快速掃瞄式光學延遲線為基礎之相位對比量測法

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(1)國立台灣師範大學光電科技研究所碩士論文. National Taiwan Normal University Institute of Electro-optical and Technology Master Thesis. 快速掃瞄式光學延遲線為基礎之相位對比量測法 RSOD-based phase contrast measurement. 指導教授 : 郭文娟博士研究生 : 周明諭中華民國九十七年七月 July 2008.

(2) 致謝感謝所有陪我走過兩年碩士生活的家人、師長、同學、朋友們，在兩年的研究生活中，有困難繁雜的實驗過程，也有得到結果的收穫喜悅，中途雖然遇到許多的問題，但藉由許多人的幫助討論而一一排解，讓我更有勇氣走向下一步。其中最要感謝的是我的指導教授郭文娟老師，總是提供新的想法不斷的與我們討論，也鼓勵同學們互相討論，並且訓練我們能夠自己解決問題的能力，讓我學習到與大學不同的自主性之外，也體會到一個團隊互相幫忙的精神。另外要感謝的是我的直屬學長大仔文竑學長，非常細心的敎我實驗，雖然已經畢業了，仍在百忙之中抽空回來與我討論實驗上的問題，也很感謝實驗室的創室元老，奐章學長和勝宗、博年、大熊學長，有了他們的努力才讓實驗室順利的運轉。感謝我的同學們：瑋晨，常常指正我的缺點，在實驗上也幫了我很大的忙，尤其時電路部分；富傑，總是很熱心的幫我想調光的問題；哲泓，程式部份找他討論就可搞定。還有四位學弟妹們；瓊瑀，在碩二下加入我的實驗，替我分擔了一些實驗上的工作；協誠，常常替別人著想，是實驗室的氣氛製造者；惠雯，總是可以聊些五花八門有趣的事；孟翰，很熱心的幫學長姐分擔一些雜事，感謝他們讓實驗室像個大家庭一樣。感謝我的大學同學，小可愛、思寰、王子及女友、孟璇、小黑，常常一起聚會吃飯，聊大家的研究生活並互相鼓勵，也謝謝我的高中死黨阿旦，常常找我去戶外走走，看許多的展覽，紓解生活上的壓力，有你們這些朋友，我真的很幸福。最後再次謝謝我的家人，有他們的支持，我才能專心的做研究，並且順利的完成碩士學位。. v.

(3) 中文摘要. 此篇研究中，我們發展一套平衡式光學同調干涉儀並且結合快速掃描光學延遲線(RSOD)和馬赫-詹德干涉儀(Mach-Zehnder interferometer)來產生相位資訊。利用相位解析 OCT (phase-resolved OCT)成功的展現奈米尺度的微小相位差異影像，這是以往傳統強度式的 OCT 所無法偵測到的。相位靈敏度的表現在相位解析 OCT 中是一個很重要的因素，相位的飄動可能是因為開放式空間環境擾動所造成的，當我們結合共路徑架構和平衡式偵測系統，共同的雜訊就可被減掉，並且改善相位靈敏度. vi.

(4) Abstract. In this research, we have developed a balanced optical low coherence interferometer to yield phase information by incorporating the rapid-scanning optical delay lines (RSOD) into a Mach-Zehnder interferometer. We have shown that small optical path differences with nanometer scale, which are invisible in conventional intensity-based OCT image, can be successfully imaged by phase-resolved OCT. The phase sensitivity is an important performance factor in phase-resolved OCT. The phase drift caused by environmental disturbance is due to the free-space system configuration. When we combined common-path configuration and balanced detection system, the common noise was canceled and the phase sensitivity can be improved.. vii.

(5) Table of Contents. Chinese Abstract …………………………………………………………………..… v English Abstract ………………………………………………….……………..……vi Chapter 1. Introduction ……………………………………………..……………...1. 1.1 Motivation …………………………………………………………………1 1.2 Brief review of time domain phase contrast optical coherence Tomography ………………………………………………….……………2 1.3 Objective and organization of the thesis ………………………………..…7 Chapter 2. Principle ………………………………………………….……………..9. 2.1. Mach-Zehnder interferometer ………………………………………….....9. 2.2. Balanced detection system ………………………………………………10. 2.3. Hilbert transformation……………………………………………………16. Chapter 3. Material and Method ………………………………………………….20. 3.1. Experimental setup ………………………………………………….…...20. 3.2. Wollaston and Nomarski prism ………………………………………….23. Chapter 4. Results ………………………………………………….…….…….....27. 4.1. Measurement of unbalanced and balanced detection systems…………...27. 4.2. Phase stability test ………………………………………………….……28. 4.3. Surface Displacement Measurements ...…………………………………29. 4.4. Normaski experiment ……………………………………………………31. Chapter 5. Discussion ………………………………………………….………….34 viii.

(6) Chapter 6. Summary ………………………………………………….…………...49. References ………………………………………………….……………..…………50. ix.

(7) List of Figures. Fig. 1.1.. Schematic of the OCT setup for differential phase contrast tomography.... 2. Fig. 1.2.. Schematic setup for DPC-OCT ……………………………………………4. Fig. 1.3.. Differential phase-contrast optical coherence microscopy (DPC OCM) ….5. Fig. 1.4.. Birefringent-fiber-based dual-channel optical low-coherence reflectmeter. …………………6. Fig. 2.1.. The Mach-Zehnder interferometer ……………………………………….10. Fig. 2.2.. Schematic of different the fiber-based interferometer configurations …...12. Fig. 2.3.. Semicircle contour ………………………………………………….……17. Fig. 3.1.. Schematic of common-path balanced optical coherence tomography system that incorporates with a RSOD ……………………………………….…21. Fig. 3.2.. The Wollaston prism ………………………………………………….…23. Fig. 3.3.. Beam splitting by a Wollaston prism ……………………………………24. Fig. 3.4.. The Nomarski prism ……………………………………………………..25. Fig. 3.5.. The specification of Nomarski prism …………………………………….26. Fig. 4.1.. Measured SNR from (a) signal detector, and (b) balance detector (BD1) …………………………………………27. x.

(8) Fig. 4.2.. Measured phase difference when the OCT system using in a common-path configuration. ………………………………………………….………...29. Fig. 4.3.. Resolution target measurement ……………………………….…………30. Fig. 4.4.. OCT test sample: schematic of the object structure ……………………..31. Fig. 4.5.. The interferometric signals of two orthogonally polarized beams ………32. Fig. 5.1. Dual channel phase-sensitive optical low coherence tomography ………...35 Fig. 5.2. DBHI of optical referenced interferometer setup …………………………37 Fig. 5.3. (A) FD SDPM interferometer. (B) SS SDPM interferometer ………….…39 Fig. 5.4. Schematics of the SD-OCPM system …………………………………..…40 Fig. 5.5.. Full-field swept-source OCT setup with a 4f common path interferometer. …………………………………42. Fig. 5.6.. Processing steps to acquire phase-sensitive data using the full-field swept -source OCT system. …………………………………………………….42. Fig. 5.7.. Setup for phase-sensitive OCT measurements …………………………...43. Fig. 5.8. Schematic of SD-OCP/MPM ……………………………………………..45 Fig. 5.9. Spatial and temporal phase stability of SD-OCP–MPM ………………….45 Fig. 5.10.. Dual beam heterodyne FDOCT ………………………………………...47. Fig. 5.11.. The standard deviation of the phase fluctuations, the dual beam signal (red) ,the standard setup (blue) …………………………………………..48. xi.

(9) List of Tables. Table 1. Displacement Sensitivity Using Conventional and FDML Lasers ………...44. xii.

(10) Chapter 1. Introduction. 1.1 Motivation. It is well known that from microscopy, many objects, especially several biological tissues, consist of rather transparent material or subwavelength changes in optical path length. These biological specimens belong to the class of phase objects have very poor contrast while imaged on a pure intensity basis that are difficult to study by using an ordinary microscopy. Before the advent of phase contrast microscopy (PCM), such objects were rendered visible by artificial staining. However, it is not possible to effectively stain living cells and tissues without causing their death or changing their structure. Besides, traditional PCM techniques, such as differential interference contrast and Zernike phase contrast, do not provide a direct quantitative measurement of the spatially varying phase in a biological specimen such as a cell. Optical coherence tomography (OCT), based on low coherence interferometry, is a powerful tool that can support non-contact and high-speed tomographic imaging in biological tissues [1]. The technique measure the intensity of backreflected or backscattered optical radiation. Recently, phase measurement and imaging techniques have gained much attention because of their high sensitivity to longitudinal displacement. The advantage of phase detection of the interference signal allows us to 1.

(11) measure very small distance changes over rather large distances. Thus, small optical path differences, which are invisible in conventional intensity-based OCT images, can be successfully imaged by phase contrast OCT. Moreover, phase information can be used to quantify other physiological parameters including birefringence, blood flow, or spatial phase variation [2-4].. 1.2 Brief review of time domain phase contrast optical coherence Tomography. Figure 1.1 shows a schematic of differential phase measurement setup which was previously reported by Christoph K. Hitzenberger in 1999. [2]. Fig. 1.1. Schematic of the OCT setup for differential phase contrast tomography developed by Christoph K. Hitzenberger.. 2.

(12) In this setup, a Wollaston prism located in the focal point of a lens splits the sample beam into two orthogonally polarized beams. The two beam components are reflected at several layers in the sample and recombined as one object beam at the Wollaston prism. The phase difference, ∆φ , between the two parts of the sample beam is measured in the same way as the phase retardation caused by a birefringent sample in polarization-sensitive OCT[5, 6]. Therefore, ∆φ as a function of depth in the object is calculated as. ∆φ = arctan( I 2 I1 ) .. A drawback of this method is that the phase information was calculated from the amplitudes of orthogonal polarization states. If the sample itself changes the polarization state of the beams (ex. birefringent or scattering samples), signal and image interpretation become complicated. The accuracy of this technique can be estimated from the residual phase error obtained at the plane anterior surface of the test object. These residual errors are ±5° and are probably caused by nonoptimized polarization components and imperfect alignment of some of these components.. At. a wavelength of 830 nm this residual error corresponds to a distance resolution of 11.5 nm.. 3.

(13) In 2001, Markus Sticker et al. presented a differential phase-contrast OCT (DPC-OCT) system (as shown in Fig. 1.3) used for quantitative imaging of pure phase objects [7]. They demonstrated the DPC-OCT imaging through scattering layers and quantified the performance and limitations of the method.. (a). (b). (c). Fig. 1.2. (a) Schematic setup for DPC-OCT, (b) schematic of the object structure, (c) phase image.. The measured interference signals use the Hilbert transformation. From the difference between the two phase functions, ∆φ ( z ) = φ2 ( z ) − φ1 ( z ) .. 4.

(14) The corresponding optical path-length difference can be calculated as ∆z = λ∆φ 720° .. Repeated measurements show a phase fluctuation of only ±2°, or ±2.3 nm. Thus, quantitative measurements with a precision of a few nanometers are possible in transparent media and through scattering layers.. However, the distance between the focal spots of the sample beams limited the transversal resolution of DPC-OCT to 17.5 mm. Other disadvantages of this method are the generation of ghost images and the dependence of the image contrast on the direction of beam separation. Therefore, Markus Sticker et al. presented a DPC-OCM setup that overcome some of these problems [8].. Fig. 1.3. Differential phase-contrast optical coherence microscopy (DPC OCM).. 5.

(15) In Fig. 1.3, the two sample beam components have orthogonal polarization states, variations of sample birefringence would also give rise to image contrast with this method. Optical path-length differences of as little as ~ 2 nm were measured and an advantage of DPC-OCM compared with the usual PCM was expected if imaging was to be performed through scattering layers.. Besides the free-space configuration , Digant P. Dave’ and Thomas E. Milner developed. a. birefringent-fiber-based. dual-channel. optical. low-coherence. reflectometer (as shown in Fig. 1.2) to extract phase information[9].. Fig. 1.4. Birefringent-fiber-based dual-channel optical low-coherence reflectometer. 6.

(16) The phase of each channel was determined by the arctangent of the measured signal divided by its Hilbert transform. For a fixed path-length separation between two channels, 50 scans recorded every 5.56 ms (180 Hz) gave a standard deviation of 0.0121 rad in ∆φ . The displacement sensitivity can achieve 1.25 nm and the advantage of this system was no cross talk exists between channels in the sample because the two polarization modes are completely decorrelated.. 1.3 Objective and organization of the thesis. Because the phase vibrations may be caused by environmental perturbations, including temperature fluctuations, air currents, vibration, and moving part of scanning and phase modulation. In order to improve the phase sensitivity, we developed a rapid-scanning optical delay line (RSOD)[10] based optical low coherence interferometers to yield phase information by incorporating the balanced detection technique into a Mach-Zehnder interferometer. The thesis is organized as follows:. Chapter two introduces the principle of Mach-Zehnder interferometer, Balanced detection system, and Hilbert transformation. Chapter three shows our experimental setup. Chapter four demonstrates signal-to-noise ratio (SNR) improvement in. 7.

(17) balanced detection system. Phase stability was also test in a common-path configuration. Finally, a surface displacement measurement of a pure phase object was presented. Chapter five discusses the advantages and disadvantages of our proposed system. Chapter six is the summary of the thesis.. 8.

(18) Chapter 2. Principle. 2.1 Mach-Zehnder interferometer. The Mach-Zehnder interferometer consists of two beamsplitters and two totally reflecting mirrors[11]. As shown in Fig. 2.1. The two waves within the apparatus travel along separate paths. A difference between the optical paths can be introduced by a slight tilt of one beamsplitters and enter two detectors. In contrast to the Michelson interferometer, there are two output ports. Since the two paths are separated, the interferometer is relatively difficult to align. For the same reason, the interferometer finds myriad applications. For example, the Mach-Zehnder interferometer is a device used to determine the phase shift caused by a small sample which is placed in the path of one of two beams from a coherent light source. If a sample is placed in the path of the sample beam, the intensities of the beams entering the two detectors will change, allowing the calculation of the phase shift caused by the sample.. 9.

(19) Fig. 2.1. The Mach-Zehnder interferometer.. 2.2 Balanced detection system. Figure 2.2 illustractes two methods of detection system: the single detection system as shown in Fig. 2.2 (a) and balanced heterodyne detection system as shown in Fig. 2.2 (b)(c)[12]. A low-coherence light source is split into the sample in one arm and a reference delay line in the other by using fiber coupler with a 50/50 splitter ratio as the beam-splitting element. In configuration (a), light retroreflected from the reference and the sample is recombined at the beam splitter, and half is collected by a photodetector in the detection arm of the interferometer. Half of the light is returned toward the source, where it is lost. In configuration (b), the optical circulator enables Michelson interferometer together with balanced heterodyne detection to be constructed. The three port optical circulator is a nonreciprocal device that couples 10.

(20) light that is incident upon port I to port II and light that is incident upon port II to port III. In configuration (c), the balanced heterodyne detection system uses a Mach-Zehnder interferometer[13].. The balanced receiver is useful for optical heterodyne detection (as in OCT), summing the heterodyne (interferometric) signals in two detectors while suppressing the common mode component of excess intensity noise. The most commonly used input elements to detector are optical fiber couplers or prism couplers.. (a). (ai). (b). E1. E2. D1. D2. (bi). 11.

(21) (c). D1. D2. Fig. 2.2. Schematic of different the fiber-based interferometer configurations: (a) standard Michelson interferometer, (b) balanced Michelson interferometer, (c) balanced Mach-Zehnder interferometer. α / 1 − α and 50/50 are beam splitters; D1, D2, photodetectors ; I, II, III, circulator ports. In free space, (ai) the single detector prism coupling. (bi) balanced detectors prism coupling.. Assume that inside the substrate optical coupler, the field that crosses the guide from the upper to lower or from the lower to upper guide experiences a 90° phase shift compared to the field that goes straight through. The 90° phase shift will be accounted for by changing the cosine function into a sine function. The input field to detectors are expressed as. E1 = S + jR =. 1 [ES (t )cos ωt + ER (t )sin ω (t − τ )] 2. 2.2.1. E2 = jS + R =. 1 [ES (t )sin ωt + ER (t )cos ω (t − τ )] 2. 2.2.2. 12.

(22) where ES (t ) and ER (t ) are the sample and the reference field, respectively. The factor 1. 2 is necessary because the light power is split 50-50. The output currents. i1 and i2 from detector D1 and D2 are. 1 2 ⎤ ⎡1 2 i1 = K1 ⎢ ES (t ) − ES (t )ER (t )sin ωτ + ER (t )⎥ 2 ⎣2 ⎦. 2.2.3. 1 2 ⎤ ⎡1 2 i2 = K 2 ⎢ ES (t ) + ES (t )ER (t )sin ωτ + ER (t )⎥ 2 ⎣2 ⎦. 2.2.4. Eq. 2.2.3 or Eq. 2.2.4 alone is expression for the output current from the single detector. If the intensity of the local oscillator laser fluctuates, the detector output fluctuates accordingly, and the fluctuations in ER (t ) become a source noise. In 2. double balanced detector, however, i1 is subtracted i2 . This means both first and third terms in Eq. 2.2.3 and Eq. 2.2.4 cancel each other, provide the quantum efficiencies of the two detectors are well balanced , K1 = K 2 = K , and the output from the double balanced detector is. i = 2 KES (t )ER (t ) sin ωτ. 2.2.5. Fluctuations in ER (t ) will cause i to fluctuate, but the amount is different for two cases. Eq. 2.2.3 for the single detector contains a term proportional to E 2 R (t ) , whereas Eq. 2.2.5 for the balanced heterodyne detector is proportional to ES (t )ER (t ) .. 13.

(23) Normally, ES (t ) is significantly weaker than ER (t ) , so that the effect of local oscillator fluctuations is significantly improved over the case of the single detector.. For the noise of a signal detector (Fig. 2.2(a)), the total photocurrent variance is given by. σ i 2 = σ re 2 + σ sh 2 + σ ex 2. 2.2.6. We express noise sources in terms of the photocurrent variance σ i [13]. The main 2. noise sources are receiver noise σ re , shot noise σ sh , and excess intensity noise σ ex . 2. Receiver noise σ re. 2. 2. 2. may be modeled as thermal noise in a resistance-limited. receiver, or, for commercial photoreceiver module, the receiver noise can be calculated directly from the manufacturer’s specification. The shot noise σ sh. 2. is. given by σ sh = 2qI dc B , where q is the electronic charge, I dc is the mean detector 2. photocurrent, and B is the electronic detector bandwidth. Excess photon noise σ ex. 2. is given by σ ex = (1 + V 2 )I dc B / ∆ν , where V is the degree of polarization of the 2. 2. source and ∆ν is the effective linewidth of the source, defined as in reference[14]. We define SNR = isignal. 2. σ i 2 . The mean-square signal photocurrent in a signal. detector is. SNRsingle - detector =. isignal. σ i2. 2. =. (ES (t ) ER (t ) sin ωτ )2 σ re 2 + σ sh 2 + σ ex 2. 14. 2.2.7.

(24) If balanced heterodyne detection is used, excess photon noise is largely canceled. Taking into account extra retroreflected power from the sample arm, however, a component of the excess photon noise remains which is called beat noise σ be . Shot 2. noise in each of the detectors composing the balanced receiver is independent, so their variances add. The total photocurrent variance in the case of balanced heterodyne detection becomes. (. σ i 2 = 2 σ re 2 + σ sh 2 + σ be 2. ). 2.2.8. Note that noise variance here should be calculated from values of total photocurrent (from both detectors) in the balanced receiver. For a balanced receiver, the total signal photocurrent is the sum of the signal photocurrent in both detectors so, 2 (KES (t )ER (t ) sin ωτ ). 2. SNRbalanced - receiver =. σ re 2 + σ sh 2 + σ be 2. 2.2.9. The SNR for a signal-detector interferometer and a balanced-receiver interferometer can be written. The result shows improved SNR that used balanced receivers indicated a more significant improvement. Even though the balanced heterodyne detection is more complicated in system than single detection, it has such adventages as;. 15.

(25) 1. The output signal current is twice as much as that from the single detector. 2. The detector is practically free from the noise generated by the intensity fluctuations of the local oscillator laser. 3. Even when the output of the detector is dc coupled to preamplifier, saturation of the preamplifier can be avoided because of the absence of the dc current.. 2.3 Hilbert transformation. There are many methods, both experimental and analytical, to extract the phase of a signal. The Hilbert transform is used widely to obtain the phase of the signal[15]. A real function f (t ) and its Hilbert transform fˆ (t ) are related to each other in such a way that they together create a so called strong analytic signal. The strong analytic signal can be written with amplitude and phase where the derivative of the phase can be identified as the instantaneous frequency. The Hilbert transform fˆ (t ) of a function f (t ) is defined for all t by. 1 ∞ f (τ ) fˆ (t ) = P ∫ dτ π −∞ t − τ. 2.3.1. when the integral exists. It is normally not possible to calculate the Hilbert transform as an ordinary improper integral because of the pole at t = τ . However, the P in front of the integral denotes. 16.

(26) the Cauchy principal value which expanding the class of functions for which the integral exist. We consider a complex function f ( z ) that is analytic in upper half-plane and on the real axis. We also require that. lim f (z ). = 0 , 0 ≤ arg z ≤ π. 2.3.2. z →∞. in order that the integral over an infinite semicircle will vanish. The point of these conditions is that we may express f ( z ) by the Cauchy integral formula[15], f ( z0 ) =. 1 f (z ) dz ∫ 2πi z − z0. 2.3.3. The integral over the upper semicircle vanishes and we have f ( z0 ) =. 1 ∞ f (x ) dz 2πi ∫− ∞ x − z0. 2.3.4. The integral over the contour show in Fig.2.3.1 has become an integral along the x-axis.. Fig. 2.3. Semicircle contour.. 17.

(27) Equation (2.3.4) assumes that z0 is in upper half-plane－interior to the closed contour. If z0 were in the lower half-plane, the integral would yield zero by the Cauchy integral theorem. Now, either letting z0 approach the real axis from above (z0 − x0 ) or placing it on real axis and taking an average of Eq. (2.3.4) and zero, we find the Eq. (2.3.4) becomes ∞ 1 f (x ) P∫ dx − ∞ πi x − x0. f ( x0 ) =. 2.3.5. where P indicates the Cauchy principle value. Splitting Eq. (2.3.5) into real and imaginary parts yields f ( x0 ) = u ( x0 ) + iv( x0 ). 2.3.6. on both sides of Eq.(2.3.5) with arguments on the real x-axis and equating real and imaginary parts then we obtain for the real part u ( x0 ) =. 1. π. P∫. ∞. −∞. v( x ) dx = Hv( x0 ) x − x0. 2.3.7. and for imaginary part v( x0 ) = −. 1. π. P∫. ∞. −∞. u(x ) dx = Hu ( x0 ) x − x0. 2.3.8. where H is the Hilbert transform operator. The real part of our complex function is expressed as an integral over the imaginary part. The imaginary part is expressed as an integral over the real part. The real and imaginary parts are Hilbert transforms of each other.. 18.

(28) The Hilbert transform can be used to create an analytic signal from a real signal. Instead of studying the signal in the frequency domain it is possible to look at a rotating vector with an instantaneous phase φ (t ) and instantaneous amplitude A(t ) in the time domain, that is z (t ) = f (t ) + ifˆ (t ) = A(t )eiφ (t ). 2.3.9. This notation is usually called the polar notation where A(t ) =. f 2 (t ) + fˆ 2 (t ). 2.3.10. ⎛ fˆ (t ) ⎞ ⎟ ⎟ ⎝ f (t ) ⎠. 2.3.11. and. φ (t ) = arctan⎜⎜. 19.

(29) Chapter 3. Material and Method. 3.1 Experimental setup. The experimental system shown in Fig. 3.1 consists of a Mach-Zender interferometer, RSOD, and the balanced detection configuration. The collimated output of a superluminescent diode has a bandwidth of 18 nm centered at 827 nm and passes through the isolator. After passing through half-wave plate (HWP), the beam has horizontal and vertical polarization and divides at a polarizing beam splitter (PBS) into signal and reference beams. A grating-based frequency-domain optical delay line (FD-ODL) is used in the reference arm for axial scanning which can control group- and phase-delay separately. The reference arm is inserted into a HWP to rotate the S-polarized light into 45° linearly polarized state. The sample arm is inserted into a quarter-wave plate (QWP) to change the linear polarization stated to circular polarization. The Wollaston prism splits the beam into two components with mutually perpendicular linear polarization states. A lens located at a distance from the Wollaston corresponding to the focal length of the lens converts the two diverging beams into parallel beams and focuses them at the sample. The polarization state of this beam depends on the phase difference between the two beams components reflected at the sample. The beams are 20.

(30) reflected at the reference mirror and the sample, and then recombined by BS3. The beam splitter then splits the interference signal into two polarization beam splitters, receives the horizontal and the vertical polarization interference signal by balance detectors 1 and 2, respectively.. Fig. 3.1 Schematic of common-path balanced optical coherence tomography system that incorporates with a RSOD.. Because for each polarization state, an independent detector records the whole ~ interference signal Ak ( z ) , the amplitude Ak ,0 ( z ) and the phase φk ( z ) of each. 21.

(31) interference signal at any depth z can be obtained from the Hilbert transformation [7] of complex function, ~ A1 ( z ) = A1 ( z ) + iΗ[ A1 ( z )] = A1,0 ( z ) exp[iφ1 ( z )] ~ A2 ( z ) = A2 ( z ) + iΗ[ A2 ( z )] = A2, 0 ( z ) exp[iφ2 ( z )] Fringe phase in each channel is calculated by computing the angle between the signal and. its. Hilbert. transform.. Computing. the. differential. phase. ∆φ ( z ) = φ p ( z ) − φs ( z ) removes common mode environmental noise. Path length change, ∆z , due to displacement can be therefore calculated from the differential phase, ∆φ , and the center wavelength of the source, λ , by ∆z = λ∆φ / 720° . Thus we obtain amplitude Ak ,0 ( z ) and phase difference ∆φ (z ) image of object, which are completely independent. In this system, a 160Hz resonant scanner is used in RSOD for axial scanning. The resonant scanning mirror with 40 kHz phase modulation is introduced during axial scanning and data acquisition rate is 400 kHz. The two differential photodetectors (BD1 and BD2) are adopted into system to obtain balanced detection of signals. Balanced detection removes the low-frequency noise caused by variations in reflection of the RSOD, and reduces the effect of intensity fluctuations from the light source. A National Instrument data acquisition card (NI-DAQ) is used in a computer to sample the signals in both channels. 22.

(32) 3.2 Wollaston and Nomarski prism. Wollaston prism. A Wollaston prism consists of two similar wedges cemented together in such a way that the combination forms a plane parallel plate[11]. The optic axes in the two component wedges are parallel to the external faces and are mutually perpendicular. It can be made of calcite or quartz in the form indicated in Fig. 3.2.. Fig. 3.2. The Wollaston prism. The two component rays separate at diagonal interface. There, the e-ray becomes an o-ray, changing its index accordingly. A Wollaston prism splits an incident ray into two rays traveling in different directions; the lateral displacement between the rays thus different distances from the Wollaston (Fig. 3.3)[16].. 23.

(33) The angular splitting α is given by the relation. α = 2(ne − n0 ) tan θ. 3.2.1. where θ is the wedge angle.. (a). (b). Fig. 3.3. (a) Beam splitting by a Wollaston prism. (b) Path difference produced by a Wollaston prism between the two split-up rays is linearly related to x .. The path difference between the OE and EO rays (Fig. 3.3(b)) emerging at a distance x from the axis y-y` of the Wollaston prism is given by ∆ = 2(ne − no )x tan θ = αx. 3.2.2. The path difference is zero along the axis, where the thicknesses of the two component wedges are equal, and increases linearly with x .. 24.

(34) In our system, the Wollaston prism is put in sample arm with the split angle. α = 5° , and split the beam into two components with mutually perpendicular linear polarization states to extract the phase difference.. Nomarski prism. Nomarski prism (Nomarski-modified Wollaston prism) has shown how to modify the design to localize the fringes outside it, as shown in fig. [17, 18]. One of the component prisms is now cut with its optic axis inclined.. Fig. 3.4. The Nomarski prism. The wedge is fabricated from two pieces of birefringent material with their respective fast optical axes orthogonal to each other. The prism separates beams polarized in and perpendicular to the plane of the paper, and the direction of beam offset is called the shear direction of the prism. The resulting beams then intersect in a plane beyond the second side of the wedge. 25.

(35) The quartz Nomarski prism (United Crystals Company: Nomarski prism). We adopted has clear aperture of 12mm×12mm and splitting angle of 0.06 degree at 830nm, the offset distance between output surface and the plane of apparent splitting is 20mm.. Fig. 3.5. The specification of Nomarski prism. (a) Top view. (b) Side view. 1-inch anodized black aluminum ring holder (with length of 20mm).. 26.

(36) Chapter 4.. Results. 4.1 Measurement of unbalanced and balanced detection systems. In order to demonstrate the advantage of balanced detection technique in this RSOD based phase contrast OCT, the signal-to-noise ratio (SNR) of signals in unbalanced and balanced detection systems were measured with single detector and balance detector (i.e. BD1), respectively. This is because one fundamental limitation on minimum detectable phase difference arises from SNR of the measurement[19, 20]. In the words, the phase difference measurement is limited by OCT system noise floor. This standard deviation directly yields the minimum detectable phase difference σ ∆φ which can be determined as. σ ∆φ = (SNR )−. 1. 2. 0. 0. -5. -5. -10. -10. -15. -15. Reflectivity (dB). Reflectivity (dB). The results of SNR for both detection systems are presented in Figure 4.1. (a) (b). -20 -25 -30 -35. -20 -25 -30 -35. -40. -40. -45. -45. -50 500. 600. 700. 800. 900. 1000. 1100. 1200. 1300. 1400. -50 500. 1500. Retrorelector Displacement (um). 600. 700. 800. 900. 1000. 1100. 1200. 1300. 1400. 1500. Retrorelector Displacement (um). Fig. 4.1. Measured SNR from (a) signal detector, and (b) balance detector (BD1).. 27.

(37) For unbalanced system, the SNR of signal was approximately 20.5 dB, whereas that is about 28 dB in balanced system. The measurement proofs the advantage of OCT when employing balance detectors.. 4.2 Phase stability test. The phase stability was measured by recording the phase at the surface of a mirror. A Wollaston prism located in the focal point of a lens （ f = 100mm ）splits the sample beam in two orthogonally polarized beams seprated by an angle of. 5° . Two parallel. beams emerge from the lens, converging to focal points separated by ∆d = 8.7 mm . The phase difference between the two object beams generated from the Wollaston prism can be measured.. Fig 4.2 shows the phase stability in the common-path. configuration. The curve indicates that the values almost maintain at a constant, and the phase variation was 1.58° which corresponds to optical path length change of nearly 1.8 nm when the wavelength of light source is centered at 830 nm. The standard deviation of phase fluctuation is 0.4447° (~ 0.51nm).. 28.

(38) Fig. 4.2. Measured phase difference when the OCT system using in a common-path configuration.. 4.3 Surface Displacement Measurements. The performance of the phase-resolved imaging was tested with a pure phase object consisting of a resolution target (Edmund: NBS 1963A RESOLUTION TARGET_ positive) covered with a chromium layer deposited upon the target surface by evaporation. The thickness of the Chrome coating is 100 nm. Fig. 4.3(a) shows a sketch of the sample, and Fig. 4.3(b) is the sample path configuration. A two-dimensional data set of 50 adjacent A-scans was recorded; the sample was moved by 10 µm in the x direction, perpendicular to chromium step, between the A-scans. Fig. 4.3(c) shows an intensity and Fig. 4.3(d) a phase-difference OCT image. The phase image shows the phase difference on a color scale. Fig. 4.3(e) shows a plot of the phase difference along the surface of air-chromium. The phase difference changes. 29.

(39) along the interface of the phase object is ~ 90° , corresponding to step heights of ~ 103 nm, can be observed, which is invisible in the intensity image [Fig. 4.3(c)]. (a). (b). (c). (d) phase difference. Intensity 3.5 200. 200. 160. 400. 140. 600. 120. 800. 100. 1000. 80. 1200. 60. 3 400 2.5. 600. 2. um. um. 800 1000. 1.5 1200 1. 1400. 1400. 0.5. 1600. 40. 1600. 20. 1800. 1800 0. 100. 200. 300. 400. 0. 500. 100. 200. 300. 400. 500. um. um. (e). Fig. 4.3. Resolution target measurement, (a) Schematic of the sample, (b) sample path configuration, (c) intensity image, (d) phase image, (e) phase difference along the Chromium layer.. 30.

(40) 4.4 Nomarski experiment. A disadvantage of Wollaston is its splits the sample beam in two orthogonally polarized beams separated by ∆d = 8.7 mm , the distance between the focal spots of two separate sample beams limited the transversal resolution. Therefore, we change to a Nomarski crystal, it can make the two orthogonally polarized beams split very small angle (0.06°) . The distance of separated beam is only 52 µm . In this condition, we measure the chromium layer resolution target (Edmund: NBS 1963A RESOLUTION TARGET_ positive) covered with a cover glass (SUPE RIOR MARIENFELD : Deckgläser Nr.1 cover glasses, 22×22mm, made in Germany ), like Fig. 4.4, and extract the phase information for multi-layer structure.. Fig. 4.4. OCT test sample: schematic of the object structure.. The interferometric signals of two orthogonally polarized beams are shown in Fig. 4.5. The first signal peak is reflected from the surface of a cover glass. The second signal peak is reflected from the interface of the bottom of cover glass and the glass plate or. 31.

(41) the chromium layer of resolution target, the phase difference between them will be measured. The third peak is from muliple reflection. (a). (b) 1. 3. signal P. 0.8. signal S 2. 0.6 0.4. 1. intensity. intensity. 0.2 0 -0.2. 0. -1. -0.4 -0.6. -2. -0.8 -1 600. 700. 800. 900. 1000. 1100. 1200. 1300. -3 600. 1400. 700. 800. 900. um. 1100. 1200. 1300. 1400. um. (c). (d) 1. 3. Hilbert S. Hilbert P. 0.9. 2.5. 0.8 0.7. 2. intensity. 0.6. intensity. 1000. 0.5 0.4. 1.5. 1. 0.3 0.2. 0.5. 0.1 0 600. 700. 800. 900. 1000. 1100. 1200. 1300. 0 600. 1400. um. 700. 800. 900. 1000. 1100. 1200. 1300. 1400. um. (e) 3. 2.5. -- P -- S. intensity. 2. 1.5. 1. 0.5. 0 700. 750. 800. 850. 900. 950 1000 1050 1100 1150 1200. um. Fig. 4.5. The interferometric signals of two orthogonally polarized beams, (a) signal P, (b)signal S, (c) Hilbert P, (d) Hilbert S, (e) combine Hilbert P and Hilbert S. 32.

(42) In Fig. 4.5(e), we discovered the signal P and signal S have displacement shift about 20µm . This is due to the light passes through Nomarski crystal to make the optical path different. According to Fig. 3.2 and Eq. 3.2.2, we must move Nomarski crystal transversely. But after calculating, the distance of adjustment is too large over the size of crystal. Two orthogonally polarized beams can not interfere in the same time, thus the adjacent phase difference can not be extracted successfully.. 33.

(43) Chapter 5.. Discussion. This present result shows that our proposed free-space phase contrast optical coherence tomography system achieves sub-nanometer scale displacement sensitivity and is better than some previous systems [2, 7-9]. It may be due to the light passes through the same path before Wollaston, the common phase noise can be canceled by differential phase measurement. Besides, some other noise that caused by environmental perturbations, including temperature fluctuations, air currents, vibration, and moving part of scanning and phase modulation can be priority canceled by using the balanced detection. However, the instability of a mechanical device (i.e. scanning mirror) in the reference arm (RSOD) of an OCT system decreases the stability of interference fringe carrier frequency thus increases the phase variance in our system. Comparing to other reports,. Taner Akkm. et al. presented a fiber-based. optical biosensor, which was capable of detecting ultra-small refractive index changes in highly scattering media with high lateral and longitudinal spatial resolution [21]. The setup is shown in Fig. 5.1(a). The LiNbO3 phase modulator implemented in the reference arm is used to generate a stable interference fringe carrier frequency. The. 34.

(44) measurement of phase difference between interferometric frings in two channels eliminates environmental phase noise in this common mode system.. (a) (b). (c). Fig. 5.1. (a) Dual channel phase-sensitive optical low coherence tomography. (b) Sample path configuration to scan spatially separated polarization channels on the sample. (c) Measured phase difference of a signal record.. Fig. 5.1(c) shows that phase sensitivity for an individual recording is as low as 10-3 radians which allowing measurement of 104 pm path length differences resolution.. 35.

(45) Therefore, with fiber-based system and implementation of external phase modulator in the reference arm of our phase contrast OCT, the similar resolution should be achieved. On the other hand, Christopher Fang-Yen et al. [22] used the dual beam heterodyne interferometer which incorporates both fiber and free-space elements (Fig. 5.2(a)) to measure phase changes of reflected light from a sample relative to a reflective surface above. The light coming out of the second output of the Michelson goes to a reference gap. The reference gap consists of two reflecting surface with adjustable distance. The optical delay created in the Michelson and the noise associated with it was canceled by taking the difference in phase between the sample and reference signals using Hilbert transform, ∆φ = k0 (∆LS − ∆LR ) where k0 is the central wave number of the source. ∆LS and ∆LR are the round-trip optical path length differences between reflections from surfaces 1 and 2 of the sample and of the reference gaps, respectively. It is similar to balanced system.. 36.

(46) (a). (b). Fig. 5.2. (a) DBHI of optical referenced interferometer setup. (b) The phase displacement on the top of coverslip.. Fig. 5.2(b) is the phase displacement on the top of coverslip, averaged over every 0.2ms and the standard deviation of the phase vibrations σ is 0.16 mrad. The stability corresponds to an OPL of 40 picometers. The experiment demonstrated that an optical referencing method has sub-nanometer noise level over a period of 50 ms. But in our balanced system, we can not avoid the optical components for horizontal and vertical polarizations have different optical path delay. The noise with different delay was not canceled completely in our balanced system. Moreover, because a Wollaston prism splits the sample beam in two orthogonally polarized beams separated by ∆d = 8.7 mm in our system, the distance between the focal spots of the sample beams limited the transversal resolution. 37.

(47) Therefore, we change to a Normaski crystal, it can make the two orthogonally polarized beams split very small angle (0.06°) . The distance of separated beam is only. 52µm . However, the Normaski causes two orthogonally polarized beams and they have large optical path difference. Two orthogonally polarized beams can not interfere in the same time, thus the adjacent phase difference can not be extracted successfully. Another disadvantage of this setup is the mechanical vibrations from the translation stage of sample arm. The sample is mounted on a stage driven by an actuator that moves the sample transversely to the beam propagation direction. The sample stage caused the mechanical vibrations, so we need to spend some time waiting for the stage stability. It makes our measurement spent long time for 2D image. In order to improve the scan speed, we can use galvanomirror scanner instead of moving stage, like Fig. 5.1(b) shows. Compared to time-domain OCT based phase measurement systems, other methods which based on frequency-domain optical coherence tomography to extract depth-resolved intensity and phase information would have significantly improved phase stability. That is because the frequency-domain OCT do not contain moving parts which can substantially increase the phase noise floor. Moreover, frequency-domain OCT do not require a scanning delay line, they can be built using a. 38.

(48) common-path configuration where virtually all phase noise is common mode between the reference and sample optical field [23-25]. For example, Michael A. Choma et al. presented a spectral-domain phase microscopy (SDPM) in 2005. A phase-sensitive functional derivative of spectral-domain OCT that allows for real-time measurement of displacements with picometer-to-nanometer-scale sensitivity was demonstrated (as shown in Fig. 5.3)[26].. ~ Since the phase of depth dependent function I (±2n∆x) is a linear function of small displacement δx , the changes in subresolution position of the sample reflector can be measured with respect to an arbitrary zero point taken at a reference time t0 : ~. ~. δx(t ) = λ0 4nπ [∠I (2n∆x, t ) − ∠I (2n∆x, t0 )]. Fig. 5.3. (A) FD SDPM interferometer. (B) SS SDPM interferometer. The insets show the displacement signals recorded from a clean coverslip.. 39.

(49) They demonstrate SDPM using both Fourier-domain and swept source OCT had a displacement sensitivity of 53pm and 780pm, respectively, and can be used to measure cellular motion with exquisite sensitivity. Its spectral-domain system has intrinsically higher phase stability than time-domain techniques. Additionally, the phase of spectral-domain common-path interferometry obviates the need for dual-beam interferometers that are necessary to achieve phase stability in time-domain systems. In the same year, Chulmin Joo et al. presented a novel quantitative phase imaging modality, referred to as spectral-domain optical coherence phase microscopy (SD-OCPM)(as shown in Fig. 5.4)[27]. Because SD-OCPM acquires depth-resolved information without mechanical scanning of the reference mirror, it can generate a three-dimensional quantitative phase-contrast image of a specimen simply by scanning the beam laterally as it measures phase profiles in depth.. Fig. 5.4. (a) Schematics of the SD-OCPM system. (b) Sample placed between a coverslip and a microscope slide. SDPM measures phase distribution of a sample referenced to the top surface of the coverslip. 40.

(50) The depth z dependent function F ( z ) is obtained by a discrete Fourier transform, so one can extract the phase as a function of z by taking the argument of. F ( z) :. ⎧ Im[F ( z )]⎫ 2π ∆p ( z ) ⎬=2 λ0 ⎩ Re[F ( z )]⎭. φ ( z ) ( x , y ) = tan −1 ⎨. The phase sensitivity is an important performance factor in SD-OCPM, and can be characterized by the standard deviation of the phase, and it is expressed as an explicit function of the signal-to-noise ratio. ∆φ 2 ≈ 1 2SNR The measured SNR was 100.4 dB, under which condition the theoretical sensitivity is 0.4 pm. The measured sensitivity in SD-OCPM was 25 pm in air. The difference between the theoretical and measured sensitivities may be due to the influence of external disturbances such as vibrations in the coverslip during the measurement. In 2006, Marinko et al. developed a full-field swept-source phase microscopy (FF SS PM) technique for quantitative nanoscale surface profiling of samples in reflection[28]. This technique utilizes swept-source optical coherence tomography in a full-field common path interferometer for phase-stable cross-sectional acquisition without scanning as illustrated in Fig. 5.5. 41.

(51) Fig. 5.5. Full-field swept-source OCT setup with a 4 f common path interferometer.. Fig. 5.6. illustrates the procedure for processing the acquired data. Each interferogram is interpolated, windowed, and zero padded prior to performing a fast Fourier transform. Subwavelength variations are observed as variations in the phase at a particular depth slice.. Fig. 5.6. Processing steps to acquire phase-sensitive data using the full-field swept-source OCT system.. The phase stability of the FF SS PM was characterized both spatially and temporally by use of a glass coverslip. The spatially phase stability was measured from standard deviation across the coverslip surface in a 100*100 pixel region near the center of the. 42.

(52) beam (i.e. 1.3 nm). The temporal phase stability was measured from the average standard deviation of the phase over 50 min. The standard deviation calculated for an area of 60 * 80 pixels was 2.5 nm. In 2007, Desmond C. Adler et al. used buffered Fourier domain mode-locked (FDML) lasers and demonstrated a dynamic phase-sensitive optical coherence tomography (OCT) and 3D OCT phase microscopy[29]. Systems are operated at sweep speeds of 42, 117, and 370 kHz, the displacement sensitivities of 39, 52, and 102 pm were achieved, respectively. Their system is shown in Fig. 5.7.. Fig. 5.7. Setup for phase-sensitive OCT measurements.. Displacement sensitivities (DS) are measured by recording the phase at the back surface of the 210 µm coverslip, relative to the front surface. Differential displacement sensitivities (DDS) are calculated by subtracting the measured phase of consecutive axial scans, ∆φ1 − ∆φ0 . The result is shown in Table 1.. 43.

(53) Table 1. Displacement Sensitivity Using Conventional and FDML Lasers.. Sensitivities are comparable to spectrometer-based OCT phase microscopy systems, but much faster acquisition speeds are possible. The differential displacement sensitivities of buffered FDML lasers are comparable with single-measurement displacement sensitivities and would provide an additional. 2 improvement in. Doppler OCT sensitivity when compared with conventional swept lasers or nonbuffered FDML lasers. Moreover, the combination of SD-OCPM and multiphoton microscopy (SD-OCP-MPM) was proposed by Chulmin Joo et al. in 2007[30]. It can obtain phase contrast and multiphonton fluorescence imaging simultaneously. The setup is shown in Fig. 5.8.. 44.

(54) Fig. 5.8. Schematic of SD-OCP/MPM.. Fig. 5.9. Spatial and temporal phase stability of SD-OCP–MPM. (a) The 2D phase repeatability map demonstrates ~ 0.5 nm repeatability in air. (b) Phase fluctuation for a stationary beam; the standard deviation was 53 pm at a SNR of 63.4 dB.. Fig. 5.9 (a) shows the spatial phase stability. The standard deviation across the field of view was measured as ~ 0.5 nm in air, which may be in part attributed to the motion. 45.

(55) jitter of the scanners. For the temporal phase stability, the phase fluctuation was recorded as all X, Y, and Z scanners are set to a fixed value (0 V) [Fig. 5.9 (b)]. The measured phase stability was ~ 53 pm in air at a SNR of 63.4 dB. In 2007, Adrian H. Bachmann et al. presented a dualbeam heterodyne Fourier domain optical coherence tomography[31]. The authors indicate any phase noise due to sample motion or mechanical beam scanning will cause signal degradation as well as insufficient suppression of mirror terms, so they developed a dual beam FDOCT variant that profits from the high phase stability of a common path configuration, without sacrificing measurement depth range, and keeping the flexibility for beam scanning as well as the possibility of dispersion balancing. A dual beam configuration is shown in Fig. 5.10., both reference and sample light share the same path and thus exhibit high relative phase stability. The Dual beam principle is that the output of an interferometer with a relative delay of. 2∆z IILS between the two light beam. intensities IR and IS is pre-compensated for the relative distance between R1 (reference surface) and R2 (sample). The configuration presents a small relative distance ∆z between reference surface (R1) and sample (R2) and up to four cross correlation terms might occur. The blue beam can be considered as the reference beam.. 46.

(56) Fig. 5.10. Dual beam heterodyne FDOCT. Inlet A depicts synchronization of the line detector. Inlet B shows the reference arm added and used for phase stability comparison between the dual beam and the standard configuration.. In order to demonstrate the advantage of dual beam versus standard FDOCT in terms of phase stability, the SNR of the signal peak was adjusted to approximately 26.5dB. For the standard FDOCT configuration, it could be measured that the phase fluctuations are strongly varying (see Fig. 5.11). The strong fluctuations of the standard signal peak intensity are mainly due to fringe washout and stress-induced polarization state changes in the perturbed fiber, resulting in reduced interference fringe contrast. These measurements proof clearly the advantage of dual beam FDOCT over standard FDOCT.. 47.

(57) (a). (b). Fig. 5.11. (a) The dual beam signal (red) remains stable even if the fiber is perturbed whereas the signal peak corresponding to the standard setup (blue) is heavily perturbed. (b) The red and blue lines indicate the standard deviation σ dual = 0.05 rad and σ std = 0.72. rad of the phase fluctuations,. respectively. The shown tomogram depth is approximately 400 µm (in air), SNR ≈ 26.5dB.. 48.

(58) Chapter 6.. Summary. We have developed balanced OCT configurations to yield phase information by incorporating the RSOD into a Mach-Zehnder interferometer. The designs that used balanced indicated a more improvement than the standard OCT interferometer. Using this system we have shown that sub-nanometer optical path differences, which are invisible in conventional intensity-based OCT image, can be successfully imaged by phase-resolved OCT. The phase drift caused by environmental disturbance may due to the free-space system configuration. With fiber-based system and implementation of external phase modulator in the reference arm of our phase contrast OCT, the better resolution should be improved. Possible fields of application might include not only biomedicine (e.g. tissue surface profilometry, cell response to various stimuli, etc.) but also, e.g., the semiconductor or the thickness of layered structures is of interest.. 49.

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