Abstract

A new heterodyne interferometer scheme which has open accesses to both the geometrically balanced probe beam (PB) and reference beam (RB) paths, for which, depending on the nature of a specific sensing mechanism, a transmission geometry or a reflection geometry can be employed, is presented. We will show that, because of a small separation between the short length PB and RB running parallel to each other our newly proposed optical arrangement allows high rejection of unlocalized environmental perturbations. In fact, the geometrically balanced optical arrangement provides 19dB rejection of any vibrations parallel to the direction of beam propagation, which cannot be achieved in a conventional interferometer scheme. Applications of this new interferometer scheme are discussed. As an example, we will show that our newly proposed interferometer scheme can be applied for high sensitivity measurements of concentration dependent refractive indexes in various solutions.

© 2013 OSA

1. Introduction

Two beam interferometry has been extensively applied for many sensor applications because of its extremely high sensitivity in measuring the effective optical path length (OPL) difference between the probe beam (PB) and the reference beam (RB) [13]. Optical arrangements in most of two beam interferometer sensors are modifications of the Michelson interferometer or the Mach-Zehnder interferometer. Since the PB and RB in the Mach-Zehnder interferometer are running parallel to each other with relatively small separation, it may be less susceptible to environmental perturbations such as atmospheric turbulence, vibrations, acoustic noises, and so forth than a Michelson interferometer. The Jamin interferometer [2,4] is specially designed to provide minimal beam separation between the PB and the RB. In a Mach-Zehnder interferometer or Jamin scheme, however, since a sample cell or a sensor head must be placed in the PB path, only a transmission geometry (TGM) is applicable in the sensor arrangement which limits application capabilities. Refractometery may be a typical application of the TGM [4,5]. It has been shown that the sample arrangement becomes more flexible if PB is taken out from the interferometer and the reflected beam from the sample is reinjected into the interferometer by using a polarizing beam splitter (PBS), λ/4-plate (QWP), and the reflecting surface or the mirror [6,7], which will be referred to the coupling mirror. This scheme can offer flexibility in the sensing arrangement: Depending on the sample or sensing mechanism, either the reflection geometry (RGM) or the TGM can be arranged. However, because of this extra beam path, the interferometer becomes unbalanced and susceptible to environmental perturbations. In fact, any acoustic and/or mechanical vibrations of the reflecting sample or the coupling mirror will be directly converted into the corresponding phase noises. Although fiber optic or integrated waveguide interferometers have been widely used in sensor applications [8,9], a bulk-optic interferometer is still important in many applications such as coherent scanning microscopy [6,7,1012], readout sensing for a bio-CD [13] or a fluidic channel [14], and precision metrologies [15,16].

In heterodyne interferometry, the PB and the RB have different frequencies. The OPL difference or the phase difference is then carried by intermediate frequency heterodyne beat which is typically in the RF range. It is the big advantage of heterodyne interferometry that state-of-the-art demodulation technologies, which have been developed in RF communications, can be employed for extracting the phase difference. It has been shown that the relative phase difference between the PB and RB can be directly measured by using the in and quadrature (I/Q) demodulation scheme [6]: The exact value of the phase difference and the amplitude change of the PB can be simultaneously measured without any rigorous calibration procedures [6,7, 11].

In this paper, we are presenting a new heterodyne I/Q-interferometer scheme, which has the balanced PB and RB paths for both transmission and reflection measurements. As will be shown, our new novel scheme cannot only provide versatility of sensing arrangements but also can reject environmental noises for which characteristic lengths are larger than the separation between the parallel PB and RB. In particular, it can provide ~19dB rejection of any vibrations of the coupling mirror. To the best of our knowledge, no interferometer scheme which can reject vibration noises of a sample has been published yet. As an application example, we will demonstrate that the proposed interferometer scheme can be used for measuring refractive index difference between the reference and sample liquids in the homemade fluidic channel.

2. Interferometer and experimental arrangement

A schematic of the experimental arrangement is shown in Fig. 1. A homemade, stabilized, dual frequency, dual polarization He-Ne laser is used as the light source. The laser has better than 33dB return loss by using two optical isolators, mirrors, and polarizing beam splitters (PBSs), which are not shown in the figure. The frequency stability is ~4MHz in the lab environment. It has two output beams for which each beam has two orthogonally polarized principal polarization modes with different frequencies. The principal polarization modes of one output beam from the laser are mixed together by using a polarizer P1 oriented at 45° to the principal polarization modes and detected by a high-speed photodiode PD1. The heterodyne beat signal from the PD1 is used as the local oscillator for RF I/Q-demodulator (PolyPhase Microwave, QD0511B). The principal polarization modes in the other output beam from the laser are split into two paths by using the PBS1: the transmitted beam with frequency ν1is used as the RB and the reflected beam with frequency ν2 is used as the PB. The plane of polarization of the PB is rotated by 90° by using the half-wave plate, HWP1, and reflected at the right angle prism (RP). The RB and PB are transmitted through the PBS2 and PBS3, respectively, circularly polarized by using a quarter-wave plate (QWP), and sent to the sample. The separation between the PB and RB is 17mm in this preliminary work but it can become much narrower if smaller optical components are closely mounted together. Depending on a specific application, either the reflection geometry or transmission geometry can be used. In the former case, generally used for scanning microscopy, the sample is a reflecting surface. The PB and RB are reflected at the sample and reference surfaces, respectively. In the latter case, the PB and RB are transmitted through the sample and reference channel, respectively. The PB and RB are reflected back into the incident beam paths by using a mirror. Because of the double pass in the QWP, the planes of polarization of the returning beams are rotated by 90°. The RB is reflected back into the interferometer at the PBS2 and the plane of polarization is rotated by 90° by using a half-wave plate HWP2. The RB is then transmitted through the PBS3 and combined with the PB which is reflected at the PBS3. The PB and RB are orthogonally polarized to each other and carefully aligned so that they are propagating along the same path. The PB and RB are mixed by using the polarizer P2 oriented at 45° to the preferred axes of the PBS3 and high frequency photodiode PD2. The output heterodyne beat signal from the PD2 carries the phase difference between the PB and RB and drives the RF input of the I/Q-demodulator. Details of the I/Q-demodulation in heterodyne interferometry can be found in [6,7]. Since the output signals from the I/Q-demodulator have 90° phase difference, it can be shown that [6]

vQvI=tan(Δϕ)
where Δϕ, vI, and vQ are the phase difference between PB and RB, low-pass filtered in-phase and quadrature outputs from the I/Q-demodulator, respectively. It should be emphasized that, since the phase difference can be obtained directly from arctangent operation of Eq. (1), it does not require any calibration to extract the phase value from the interference signal. Moreover, since the tangent function always has a large slope, it does not require any feedback control for maintaining the optimum phase demodulation condition. These two are the major advantages of the I/Q-interferometer scheme over the conventional schemes. Equation (1) is switched to cotangent function if |Δϕ|>π/4 or, equivalently, if |vI|<|vQ|, and add or subtract π/2 to the cot−1 value to avoid the discontinuity of tangent function at |Δϕ|=±π/2.

 

Fig. 1 Schematic of experimental arrangement (a), and a photograph of the interferometer and sample part of the experimental arrangement (b).

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It is clear from Fig. 1 that two arms of the interferometer are almost exactly balanced, i.e., the corresponding OPLs of the PB and RB from the laser to the PD2 except for the roundtrip OPL difference induced by the sample have the same lengths larm. Therefore, the phase difference between the PB and RB is given by

Δϕ=2πc[(ν1ν2)larm+ΔϕQS+ΔϕNL+2(nRlRν1nslsν2)]
where nsls, and nRlR are OPLs of the sample and the reference channels, respectively. Any other quasi-static phase drifts are included in ΔϕQ-S. The periodic nonlinearity, ΔϕNL, which is a small but significant systematic error resulted from the crosstalk between principal polarization modes caused by ellipticity on the principal polarization modes and imperfect alignment of polarization components used in the interferometer [17]. The cross talk between the principal modes of our stabilized laser is approximately −40dB, but, because of the misalignment between polarization components and the principal polarizations, overall crosstalk at the PD2 is approximately −30dB. The crosstalk has been measured by rotating the polarizer at each output port and derived from the amplitude ratio between the maximum and minimum beat signal. All polarization components are carefully aligned to minimize the crosstalk between the channels and, thereby, the periodic nonlinearity. It has been shown that the magnitude of this periodic displacement is in the nm range if the principal axes of the light source and polarization components are properly aligned. The periodic nonlinearity may be an important parameter for determining the accuracy of phase measurements. If it is small, however, it may be less important in measuring a phase difference, because the phase change caused by the nonlinearity may not be significant for a small displacement in effective OPL. More rigorous studies about the periodic nonlinearity will be performed in the future work. The first term in Eq. (2) is the phase difference due to the frequency difference ν1ν2 between the PB and RB which is 820MHz in our present work, which is responsible for slow and small thermal drift caused by ambient temperature change. The center wave length of the laser is 632.8nm. All components are fixed on top of the optical table and placed inside a plexiglass enclosure on the table. In addition, as shown in Fig. 1.(b), optical components of the interferometer part of the experimental setup are carefully aligned and firmly mounted in a small black anodized aluminum box to isolate the sensitive components from the environment. Characteristic sizes of atmospheric turbulences may be larger than the beam separation in this environment for which interferences caused by environmental perturbations can be minimized. Moreover, any longitudinal motion of the sample and/or the coupling mirror results in the corresponding changes of larmwhich gives a negligible phase modulation through the first term in Eq. (1). As a result, the interferometer becomes highly immune to acoustic or mechanical vibrations parallel or orthogonal to the propagation directions of the PB and RB. To prove this immunity, mirror vibrations were measured in two different arrangements. In the first arrangement, as shown in the inset of Fig. 2(a), the PB and RB are reflected by two independent mirrors for which one mirror, the PB mirror, is vibrated at 40Hz by using a PZT actuator. The frequency spectrum of the vibration measurement results is shown in Fig. 2(a). In the second arrangement, as shown in the inset of Fig. 2(b), both the PB and RB are reflected by one vibrating mirror which is mounted on the same PZT stage as the first arrangement with the identical driving conditions. The frequency spectrum of the vibration measurements for the second arrangement is shown in Fig. 2(b). It is clear from these results that our new interferometer scheme can provide almost 19dB rejection of common mode vibrations. It can also be noted from these results that there are significant reduction of low frequency noises which may be coupled to the mirrors by environmental perturbations. The interferometer, however, is sensitive to tip-tilt motion of the coupling mirror, which may be responsible for a slow drift in the measured phase. Asymmetric phase changes due to localized environmental perturbations may be another reason for the drift and/or noise. In our lab environment, the drift is slow enough that it can be compensated for by eliminating the slope in the measured results if necessary. It should be noted that the susceptibility to environmental perturbations can be significantly reduced if all of optical components in the interferometer are aligned and glued together.

 

Fig. 2 Frequency spectra of vibration measurements: (a) the RB and PB are reflected by the two independent mirrors of which a 40Hz vibration is applied to one mirror by using a PZT, and (b) the PB and RB are reflected by one mirror which is driven by the same PZT with the identical driving signal with (a).

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3. Applications of the proposed interferometer

One potential application of our new interferometer scheme in TGM may be readout sensing of fluidic channels in which a sample and reference fluid are flowing. As a feasibility study, a homemade dual-channel fluidic cell is used for measuring refractive indexes of various solutions with different concentrations. For this purpose, the sample in Fig. 1(a) bis replaced by the fluidic cell for which a schematic diagram is shown in the inset of Fig. 3(a). The fluidic cell is constructed by inserting properly tailored silicon pads in between the glass window and mirror. It has two channels in which the reference and sample liquids are flowing. We will refer to these two channels as the sample channel (SCH) and reference channel (RCH), respectively. The gap between the mirror and window is ~.5mm and the width of each channel is approximately 1mm. The RB and PB are reflected at the mirrored back plane of the corresponding fluidic channels. The phase difference between the PB and RB can be directly measured by using our interferometer scheme without any calibration procedure, from which the refractive index of the liquid in the SCH can be derived.

 

Fig. 3 Experimental results on RI measurements: (a) Phase measurements for various concentrations of ethylene glycol solution. The phase was modulated by altering flow of the water and the ethylene glycol solutions in the SCH. (b) Measurement results for concentration dependent ΔRIs for various solutions. The solid lines are extrapolations of the corresponding linear fitting results of previous measurements in [1820].

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The second term in Eq. (2) represents the phase difference induced by the OPL difference between the SCH and RCH, which can be rewritten as

ΔϕFC=4πc(nRν1nsν2)lFC,
where ns, nR, and lFCare refractive index of sample liquid, reference liquid, and the thickness of the fluidic channels, respectively. Therefore, the RI difference (ΔRI) between the sample and reference liquids can be derived from the phase measurements. The absolute value of the RI of a sample liquid can be obtained if the RI of reference liquid is known. In this study, deionized water was used as the reference liquid, for which we use 1.33 as the RI at 633nm and at room temperature [18]. The RCH is filled with the DI water all the time during the measurements. Different concentrations of ethylene glycol, sodium chloride, and ethanol solutions were carefully prepared and used as sample liquids. Resulting phase difference is then determined by the solute and its concentration. The reference liquid and the sample liquids are alternately flowing through the SCH to modulate the phase of the PB. Phase differences between the sample liquids and the reference liquid are measured from these modulated signals. For example, the measurement results for various concentrations of ethylene glycol solutions are shown in Fig. 3(a), in which, if necessary, the slope due to a slow drift was removed. An enlarged view of the measurement results for 0.003wt% is shown in another inset specified as dotted ellipse in Fig. 3(a). Since the RI does not change abruptly because of the diffusion at the boundary between water and liquids under test in the SCH, there may be no upper bound for measuring concentration dependent RIs of sample liquids by employing a standard fringe counting algorithm. Measurement results for various concentrations of ethylene glycol, sodium chloride, and ethanol are shown in Fig. Theoretical plots based on the corresponding empirical law for ethylene glycol given in the literature [19] is shown as solid line in Fig. 3(b). The corresponding best fits of the published data on measurements of concentration dependent refractive indexes for ethyl alcohol [20] and sodium chloride [21] solutions to the linear function are extrapolated to low concentrations and plotted as the corresponding solid lines in Fig. 3(b). It can be seen in the figure that our measurement results show good agreement with the previous measurement results, for which we can make sure that our new interferometer scheme is ideal for precision measurements of the RI difference between the reference and sample liquids.

Typical output signal from the interferometer when both channels are filled with water in a lab environment for 10 minutes is shown in Fig. 4(a). RMS phase noise,(ΔϕΔϕ)21/2, where stands for average, for this particular measurements is 1.58 × 10−4 rad. Same measurements were made for 20 times and the corresponding RMS phase noises for these measurements are shown in Fig. 2(b). A slope due to slow drift is removed for each measurement if necessary. Our preliminary results show that the average RMS noise for 20 measurements is 2.2 × 10−4 rad. There may be better way to claim the sensitivity for RI measurements, but we think that it is reasonable to assume that the average RMS phase noise or the average standard deviation is the system noise for our measurements. Based on this assumption, we may say that the minimum ΔRI which can be measured in our new scheme is 2.3 × 10−8. We are now in the process of constructing a better quality fluidic channel and we will return to this issue in the near future. We believe that the measurement speed can be increased significantly if professionally fabricated micro-fluidic channels are used. Noise characteristics may be improved for short-time measurements and, thereby, improve the sensitivity.

 

Fig. 4 Phase noise measurements when both channels are filled with water for 10 minutes (a), and RMS noises for 20 same consecutive measurements.

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The RGM scheme also has many applications such as scanning microscopy, wafer inspection, readout sensing for a biochip, and so forth. In these applications, vibration modes perpendicular to the scanning directions can be excited on the sample mount during the scanning, which results in the corresponding phase noises in a conventional interferometer scheme, which limits the scanning speed and the depth resolution. In our new interferometer scheme, however, both the sample surface and the reference surface can be mounted on the same scanning stage, which can reject phase noises caused by vibrations of the scanning stage. Therefore, we believe that our newly proposed interferometer is ideal for these applications utilizing the RGM. We are now doing researches on applying the proposed interferometer scheme to a scanning microscopy and will not be discussed in this paper.

4. Summary and conclusions

In summary, a new heterodyne interferometer scheme has been introduced. The interferometer was specially designed so that the PB and RB have the geometrically balanced paths. Moreover, depending on the nature of a specific application, our new optical design allows for choosing either the TGM or RGM, which gives flexibilities in interferometric measurements. We showed that, because of the geometrical balance and small separation between the PB and RB, 17mm in this preliminary study, the interferometer was inherently immune to environmental perturbations. The phase difference between the PB and RB was measured without calibration procedure by using the I/Q-demodulation scheme. Since the output signal is proportional to the tangent of the phase difference and the tangent function has a large slope with respect to a phase change, it does not require any additional feedback control for maintaining optimum phase demodulation condition. As an application example, we applied our new scheme for measuring RIs of the liquids flowing through the SCH of the fluidic cell. In general, for RI measurements in SCH, because of diffusion at the boundaries between water and the other solutions, RIs or OPLs do not change abruptly, for which a standard fringe counting or a phase unwrapping algorithm can be used for wide dynamic range measurements. In all, by using our new scheme, we were able to achieve both very high sensitivity and wide dynamic range measurements of RI difference between the reference and sample liquids. It should be noted that we can use any combinations of reference and sample liquids, which may offer flexibility and versatility for designing a biosensor.

Acknowledgment

This research was supported by National R&D Program through the National Research Foundation of Korea (NRF) funded by Ministry of Education, Science and Technology (NRF-20120005921).

References and links

1. M. J. Collett, R. Loudon, and C. W. Gardiner, “Quantum theory of optical homodyne and heterodyne detection,” J. Mod. Opt. 34(6-7), 881–902 (1987). [CrossRef]  

2. C. C. Davis, “Building small, extremely sensitive practical laser interferometers for sensor applications,” Nucl. Phys. B 6, 290–297 (1989). [CrossRef]  

3. N. Bobroff, “Recent advanced in displacement measuring interferometry,” Meas. Sci. Technol. 4(9), 907–926 (1993). [CrossRef]  

4. P. Schellekens, G. Wilkening, F. Reinboth, M. J. Downs, K. P. Birch, and J. Spronck, “Measurements of the Refractive Index of Air Using Interference Refractormeters,” Metrologia 22(4), 279–287 (1986). [CrossRef]  

5. J. Terrien, “An air refractometer for interference length metrology,” Metrologia 1(3), 80–83 (1965). [CrossRef]  

6. K. H. Kwon, B. S. Kim, and K. Cho, “A new scanning heterodyne interferometer scheme for mapping both surface structure and effective local reflection coefficient,” Opt. Express 16(17), 13456–13464 (2008). [CrossRef]   [PubMed]  

7. Y. Park, K.-E. Kim, S.-J. Kim, J.-G. Park, Y.-H. Joo, B. Hyun Shin, S.-Y. Lee, and K. Cho, “Wide measurement range scanning heterodyne interferometer utilizing astigmatic position sensing scheme,” Opt. Lett. 36(16), 3112–3114 (2011). [CrossRef]   [PubMed]  

8. D. L. Mazzoni, K. Cho, and C. C. Davis, “A Coherent hybrid fiber-optic probe for mapping induced birefringence in GaAs structures,” J. Lightwave Technol. 11(7), 1158–1161 (1993). [CrossRef]  

9. X. Fan, I. M. White, S. I. Shopova, H. Zhu, J. D. Suter, and Y. Sun, “Sensitive optical biosensors for unlabeled targets: A review,” Anal. Chim. Acta 620(1-2), 8–26 (2008). [CrossRef]   [PubMed]  

10. K. Cho, D. L. Mazzoni, and C. C. Davis, “Measurement of the local slope of a surface by vibrating-sample interferometry: a new method in scanning microscopy,” Opt. Lett. 18(3), 232–234 (1993). [CrossRef]   [PubMed]  

11. Y. Park and K. Cho, “Heterodyne interferometer scheme using a double pass in an acousto-optic modulator,” Opt. Lett. 36(3), 331–333 (2011). [CrossRef]   [PubMed]  

12. M. Chiu, B. Shih, and C. Lai, “Laser-scanning angle deviation microscopy,” Appl. Phys. Lett. 90(2), 021111 (2007). [CrossRef]  

13. M. M. Varma, D. D. Nolte, H. D. Inerowicz, and F. E. Regnier, “Spinning-disk self-referencing interferometry of antigen-antibody recognition,” Opt. Lett. 29(9), 950–952 (2004). [CrossRef]   [PubMed]  

14. D. J. Bornhop, J. C. Latham, A. Kussrow, D. A. Markov, R. D. Jones, and H. S. Sørensen, “Free-solution, label-free molecular interactions studied by back-scattering interferometry,” Science 317(5845), 1732–1736 (2007). [CrossRef]   [PubMed]  

15. C. M. Wu, “Heterodyne interferometric system with subnanometer accuracy for measurement of straightness,” Appl. Opt. 43(19), 3812–3816 (2004). [CrossRef]   [PubMed]  

16. N. A. Riza and M. A. Arain, “Angstrom-range optical path-length measurement with a high-speed scanning heterodyne optical interferometer,” Appl. Opt. 42(13), 2341–2345 (2003). [CrossRef]   [PubMed]  

17. S. J. A. G. Cosijns, H. Haitjema, and P. H. J. Schellekens, “Modeling and verifying non-linearities in heterodyne displacement interferometry,” Precis. Eng. 26(4), 448–455 (2002). [CrossRef]  

18. A. N. Bashkatov and E. A. Genina, “Water refractive index in dependence on temperature and wavelength: a simple approximation,” Proc. SPIE 5068, 393–395 (2003). [CrossRef]  

19. W. M. Yunus and A. B. Rahman, “Refractive index of solutions at high concentrations,” Appl. Opt. 27(16), 3341–3343 (1988). [CrossRef]   [PubMed]  

20. R. J. Jiménez Riobóo, M. Philipp, M. A. Ramos, and J. K. Krüger, “Concentration and temperature dependence of the refractive index of ethanol-water mixtures: Influence of intermolecular interactions,” Eur Phys J E Soft Matter 30(1), 19–26 (2009). [CrossRef]   [PubMed]  

21. O. Esteban, M. Cruz-Navarrete, A. González-Cano, and E. Bernabeu, “Measurement of the degree of salinity of water with a fiber-optic sensor,” Appl. Opt. 38(25), 5267–5271 (1999). [CrossRef]   [PubMed]  

References

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  1. M. J. Collett, R. Loudon, and C. W. Gardiner, “Quantum theory of optical homodyne and heterodyne detection,” J. Mod. Opt.34(6-7), 881–902 (1987).
    [CrossRef]
  2. C. C. Davis, “Building small, extremely sensitive practical laser interferometers for sensor applications,” Nucl. Phys. B6, 290–297 (1989).
    [CrossRef]
  3. N. Bobroff, “Recent advanced in displacement measuring interferometry,” Meas. Sci. Technol.4(9), 907–926 (1993).
    [CrossRef]
  4. P. Schellekens, G. Wilkening, F. Reinboth, M. J. Downs, K. P. Birch, and J. Spronck, “Measurements of the Refractive Index of Air Using Interference Refractormeters,” Metrologia22(4), 279–287 (1986).
    [CrossRef]
  5. J. Terrien, “An air refractometer for interference length metrology,” Metrologia1(3), 80–83 (1965).
    [CrossRef]
  6. K. H. Kwon, B. S. Kim, and K. Cho, “A new scanning heterodyne interferometer scheme for mapping both surface structure and effective local reflection coefficient,” Opt. Express16(17), 13456–13464 (2008).
    [CrossRef] [PubMed]
  7. Y. Park, K.-E. Kim, S.-J. Kim, J.-G. Park, Y.-H. Joo, B. Hyun Shin, S.-Y. Lee, and K. Cho, “Wide measurement range scanning heterodyne interferometer utilizing astigmatic position sensing scheme,” Opt. Lett.36(16), 3112–3114 (2011).
    [CrossRef] [PubMed]
  8. D. L. Mazzoni, K. Cho, and C. C. Davis, “A Coherent hybrid fiber-optic probe for mapping induced birefringence in GaAs structures,” J. Lightwave Technol.11(7), 1158–1161 (1993).
    [CrossRef]
  9. X. Fan, I. M. White, S. I. Shopova, H. Zhu, J. D. Suter, and Y. Sun, “Sensitive optical biosensors for unlabeled targets: A review,” Anal. Chim. Acta620(1-2), 8–26 (2008).
    [CrossRef] [PubMed]
  10. K. Cho, D. L. Mazzoni, and C. C. Davis, “Measurement of the local slope of a surface by vibrating-sample interferometry: a new method in scanning microscopy,” Opt. Lett.18(3), 232–234 (1993).
    [CrossRef] [PubMed]
  11. Y. Park and K. Cho, “Heterodyne interferometer scheme using a double pass in an acousto-optic modulator,” Opt. Lett.36(3), 331–333 (2011).
    [CrossRef] [PubMed]
  12. M. Chiu, B. Shih, and C. Lai, “Laser-scanning angle deviation microscopy,” Appl. Phys. Lett.90(2), 021111 (2007).
    [CrossRef]
  13. M. M. Varma, D. D. Nolte, H. D. Inerowicz, and F. E. Regnier, “Spinning-disk self-referencing interferometry of antigen-antibody recognition,” Opt. Lett.29(9), 950–952 (2004).
    [CrossRef] [PubMed]
  14. D. J. Bornhop, J. C. Latham, A. Kussrow, D. A. Markov, R. D. Jones, and H. S. Sørensen, “Free-solution, label-free molecular interactions studied by back-scattering interferometry,” Science317(5845), 1732–1736 (2007).
    [CrossRef] [PubMed]
  15. C. M. Wu, “Heterodyne interferometric system with subnanometer accuracy for measurement of straightness,” Appl. Opt.43(19), 3812–3816 (2004).
    [CrossRef] [PubMed]
  16. N. A. Riza and M. A. Arain, “Angstrom-range optical path-length measurement with a high-speed scanning heterodyne optical interferometer,” Appl. Opt.42(13), 2341–2345 (2003).
    [CrossRef] [PubMed]
  17. S. J. A. G. Cosijns, H. Haitjema, and P. H. J. Schellekens, “Modeling and verifying non-linearities in heterodyne displacement interferometry,” Precis. Eng.26(4), 448–455 (2002).
    [CrossRef]
  18. A. N. Bashkatov and E. A. Genina, “Water refractive index in dependence on temperature and wavelength: a simple approximation,” Proc. SPIE5068, 393–395 (2003).
    [CrossRef]
  19. W. M. Yunus and A. B. Rahman, “Refractive index of solutions at high concentrations,” Appl. Opt.27(16), 3341–3343 (1988).
    [CrossRef] [PubMed]
  20. R. J. Jiménez Riobóo, M. Philipp, M. A. Ramos, and J. K. Krüger, “Concentration and temperature dependence of the refractive index of ethanol-water mixtures: Influence of intermolecular interactions,” Eur Phys J E Soft Matter30(1), 19–26 (2009).
    [CrossRef] [PubMed]
  21. O. Esteban, M. Cruz-Navarrete, A. González-Cano, and E. Bernabeu, “Measurement of the degree of salinity of water with a fiber-optic sensor,” Appl. Opt.38(25), 5267–5271 (1999).
    [CrossRef] [PubMed]

2011 (2)

2009 (1)

R. J. Jiménez Riobóo, M. Philipp, M. A. Ramos, and J. K. Krüger, “Concentration and temperature dependence of the refractive index of ethanol-water mixtures: Influence of intermolecular interactions,” Eur Phys J E Soft Matter30(1), 19–26 (2009).
[CrossRef] [PubMed]

2008 (2)

X. Fan, I. M. White, S. I. Shopova, H. Zhu, J. D. Suter, and Y. Sun, “Sensitive optical biosensors for unlabeled targets: A review,” Anal. Chim. Acta620(1-2), 8–26 (2008).
[CrossRef] [PubMed]

K. H. Kwon, B. S. Kim, and K. Cho, “A new scanning heterodyne interferometer scheme for mapping both surface structure and effective local reflection coefficient,” Opt. Express16(17), 13456–13464 (2008).
[CrossRef] [PubMed]

2007 (2)

M. Chiu, B. Shih, and C. Lai, “Laser-scanning angle deviation microscopy,” Appl. Phys. Lett.90(2), 021111 (2007).
[CrossRef]

D. J. Bornhop, J. C. Latham, A. Kussrow, D. A. Markov, R. D. Jones, and H. S. Sørensen, “Free-solution, label-free molecular interactions studied by back-scattering interferometry,” Science317(5845), 1732–1736 (2007).
[CrossRef] [PubMed]

2004 (2)

2003 (2)

N. A. Riza and M. A. Arain, “Angstrom-range optical path-length measurement with a high-speed scanning heterodyne optical interferometer,” Appl. Opt.42(13), 2341–2345 (2003).
[CrossRef] [PubMed]

A. N. Bashkatov and E. A. Genina, “Water refractive index in dependence on temperature and wavelength: a simple approximation,” Proc. SPIE5068, 393–395 (2003).
[CrossRef]

2002 (1)

S. J. A. G. Cosijns, H. Haitjema, and P. H. J. Schellekens, “Modeling and verifying non-linearities in heterodyne displacement interferometry,” Precis. Eng.26(4), 448–455 (2002).
[CrossRef]

1999 (1)

1993 (3)

D. L. Mazzoni, K. Cho, and C. C. Davis, “A Coherent hybrid fiber-optic probe for mapping induced birefringence in GaAs structures,” J. Lightwave Technol.11(7), 1158–1161 (1993).
[CrossRef]

K. Cho, D. L. Mazzoni, and C. C. Davis, “Measurement of the local slope of a surface by vibrating-sample interferometry: a new method in scanning microscopy,” Opt. Lett.18(3), 232–234 (1993).
[CrossRef] [PubMed]

N. Bobroff, “Recent advanced in displacement measuring interferometry,” Meas. Sci. Technol.4(9), 907–926 (1993).
[CrossRef]

1989 (1)

C. C. Davis, “Building small, extremely sensitive practical laser interferometers for sensor applications,” Nucl. Phys. B6, 290–297 (1989).
[CrossRef]

1988 (1)

1987 (1)

M. J. Collett, R. Loudon, and C. W. Gardiner, “Quantum theory of optical homodyne and heterodyne detection,” J. Mod. Opt.34(6-7), 881–902 (1987).
[CrossRef]

1986 (1)

P. Schellekens, G. Wilkening, F. Reinboth, M. J. Downs, K. P. Birch, and J. Spronck, “Measurements of the Refractive Index of Air Using Interference Refractormeters,” Metrologia22(4), 279–287 (1986).
[CrossRef]

1965 (1)

J. Terrien, “An air refractometer for interference length metrology,” Metrologia1(3), 80–83 (1965).
[CrossRef]

Arain, M. A.

Bashkatov, A. N.

A. N. Bashkatov and E. A. Genina, “Water refractive index in dependence on temperature and wavelength: a simple approximation,” Proc. SPIE5068, 393–395 (2003).
[CrossRef]

Bernabeu, E.

Birch, K. P.

P. Schellekens, G. Wilkening, F. Reinboth, M. J. Downs, K. P. Birch, and J. Spronck, “Measurements of the Refractive Index of Air Using Interference Refractormeters,” Metrologia22(4), 279–287 (1986).
[CrossRef]

Bobroff, N.

N. Bobroff, “Recent advanced in displacement measuring interferometry,” Meas. Sci. Technol.4(9), 907–926 (1993).
[CrossRef]

Bornhop, D. J.

D. J. Bornhop, J. C. Latham, A. Kussrow, D. A. Markov, R. D. Jones, and H. S. Sørensen, “Free-solution, label-free molecular interactions studied by back-scattering interferometry,” Science317(5845), 1732–1736 (2007).
[CrossRef] [PubMed]

Chiu, M.

M. Chiu, B. Shih, and C. Lai, “Laser-scanning angle deviation microscopy,” Appl. Phys. Lett.90(2), 021111 (2007).
[CrossRef]

Cho, K.

Collett, M. J.

M. J. Collett, R. Loudon, and C. W. Gardiner, “Quantum theory of optical homodyne and heterodyne detection,” J. Mod. Opt.34(6-7), 881–902 (1987).
[CrossRef]

Cosijns, S. J. A. G.

S. J. A. G. Cosijns, H. Haitjema, and P. H. J. Schellekens, “Modeling and verifying non-linearities in heterodyne displacement interferometry,” Precis. Eng.26(4), 448–455 (2002).
[CrossRef]

Cruz-Navarrete, M.

Davis, C. C.

K. Cho, D. L. Mazzoni, and C. C. Davis, “Measurement of the local slope of a surface by vibrating-sample interferometry: a new method in scanning microscopy,” Opt. Lett.18(3), 232–234 (1993).
[CrossRef] [PubMed]

D. L. Mazzoni, K. Cho, and C. C. Davis, “A Coherent hybrid fiber-optic probe for mapping induced birefringence in GaAs structures,” J. Lightwave Technol.11(7), 1158–1161 (1993).
[CrossRef]

C. C. Davis, “Building small, extremely sensitive practical laser interferometers for sensor applications,” Nucl. Phys. B6, 290–297 (1989).
[CrossRef]

Downs, M. J.

P. Schellekens, G. Wilkening, F. Reinboth, M. J. Downs, K. P. Birch, and J. Spronck, “Measurements of the Refractive Index of Air Using Interference Refractormeters,” Metrologia22(4), 279–287 (1986).
[CrossRef]

Esteban, O.

Fan, X.

X. Fan, I. M. White, S. I. Shopova, H. Zhu, J. D. Suter, and Y. Sun, “Sensitive optical biosensors for unlabeled targets: A review,” Anal. Chim. Acta620(1-2), 8–26 (2008).
[CrossRef] [PubMed]

Gardiner, C. W.

M. J. Collett, R. Loudon, and C. W. Gardiner, “Quantum theory of optical homodyne and heterodyne detection,” J. Mod. Opt.34(6-7), 881–902 (1987).
[CrossRef]

Genina, E. A.

A. N. Bashkatov and E. A. Genina, “Water refractive index in dependence on temperature and wavelength: a simple approximation,” Proc. SPIE5068, 393–395 (2003).
[CrossRef]

González-Cano, A.

Haitjema, H.

S. J. A. G. Cosijns, H. Haitjema, and P. H. J. Schellekens, “Modeling and verifying non-linearities in heterodyne displacement interferometry,” Precis. Eng.26(4), 448–455 (2002).
[CrossRef]

Hyun Shin, B.

Inerowicz, H. D.

Jiménez Riobóo, R. J.

R. J. Jiménez Riobóo, M. Philipp, M. A. Ramos, and J. K. Krüger, “Concentration and temperature dependence of the refractive index of ethanol-water mixtures: Influence of intermolecular interactions,” Eur Phys J E Soft Matter30(1), 19–26 (2009).
[CrossRef] [PubMed]

Jones, R. D.

D. J. Bornhop, J. C. Latham, A. Kussrow, D. A. Markov, R. D. Jones, and H. S. Sørensen, “Free-solution, label-free molecular interactions studied by back-scattering interferometry,” Science317(5845), 1732–1736 (2007).
[CrossRef] [PubMed]

Joo, Y.-H.

Kim, B. S.

Kim, K.-E.

Kim, S.-J.

Krüger, J. K.

R. J. Jiménez Riobóo, M. Philipp, M. A. Ramos, and J. K. Krüger, “Concentration and temperature dependence of the refractive index of ethanol-water mixtures: Influence of intermolecular interactions,” Eur Phys J E Soft Matter30(1), 19–26 (2009).
[CrossRef] [PubMed]

Kussrow, A.

D. J. Bornhop, J. C. Latham, A. Kussrow, D. A. Markov, R. D. Jones, and H. S. Sørensen, “Free-solution, label-free molecular interactions studied by back-scattering interferometry,” Science317(5845), 1732–1736 (2007).
[CrossRef] [PubMed]

Kwon, K. H.

Lai, C.

M. Chiu, B. Shih, and C. Lai, “Laser-scanning angle deviation microscopy,” Appl. Phys. Lett.90(2), 021111 (2007).
[CrossRef]

Latham, J. C.

D. J. Bornhop, J. C. Latham, A. Kussrow, D. A. Markov, R. D. Jones, and H. S. Sørensen, “Free-solution, label-free molecular interactions studied by back-scattering interferometry,” Science317(5845), 1732–1736 (2007).
[CrossRef] [PubMed]

Lee, S.-Y.

Loudon, R.

M. J. Collett, R. Loudon, and C. W. Gardiner, “Quantum theory of optical homodyne and heterodyne detection,” J. Mod. Opt.34(6-7), 881–902 (1987).
[CrossRef]

Markov, D. A.

D. J. Bornhop, J. C. Latham, A. Kussrow, D. A. Markov, R. D. Jones, and H. S. Sørensen, “Free-solution, label-free molecular interactions studied by back-scattering interferometry,” Science317(5845), 1732–1736 (2007).
[CrossRef] [PubMed]

Mazzoni, D. L.

D. L. Mazzoni, K. Cho, and C. C. Davis, “A Coherent hybrid fiber-optic probe for mapping induced birefringence in GaAs structures,” J. Lightwave Technol.11(7), 1158–1161 (1993).
[CrossRef]

K. Cho, D. L. Mazzoni, and C. C. Davis, “Measurement of the local slope of a surface by vibrating-sample interferometry: a new method in scanning microscopy,” Opt. Lett.18(3), 232–234 (1993).
[CrossRef] [PubMed]

Nolte, D. D.

Park, J.-G.

Park, Y.

Philipp, M.

R. J. Jiménez Riobóo, M. Philipp, M. A. Ramos, and J. K. Krüger, “Concentration and temperature dependence of the refractive index of ethanol-water mixtures: Influence of intermolecular interactions,” Eur Phys J E Soft Matter30(1), 19–26 (2009).
[CrossRef] [PubMed]

Rahman, A. B.

Ramos, M. A.

R. J. Jiménez Riobóo, M. Philipp, M. A. Ramos, and J. K. Krüger, “Concentration and temperature dependence of the refractive index of ethanol-water mixtures: Influence of intermolecular interactions,” Eur Phys J E Soft Matter30(1), 19–26 (2009).
[CrossRef] [PubMed]

Regnier, F. E.

Reinboth, F.

P. Schellekens, G. Wilkening, F. Reinboth, M. J. Downs, K. P. Birch, and J. Spronck, “Measurements of the Refractive Index of Air Using Interference Refractormeters,” Metrologia22(4), 279–287 (1986).
[CrossRef]

Riza, N. A.

Schellekens, P.

P. Schellekens, G. Wilkening, F. Reinboth, M. J. Downs, K. P. Birch, and J. Spronck, “Measurements of the Refractive Index of Air Using Interference Refractormeters,” Metrologia22(4), 279–287 (1986).
[CrossRef]

Schellekens, P. H. J.

S. J. A. G. Cosijns, H. Haitjema, and P. H. J. Schellekens, “Modeling and verifying non-linearities in heterodyne displacement interferometry,” Precis. Eng.26(4), 448–455 (2002).
[CrossRef]

Shih, B.

M. Chiu, B. Shih, and C. Lai, “Laser-scanning angle deviation microscopy,” Appl. Phys. Lett.90(2), 021111 (2007).
[CrossRef]

Shopova, S. I.

X. Fan, I. M. White, S. I. Shopova, H. Zhu, J. D. Suter, and Y. Sun, “Sensitive optical biosensors for unlabeled targets: A review,” Anal. Chim. Acta620(1-2), 8–26 (2008).
[CrossRef] [PubMed]

Sørensen, H. S.

D. J. Bornhop, J. C. Latham, A. Kussrow, D. A. Markov, R. D. Jones, and H. S. Sørensen, “Free-solution, label-free molecular interactions studied by back-scattering interferometry,” Science317(5845), 1732–1736 (2007).
[CrossRef] [PubMed]

Spronck, J.

P. Schellekens, G. Wilkening, F. Reinboth, M. J. Downs, K. P. Birch, and J. Spronck, “Measurements of the Refractive Index of Air Using Interference Refractormeters,” Metrologia22(4), 279–287 (1986).
[CrossRef]

Sun, Y.

X. Fan, I. M. White, S. I. Shopova, H. Zhu, J. D. Suter, and Y. Sun, “Sensitive optical biosensors for unlabeled targets: A review,” Anal. Chim. Acta620(1-2), 8–26 (2008).
[CrossRef] [PubMed]

Suter, J. D.

X. Fan, I. M. White, S. I. Shopova, H. Zhu, J. D. Suter, and Y. Sun, “Sensitive optical biosensors for unlabeled targets: A review,” Anal. Chim. Acta620(1-2), 8–26 (2008).
[CrossRef] [PubMed]

Terrien, J.

J. Terrien, “An air refractometer for interference length metrology,” Metrologia1(3), 80–83 (1965).
[CrossRef]

Varma, M. M.

White, I. M.

X. Fan, I. M. White, S. I. Shopova, H. Zhu, J. D. Suter, and Y. Sun, “Sensitive optical biosensors for unlabeled targets: A review,” Anal. Chim. Acta620(1-2), 8–26 (2008).
[CrossRef] [PubMed]

Wilkening, G.

P. Schellekens, G. Wilkening, F. Reinboth, M. J. Downs, K. P. Birch, and J. Spronck, “Measurements of the Refractive Index of Air Using Interference Refractormeters,” Metrologia22(4), 279–287 (1986).
[CrossRef]

Wu, C. M.

Yunus, W. M.

Zhu, H.

X. Fan, I. M. White, S. I. Shopova, H. Zhu, J. D. Suter, and Y. Sun, “Sensitive optical biosensors for unlabeled targets: A review,” Anal. Chim. Acta620(1-2), 8–26 (2008).
[CrossRef] [PubMed]

Anal. Chim. Acta (1)

X. Fan, I. M. White, S. I. Shopova, H. Zhu, J. D. Suter, and Y. Sun, “Sensitive optical biosensors for unlabeled targets: A review,” Anal. Chim. Acta620(1-2), 8–26 (2008).
[CrossRef] [PubMed]

Appl. Opt. (4)

Appl. Phys. Lett. (1)

M. Chiu, B. Shih, and C. Lai, “Laser-scanning angle deviation microscopy,” Appl. Phys. Lett.90(2), 021111 (2007).
[CrossRef]

Eur Phys J E Soft Matter (1)

R. J. Jiménez Riobóo, M. Philipp, M. A. Ramos, and J. K. Krüger, “Concentration and temperature dependence of the refractive index of ethanol-water mixtures: Influence of intermolecular interactions,” Eur Phys J E Soft Matter30(1), 19–26 (2009).
[CrossRef] [PubMed]

J. Lightwave Technol. (1)

D. L. Mazzoni, K. Cho, and C. C. Davis, “A Coherent hybrid fiber-optic probe for mapping induced birefringence in GaAs structures,” J. Lightwave Technol.11(7), 1158–1161 (1993).
[CrossRef]

J. Mod. Opt. (1)

M. J. Collett, R. Loudon, and C. W. Gardiner, “Quantum theory of optical homodyne and heterodyne detection,” J. Mod. Opt.34(6-7), 881–902 (1987).
[CrossRef]

Meas. Sci. Technol. (1)

N. Bobroff, “Recent advanced in displacement measuring interferometry,” Meas. Sci. Technol.4(9), 907–926 (1993).
[CrossRef]

Metrologia (2)

P. Schellekens, G. Wilkening, F. Reinboth, M. J. Downs, K. P. Birch, and J. Spronck, “Measurements of the Refractive Index of Air Using Interference Refractormeters,” Metrologia22(4), 279–287 (1986).
[CrossRef]

J. Terrien, “An air refractometer for interference length metrology,” Metrologia1(3), 80–83 (1965).
[CrossRef]

Nucl. Phys. B (1)

C. C. Davis, “Building small, extremely sensitive practical laser interferometers for sensor applications,” Nucl. Phys. B6, 290–297 (1989).
[CrossRef]

Opt. Express (1)

Opt. Lett. (4)

Precis. Eng. (1)

S. J. A. G. Cosijns, H. Haitjema, and P. H. J. Schellekens, “Modeling and verifying non-linearities in heterodyne displacement interferometry,” Precis. Eng.26(4), 448–455 (2002).
[CrossRef]

Proc. SPIE (1)

A. N. Bashkatov and E. A. Genina, “Water refractive index in dependence on temperature and wavelength: a simple approximation,” Proc. SPIE5068, 393–395 (2003).
[CrossRef]

Science (1)

D. J. Bornhop, J. C. Latham, A. Kussrow, D. A. Markov, R. D. Jones, and H. S. Sørensen, “Free-solution, label-free molecular interactions studied by back-scattering interferometry,” Science317(5845), 1732–1736 (2007).
[CrossRef] [PubMed]

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Figures (4)

Fig. 1
Fig. 1

Schematic of experimental arrangement (a), and a photograph of the interferometer and sample part of the experimental arrangement (b).

Fig. 2
Fig. 2

Frequency spectra of vibration measurements: (a) the RB and PB are reflected by the two independent mirrors of which a 40Hz vibration is applied to one mirror by using a PZT, and (b) the PB and RB are reflected by one mirror which is driven by the same PZT with the identical driving signal with (a).

Fig. 3
Fig. 3

Experimental results on RI measurements: (a) Phase measurements for various concentrations of ethylene glycol solution. The phase was modulated by altering flow of the water and the ethylene glycol solutions in the SCH. (b) Measurement results for concentration dependent ΔRIs for various solutions. The solid lines are extrapolations of the corresponding linear fitting results of previous measurements in [1820].

Fig. 4
Fig. 4

Phase noise measurements when both channels are filled with water for 10 minutes (a), and RMS noises for 20 same consecutive measurements.

Equations (3)

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v Q v I =tan( Δϕ )
Δϕ= 2π c [ ( ν 1 ν 2 ) l arm +Δ ϕ QS +Δ ϕ NL +2( n R l R ν 1 n s l s ν 2 ) ]
Δ ϕ FC = 4π c ( n R ν 1 n s ν 2 ) l FC ,

Metrics