instrument. Since the period of interferometric oscillations in the spectrum is inversely proportional to the optical path difference, if the path difference in the spectral domain OCT system is too large, oscillations may not be resolvable by either the spectrograph or the linear detector array. In the swept source OCT systems [5,6,12–17], limitations on the maximum optical path difference come from the finite bandwidth of the source tuning range and related accuracy of the tuning. To compensate for the large optical path differences the bandwidth of the source should be sufficiently narrow and, as a result, spectral tuning needs to be more precise. In addition, since the frequency of the interferometric signal in spectral

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domain OCT is proportional to the product of the spectral tuning sweep speed and optical path difference, this results in a limitation on the sweep speed and as a result on the overall speed of the instrument. A number of attempts have been made to overcome these limitations in standard time domain setups [4,5,18] and spectral setups [5,19–23] where the sample is located outside of the interferometer or outside of the spectral system. To overcome the problems outlined above we developed a new type of the LCI system that has no reference arm whatsoever. We call it autocorrelation LCI or ALCI. The ALCI system employs a Michelson interferometer to measure the longitudinal autocorrelation properties of the sample optical field and has no restrictions on the distance between the sample and the ALCI system. Moreover, this distance can continuously change during the measurements. 2. Theoretical principles of autocorrelation low coherence interferometry The basic principle of the ALCI system can be better understood by using the schematic provided in Fig. 1. Here the sample is illuminated by a low coherence light source

with a wide frequency spectrum and the light field reflected from the sample is directed to the entrance of the Michelson interferometer. The interferometer has two flat mirrors, one of them fixed and the other one capable of scanning the field along the optical axis. By moving the second mirror we delay part of the sample optical field and by combining it with the optical field reflected from the first mirror we create an interference pattern on the detector related to the longitudinal autocorrelation function of the sample field. Since the autocorrelation function of the sample field is related to the optical properties of the sample it should be possible in principle to reconstruct the composition of the sample from the ALCI measurements. To describe ALCI we will employ a simple scalar model in which the field Es(z, t) reflected from an arbitrary point z along the optical axis inside the layered or turbid sample can be approximated by  Z z    2zn Es ðz; tÞ ¼ rðzÞ exp 2 ltr ðez Þdez U 0 t  c 0  exp ½ið2k 0 nz  x0 tÞ:

ð1Þ

Here U0(t) is the complex amplitude of the incident field originated at the low coherence source, x0 and k0 are the angular frequency and the wave number of the incident field, c is the speed of light, and n is the mean refractive index of the sample. Here the optical properties of the sample are defined by the local amplitude reflection coefficient r(z) and transport coefficient ltr(z) which describes the attenuation of light in the sample. To simplify the equation we combine the terms that describe the optical properties of the sample in a single term  Z z  RðzÞ ¼ rðzÞ exp 2 ltr ðez Þdez : ð2Þ 0

The total sample field can be written as sum or an integral of the fields (1) over all points inside the sample. In the Michelson interferometer this total optical field will be split in two and then re-combined with a time delay s = 2Dz/c between two identical parts of the same field. Here Dz is the difference in the positions of the two mirrors in the interferometer. The intensity recorded by the photodetector will than be equal to * Z 2 + Z 1  1  0 0 00 00 I ð2DzÞ   Es ðz ; tÞdz þ Es ðz ; t þ 2Dz=cÞdz  ; 0

0

ð3Þ where the angle brackets denote a time-average [24]. The cross term in the Eq. (3) describes the autocorrelation properties of the sample field and is equal to Z 1Z 1 

eI ð2DzÞ  2Re Es ðz0 ; tÞEs ðz00 ; t þ 2Dz=cÞdz0 dz00 : 0

0

ð4Þ Fig. 1. Basic principle of the ALCI system. Fields originating from two arbitrary points inside the sample are split and delayed in the interferometer. The four fields interfere at the ALCI detector.

Using the expression for Es ðz; tÞ from Eq. (1) we get the following expression for the autocorrelation term eI ð2DzÞ

M.D. Modell et al. / Optics Communications 281 (2008) 1991–1996

eI ð2DzÞ  2Re

Z 1Z

1

Rðz0 ÞRðz00 ÞGð2Dz þ 2ðz0  z00 ÞnÞ

 exp½ik 0 2ðz0  z00 Þn  ik 0 2Dzdz0 dz00 ; 0

0

ð5Þ

 where GðDÞ  U 0 ðtÞ  U 0 ðt þ D=cÞ is the temporal coherence function of complex amplitude U 0 ðtÞ of the incident

light2 [24],  which approximately equals GðDÞ  G0 exp D =l2c , where D is an optical path difference and lc is the temporal coherence length of the source. The autocorrelation term eI ð2DzÞ describes the longitudinal autocorrelation properties of the sample optical field. Substitution of variables n = z0  z00 and z ¼ ðz0 þ z00 Þ=2 provides greater physical insight into the meaning of the function eI ð2DzÞ Z 1

eI ð2DzÞ  2Re BðnÞGð2Dzþ2nnÞexp ½ik 0 ð2nn2DzÞdn :

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tor3  0.9, where the refractive indices of the layers are n12  1.4 for the fluoroplastic film and n23 = 1.0 for the air gap. The thicknesses of the layers are d12  50 lm and d23  145 lm. The signal measured with the ALCI system is presented in Fig. 2b. The leftmost interference peak is related to the zero optical path-length difference in the interferometer, 2Dz = 0. The side peaks are related to the interference of waves reflected from different surfaces inside the phantom. The first side peak is related to the interference of the waves reflected from the surfaces of the thinnest optical layer

0

ð6Þ 

 R1 Here BðnÞ ¼ 0 R z þ n2 R z  n2 dz is the autocorrelation integral of the optical structure of the sample. Thus the ALCI signal eI ð2DzÞ is the convolution of the autocorrelation integral of the optical structure of the sample and the complex degree of coherence of the source CðDÞ ¼ GðDÞ expðik 0 DÞ. It also confirms that, by analogy to the regular time domain LCI, the spatial resolution of the ALCI technique is defined by the temporal coherence length of the source lc. Since the low coherence light source is characterized by short coherence length lc the temporal coherence function GðDÞ can be approximated by a d-function, GðDÞ  G0 dðDÞ and, in principle, it should be possible to reconstruct R(z) and hence local refractive indices from the ALCI measurements. 3. Experiment and comparison with theory To test the feasibility of the ALCI we developed a proofof-principle ALCI system and tested it on a turbid phantom with two refractive and one reflective surfaces. The system shown in Fig. 1 consisted of a super-luminescent diode (SLD) source (Inject, Ltd) with coherence length lc  15 lm emitting at a central wavelength of 850 nm and a Michelson interferometer. The interferometer has a continuously scanning mirror operating at a low frequency of 3 Hz with a sufficiently large scanning amplitude l0  1 mm. Note that this scanning amplitude is significantly larger than the various optical thicknesses of the phantom. The phantom used in the experiments consists of a 50 lm thick fluoroplastic film placed on the surface of a metal mirror. The fluoroplastic film is turbid and the transport coefficient in this layer ltr  4 mm1. There is an air gap between the top of the mirror and the bottom of the fluoroplastic film which results in the phantom having two refractive surfaces with amplitude reflection coeffi1 12 cients r1 ¼j nn12 j 0:17 and r2 ¼j nn2323 n j 0:17, and a mirþn12 12 þ1 ror with amplitude reflection coefficient at 850 nm equal

Fig. 2. Structure of the phantom with two refractive and one reflective surfaces (a). The signal measured with the ALCI system (b). The longitudinal autocorrelation term calculated for the same phantom using the theoretical model and nominal thicknesses and refractive indices (c).

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within the phantom, the fluoroplastic film. The second side peak is related to the interference from the surfaces of the thicker phantom layer, the air gap. Finally the position of the last side peak is determined by the combined thickness of both layers. The high frequency oscillations of the signals in Fig. 2b are not resolved because of the high density of those oscillations. To model this signal we applied Eq. (6) to the phantom described above. The light absorption within the phantom layers is negligible. In this case the autocorrelation term eI ð2DzÞ is equal to eI ð2DzÞ 

3 X

R2j jGð2DzÞj cos ½2k 0 Dz

j¼1

þ R1 R2 jGð2Dz 2d 12 n12 Þj cos ½2k 0 Dz þ 2k 0 d 12 n12  þ R1 R3 jGð2Dz 2ðd 12 n12 þ d 23 n23 ÞÞj cos ½2k 0 Dz þ 2k 0 ðd 12 n12 þ d 23 n23 Þ þ R2 R3 jGð2Dz 2d 23 n23 Þj  cos ½2k 0 Dz þ 2k 0 d 23 n23 ;

ð7Þ 2

where the coefficients R1 = r1, R2 ¼ r2 ð1  r1 Þ exp ð2ltr 2 2 d 12 Þ, and R3 ¼ r3 ð1  r2 Þ ð1  r1 Þ exp ð2ltr d 12 Þ. Here the second and third terms in the cosines are just the initial phase terms and can be neglected. In this case 3 n X eI ð2DzÞ  R2j jGð2DzÞj þ R1 R2 jGð2Dz 2d 12 n12 Þj j¼1

þ R1 R3 jGð2Dz 2ðd 12 n12 þ d 23 n23 ÞÞj o þ R2 R3 jGð2Dz 2d 23 n23 Þj cos ð2k 0 DzÞ:

ð8Þ

The autocorrelation term calculated using Eq. (8) with the values for positions of the refractive surfaces and the refractive indices of the layers provided above is presented in Fig. 2c. Note that the theory is in good agreement with the experiment. For example, the theory predicts that the peaks related to the refractive and reflective surfaces of the phantom are located at Dz ¼ 0; d 1 2n12 ; ðd 12 n12 þ d 23 n23 Þ and d 23 n23 and their positions are in excellent agreement with the experiment. The amplitudes of the first and second side peaks are also in excellent agreement with the experiment. The small discrepancy in the experimental and theoretically predicted amplitudes of the third side peak may be due to some phase mismatch of the interfering fields on the photodetector aperture because of the presence of slightly non-parallel surfaces in the phantom. A similar effect can be observed by inducing a small inclination in the interferometer mirrors which results in changes in the relative amplitudes of the peaks. We also found that the relative amplitudes of all side peaks measured in the experiment are always slightly smaller than those predicted by theory. This may be due to several effects. Small variations in the film thickness and its refractive index could result in a random speckle-type modulation of the waves reflected from the surfaces within the phantom resulting in the reduction of the interference signal on the detector aperture. Also, as shown below the amplitudes of the peaks

Table 1 Nominal and reconstructed parameters of the phantom

Nominal phantom parameters Reconstructed phantom parameters Difference from nominal

d12 (lm)

d23 (lm)

n12

n23

50.0 49.3 0.7

145.0 143.5 1.5

1.40 1.37 0.03

1.00 1.00 0

are a strong function of the position of the reflecting surfaces and the refractive indices in the phantom. Thus incomplete knowledge of the values used in the calculation would result in the calculated amplitudes differing from the experimental values. This dependence of the amplitudes on the precise position of the refractive surfaces within the sample provides a method for reconstructing the actual position of the refractive surface. The arguments of the coherence function in Eq. (8) define the positions of the refractive surfaces and the amplitudes define the refractive indices of the layers. Thus, we can reconstruct those values from the experimental data. We calculated the relative amplitudes of the side peaks A1, A2, and A3 by dividing the amplitudes of those peaks in Fig. 2b by the amplitude of the central interference peak. We then compared those amplitudes with the amplitudes predicted by the model Eq. (8). This gives the following system of equations for the coefficients R1, R2, and R3 8 RR 1 2 ¼ A1 > > R21 þR22 þR23 > < R2 R3 ¼ A2 : ð9Þ R21 þR22 þR23 > > > : 2 R1 R2 3 2 ¼ A3 R þR þR 1

2

3

From this system of equations we reconstructed the refractive indices of the layers and from the positions of the peaks we found the thicknesses of the layers. The results of those calculations are presented in Table 1, where they are compared with the nominal parameters characterizing the phantom. The reconstructed parameters are in good agreement with the phantom parameters. Using the reconstructed parameters, the amplitude of the last side peak is in much better agreement with experiment. 4. Discussion and conclusion In regular LCI and OCT the order of the peaks and valleys in the interference signal follows the order in which the corresponding structures appear in the sample. On the contrary in the ALCI signal the position of the peaks and valleys is related to the thickness of the structures with their order seemingly unrelated to the actual position of the structures. Thus the first side peak in the experiments with the phantom is formed not by the most superficial layer, but by the thinnest layer in the phantom. This unique feature of ALCI can be an advantage but also a disadvantage of the technique. In applications where the overall structure of the sample is known, for example in ophthalmology,

M.D. Modell et al. / Optics Communications 281 (2008) 1991–1996

where the order of the layers in the eye is well studied and understood and where precise measurement of thicknesses of some of the layers is an important task [2,4–7] ALCI would provide very transparent and unambiguous information. Another use of ALCI could be measuring properties of a highly dispersive layer located deep within a sample. In that situation regular time domain LCI and OCT systems employ a compensating medium within the reference arm of the interferometer with the same dispersive properties as the layer in the sample [25–27]. ALCI does not need any compensating medium since both interference waves in ALCI pass through the same dispersive layer within the sample. This is especially important if the dispersive properties of the layer are not known. However, when the structure of the sample is not known a priori the reconstruction of the spatial distribution of the refractive indices within the sample from the ALCI signal might not be as straightforward. To demonstrate that ALCI not only provides the thicknesses and the refractive indexes of various layers within the turbid sample but, indeed, uniquely depends on the order those layers are positioned within the sample we calculated the ALCI signal for two samples consisting of two layers placed on a mirror in a manner similar to the way our phantom was constructed. The first sample consists of a 40 lm thick layer with refractive index 1.5 placed above a 30 lm thick layer with refractive index 1.2. Both layers are placed on a mirror. The second sample has the same two layers but their order is reversed with the 30 lm layer with refractive index 1.2 placed above the 40 lm thick layer with refractive index 1.5. If the ALCI signal would only depend on the thickness of the layers but not on their position, the signals from those two samples should be identical. However, as seen in Fig. 3, although the locations of the peaks are the same, the amplitudes are quite different. This dependence of the amplitudes of the ALCI signal on the location of the layers provides enough information to uniquely reconstruct the spatial distribution of the refractive indices within the turbid sample. Indeed, the reconstruction of the phantom properties in the previous section of this paper shows how it can be done. To summarize, ALCI has several advantages over regular LCI and OCT systems. ALCI does not constrain the distance between the instrument and the sample as does LCI. This distance can change during the measurement time and any movement and/or vibrations in the sample which can be a major problem during in vivo data collection does not affect these results. In addition, because of that insensitivity to motion artifacts, the requirements on the interferometer needed to perform ALCI are not as stringent as in the standard LCI setup. On the other hand, the absence of the reference arm in ALCI is a mixed blessing and it creates certain disadvantages when compared to conventional LCI and OCT systems. The strength of the interference signal in ALCI is smaller than that in regular LCI and OCT systems. As the result it would likely have a smaller dynamic range and decreased penetration depth.

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Fig. 3. ALCI signals calculated for two samples consisted of two layers placed on a mirror. (a) Sample with a 40 lm thick, 1.5 refractive index layer above a 30 lm thick, 1.2 refractive index layer. (b) Sample with a 30 lm thick, 1.2 refractive index layer above a 40 lm thick, 1.5 refractive index layer.

However, for many applications the advantages of ALCI can outweigh this disadvantage. The results reported here indicate the potential and promise of ALCI measurements. ALCI should be especially promising for in vivo measurements in clinical settings. Because both the distance to the target and motion artifacts are not critical to the ALCI system it can be easily adapted to an endoscopic fiber instrument which can be used during routine endoscopic or videoscopic procedures in various organs. Acknowledgments The authors would like to thank Dr. V.V. Tuchin for fruitful discussions. This study was supported in part by the National Institutes of Health Grant RR017361, the National Science Foundation Grant BES0116833, the US Civilian Research and Development Foundation Grant RUX0-006-SR-06 (ANNEX BP1M06), the Russian Foundation for Basic Research Grant 05-08-65514-a and the

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