## Quadrature phase-shift error analysis using a homodyne laser interferometer

Optics Express, Vol. 17, Issue 18, pp. 16322-16331 (2009)

http://dx.doi.org/10.1364/OE.17.016322

Acrobat PDF (361 KB)

### Abstract

The influence of quadrature phase shift on the measured displacement error was experimentally investigated using a two-detector polarizing homodyne laser interferometer with a quadrature detection system. Common nonlinearities, including the phase-shift error, were determined and effectively corrected by a robust data-processing algorithm. The measured phase-shift error perfectly agrees with the theoretically determined phase-shift error region. This error is systematic, periodic and severely asymmetrical around the nominal displacement value. The main results presented in this paper can also be used to assess and correct the detector errors of other interferometric and non-interferometric displacement-measuring devices based on phase-quadrature detection.

© 2009 OSA

## 1. Introduction

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

5. M. Novak, J. Millerd, N. Brock, M. North-Morris, J. Hayes, and J. Wyant, “Analysis of a micropolarizer array-based simultaneous phase-shifting interferometer,” Appl. Opt. **44**(32), 6861–6868 (2005). [CrossRef] [PubMed]

6. L. M. Sanchez-Brea and T. Morlanes, “Metrological errors in optical encoders,” Meas. Sci. Technol. **19**(11), 115104 (2008). [CrossRef]

7. T. Požar, R. Petkovšek, and J. Možina, “Dispersion of an optodynamic wave during its multiple transitions in a rod,” Appl. Phys. Lett. **92**(23), 234101–234103 (2008). [CrossRef]

8. G. L. Dai, F. Pohlenz, H. U. Danzebrink, K. Hasche, and G. Wilkening, “Improving the performance of interferometers in metrological scanning probe microscopes,” Meas. Sci. Technol. **15**(2), 444–450 (2004). [CrossRef]

5. M. Novak, J. Millerd, N. Brock, M. North-Morris, J. Hayes, and J. Wyant, “Analysis of a micropolarizer array-based simultaneous phase-shifting interferometer,” Appl. Opt. **44**(32), 6861–6868 (2005). [CrossRef] [PubMed]

9. N.-I. Toto-Arellano, G. Rodriguez-Zurita, C. Meneses-Fabian, and J. F. Vazquez-Castillo, “Phase shifts in the Fourier spectra of phase gratings and phase grids: an application for one-shot phase-shifting interferometry,” Opt. Express **16**(23), 19330–19341 (2008). [CrossRef]

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

10. P. L. M. Heydemann, “Determination and correction of quadrature fringe measurement errors in interferometers,” Appl. Opt. **20**(19), 3382–3384 (1981). [CrossRef] [PubMed]

11. C.-M. Wu and C.-S. Su, “Nonlinearity in measurements of length by optical interferometry,” Meas. Sci. Technol. **7**(1), 62–68 (1996). [CrossRef]

12. T. Usuda and T. Kurosawa, “Calibration methods for vibration transducers and their uncertainties,” Metrologia **36**(4), 375–383 (1999). [CrossRef]

6. L. M. Sanchez-Brea and T. Morlanes, “Metrological errors in optical encoders,” Meas. Sci. Technol. **19**(11), 115104 (2008). [CrossRef]

11. C.-M. Wu and C.-S. Su, “Nonlinearity in measurements of length by optical interferometry,” Meas. Sci. Technol. **7**(1), 62–68 (1996). [CrossRef]

10. P. L. M. Heydemann, “Determination and correction of quadrature fringe measurement errors in interferometers,” Appl. Opt. **20**(19), 3382–3384 (1981). [CrossRef] [PubMed]

13. K. P. Birch, “Optical fringe subdivision with nanometric accuracy,” Precis. Eng. **12**(4), 195–198 (1990). [CrossRef]

14. C.-M. Wu, C.-S. Su, and G.-S. Peng, “Correction of nonlinearity in one-frequency optical interferometry,” Meas. Sci. Technol. **7**(4), 520–524 (1996). [CrossRef]

15. T. Eom, J. Kim, and K. Jeong, “The dynamic compensation of nonlinearity in a homodyne laser interferometer,” Meas. Sci. Technol. **12**(10), 1734–1738 (2001). [CrossRef]

16. J.-A. Kim, J. W. Kim, C.-S. Kang, T. B. Eom, and J. Ahn, “A digital signal processing module for real-time compensation of nonlinearity in a homodyne interferometer using a field-programmable gate array,” Meas. Sci. Technol. **20**(1), 017003 (2009). [CrossRef]

## 2. Homodyne quadrature laser interferometer

3. V. Greco, G. Molesini, and F. Quercioli, “Accurate polarization interferometer,” Rev. Sci. Instrum. **66**(7), 3729–3734 (1995). [CrossRef]

11. C.-M. Wu and C.-S. Su, “Nonlinearity in measurements of length by optical interferometry,” Meas. Sci. Technol. **7**(1), 62–68 (1996). [CrossRef]

**7**(1), 62–68 (1996). [CrossRef]

14. C.-M. Wu, C.-S. Su, and G.-S. Peng, “Correction of nonlinearity in one-frequency optical interferometry,” Meas. Sci. Technol. **7**(4), 520–524 (1996). [CrossRef]

18. I. Dániel, “Advanced successive phase unwrapping algorithm for quadrature output Michelson interferometers,” Measurement **37**(2), 95–102 (2005). [CrossRef]

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

3. V. Greco, G. Molesini, and F. Quercioli, “Accurate polarization interferometer,” Rev. Sci. Instrum. **66**(7), 3729–3734 (1995). [CrossRef]

*λ*= 632.8 nm; amplitude stability over 1 min < 0.2% and amplitude noise (0 – 10 MHz) < 0.2%) is rotated so that the linearly polarized beam exiting the laser forms a 45

*°*angle with respect to the plane of the optical table (

*x*-plane). This polarization can be decomposed into two orthogonal polarizations with equal intensities, one in the plane of the paper (

*x*-axis) and the other perpendicular to it (

*y*-axis). The beamsplitter (BS) evenly splits the beam into the reference and measurement arms. The first transition through the octadic-wave plate (OWP), which is placed in the reference arm, gives rise to the 45

*°*(

*λ*/8) phase difference between the orthogonal polarizations. The beam is then reflected from a high-reflectivity (HR) mirror and another 45

*°*are added on the returning passage through the OWP. The orthogonal polarizations in the measurement arm experience an equal phase shift

*δ*(

*u*) due to the movement of the measuring surface, which is driven by a piezoelectric transducer (PZT). The light in phase opposition is not detected since it returns towards the laser. The PBS transmits the

*x*-polarization and reflects the

*y*-polarization. An optically narrow band-pass filter (BPF) is placed before the photodiodes labeled PDx and PDy to attenuate the scattered light at different wavelengths.

3. V. Greco, G. Molesini, and F. Quercioli, “Accurate polarization interferometer,” Rev. Sci. Instrum. **66**(7), 3729–3734 (1995). [CrossRef]

4. T. Keem, S. Gonda, I. Misumi, Q. Huang, and T. Kurosawa, “Removing nonlinearity of a homodyne interferometer by adjusting the gains of its quadrature detector systems,” Appl. Opt. **43**(12), 2443–2448 (2004). [CrossRef] [PubMed]

**E**and the ideal optical components found in the HQLI can be written as:

*bs*corresponds to the 50%-50% BS,

*pbs*to the PBS’s transmitted output with the polarization in the

_{x}*x*-plane,

*pbs*to the reflected output with the polarization in the perpendicular direction (

_{y}*y*-plane),

*opd*is the phase factor that accounts for the optical phase difference between the two arms arising from the displacement of the measuring surface, and

*owp*is an octadic-wave plate that can be rotated by an arbitrary angle

*ϕ*, measured from the

*y*-plane to the OWP’s fast axis.

*x*-plane, one from the reference arm and the other from the measurement arm, reach the photodiode PDx. Similarly, the perpendicular polarizations coming from both arms illuminate the photodiode PDy. Ideally, the interference signals on the photodiodes are shifted by 90

*°*, which can be achieved with a properly rotated OWP. The measured displacement

*u*is encoded in the phase

*δ*(

*u*) = 4

*πu*/

*λ*, where

*λ*is the wavelength of the interferometric laser. The electric fields arising from the reference (index

*r*) and the measurement (index

*m*) arms can be calculated using the Jones matrix formalism as:

*I*and

_{x}*I*, as a function of the optical phase difference

_{y}*δ*and the angle of the OWP rotation

*ϕ*. The dagger denotes the conjugate transpose and

*I*

_{0}stands for the laser output intensity. The intensities on the photodiodes are:

*x*-plane, the signals are in phase quadrature:

*ϕ*on the detected displacement will be discussed later. The displacement of the measuring surface along the line of the laser beam can be derived from the ideal quadrature signals

*I*and

_{x}*I*(Eq. (2)) by subtracting the DC offset as:

_{y}*m*has to be chosen correctly so that the function

*u*(

*t*) becomes continuous.

## 3. Signal processing

*I*and

_{x}*I*, are detected as the photodiode output signals,

_{y}*V*(

_{x}*t*) and

*V*(

_{y}*t*), which were equidistantly sampled by a 500-MHz oscilloscope with a sampling capacity of 2 MS per channel. The sampling is limited either by the oscilloscope’s sampling rate or by the frequency response of the photodiodes. Assuming that the acquired (raw) signals take the following distorted formthey have to be corrected so that the phase unwrapping (Eq. (3)) yields an accurate displacement. Here,

*V*

_{x}_{0,}

_{y}_{0}stands for the AC amplitudes of the detected voltage,

*δ*

_{x}_{,}

*are the corresponding initial phases and*

_{y}*V*are the DC offsets.

_{xoff,yoff}*V*

_{x}_{0,}

_{y}_{0}and

*V*are constants, because we ensured a highly constant laser output power within the acquisition time. A few error sources influence only a single parameter in Eq. (4), while, for example, rotating the OWP contributes to a change in all the parameters in Eq. (4), as described by Eqs. (1).

_{xoff,yoff}*f*= 100 Hz harmonically moves the measuring HR mirror by an amplitude

*A*= 270 nm are shown in Fig. 2(a) . After the raw signals were acquired from photodiodes, the data processing was done offline in two steps: in the first place, the acquired (raw) signals

*V*and

_{x}*V*were transformed into the processed error-corrected signals

_{y}*s*and

_{x}*s*(the arrow from Fig. 2(a) to Fig. 2(b)); this was followed by a phase-unwrapping transformation (the arrow from Fig. 2(b) to Fig. 2(c)).

_{y}*V*(

*I*) nonlinearities of both photodiodes. The signals were then low-pass filtered, thereby eliminating the unwanted high-frequency noise with an adjustable cutoff. The filtered signals were later separately normalized according to the maximum and minimum filtered voltage in each channel and shifted to eliminate the offsets. This procedure can be expanded to the case when the amplitudes

*V*

_{x}_{0,}

_{y}_{0}and the zero offsets

*V*vary with time, which cannot be done with the standard least-squares fitting methods [10

_{xoff,yoff}10. P. L. M. Heydemann, “Determination and correction of quadrature fringe measurement errors in interferometers,” Appl. Opt. **20**(19), 3382–3384 (1981). [CrossRef] [PubMed]

13. K. P. Birch, “Optical fringe subdivision with nanometric accuracy,” Precis. Eng. **12**(4), 195–198 (1990). [CrossRef]

14. C.-M. Wu, C.-S. Su, and G.-S. Peng, “Correction of nonlinearity in one-frequency optical interferometry,” Meas. Sci. Technol. **7**(4), 520–524 (1996). [CrossRef]

*λ*/2, the intensity extremes may not be reached, so in this case the raw signals have to be normalized and zero-shifted with the values of the extremes obtained before the real measurement is performed. The lack-of-quadrature correction was carried out last; it was derived from the work of Heydemann [10

**20**(19), 3382–3384 (1981). [CrossRef] [PubMed]

*s*(

_{x}*t*),

*s*(

_{y}*t*)). One revolution of the rotating vector path corresponds to a phase change of 2

*π*. This is equivalent to a displacement by

*λ*/2 of the measuring surface, so the measurement of the displacement becomes possible by following the phase of the rotating vector. As the measuring surface moves forward (towards the BS), the vector rotates in the counterclockwise direction. If it moves backward (away from the BS), the vector rotates in the clockwise direction.

*s*and

_{x}*s*using a phase-unwrapping algorithm depicted in the flowchart in Ref [12

_{y}12. T. Usuda and T. Kurosawa, “Calibration methods for vibration transducers and their uncertainties,” Metrologia **36**(4), 375–383 (1999). [CrossRef]

*s*and

_{x}*s*(the dashed line A in Fig. 2). The opposite transition (the dashed line B in Fig. 2) corresponds to the alternating second derivatives of

_{y}*s*and

_{x}*s*.

_{y}### 3.1 Correction of the quadrature phase-shift error

*V*

_{x}_{0,}

_{y}_{0}= 1) and the zero offsets are set to null (

*V*= 0), the only significant remaining error in Eq. (4) is the quadrature phase-shift error. Setting

_{xoff,yoff}*δ*= 0 and

_{x}*δ*=

_{y}*α*, the ideal signals

*s*and

_{x}*s*are distorted by the phase-shift error and can be described bywhere

_{y}*s*and

_{xe}*s*correspond to the preprocessed signals before the phase correction is made.

_{ye}*u*(the black line) was obtained with an adjusted HQLI. Then we rotated the OWP to change the phase shift to

_{r}*α*= 117

*°*. An inaccurate measurement

*u*performed with a HQLI lacking the phase quadrature is shown as the red line in Fig. 3(a). Both measurements were made at the same frequency and displacement amplitude of the vibrating mirror (

_{m}*f*= 100 Hz,

*A*= 270 nm).

*α*and correct the signals of an inaccurate measurement (

*s*and

_{xe}*s*) as follows.

_{ye}*r*as a function of the angle

*θ*. The phase shift

*α*is obtained by fitting the expressionto the transformed data pairsusing the method of least squares. Now that

*α*has been determined by the fitting procedure the inaccurate data is corrected by inserting

*α*back into Eqs. (5). Thus, the corrected signals

*s*and

_{xc}*s*are obtained (the blue dots in Fig. 3(b)). The corrected displacement

_{yc}*u*(the blue dots in Fig. 3(a)) was calculated using the phase-unwrapping algorithm on the phase-shift-corrected data.

_{c}*u*−

_{m}*u*(the red line) and the one between the software-corrected and the reference displacement

_{r}*u*−

_{c}*u*(the blue line). The maximum displacement error was significantly reduced from the original 25 nm to the improved 3 nm. Moreover, the original error was periodic and severely unidirectional, while the corrected one is symmetrical around zero and without a period. Apart from the purpose of the software error correction, knowing the phase shift between the signals also helps as a guide for the manual adjustment of the OWP angle

_{r}*ϕ*.

*α*and the angle of the OWP rotation

*ϕ*is established from Eqs. (1) as

*°*. The sensitivity of the phase shift on the OWP rotation corresponds to the slope

*dα*/

*dϕ*, which can be extracted from Eq. (6). In the ideal case, when the phase retardation between the two polarizations is achieved only through the OWP, the HQLI’s accuracy is insensitive to slight variations of the OWP rotation. However, this may pose a problem if an additional phase shift originates from the polarization-sensitive light reflections, such as the reflection at the BS [17]. In the latter case, the OWP is used to add the remaining phase shift needed to achieve the phase quadrature, which may set the angle of the OWP rotation

*ϕ*to the point where

*dα*/

*dϕ*is large. This effect may, therefore, undermine the robustness of the interferometer and should be avoided.

## 4. Analysis of the quadrature phase-shift error

*u*is defined by subtracting the reference displacement

_{err}*u*from the inaccurately measured one

_{r}*u*as

_{m}*u*of the measuring mirror, mounted on a PZT, at various angles of OWP rotation

_{m}*ϕ*, i.e., for different phase shifts

*α*(Eq. (6)). The PZT moved the mirror in two distinct modes: harmonic (e.g., the red line in Fig. 4(a) ) and triangular (e.g., the red line in Fig. 4(b)). The mirror vibrating in harmonic mode had a frequency of 100 Hz and an amplitude of 270 nm. The triangular displacement’s frequency and amplitude were 70 Hz and 330 nm, respectively. Using the above-described phase-correction algorithm we obtained the corrected displacement

*u*(the black lines in Figs. 4(a) and 4(b)). The difference between the measured and corrected displacement

_{c}*u*−

_{m}*u*is shown as a green line in Figs. 4(a) and 4(b). We assumed that this difference equals the displacement error defined in Eq. (6). The extremes of the error in Fig. 4(a) are shown as two circles in Fig. 4(c) at

_{c}*α*= 43

*°*and the error interval is indicated by “error H”. Similarly, the two squares in Fig. 4(c) at

*α*= 140

*°*correspond to the error extremes of the triangular mode shown in Fig. 4(b). This error interval is indicated by “error T”. The other circles and squares in Fig. 4(c) represent the error extremes of the displacement error obtained at various phase shifts for the harmonic and triangular modes, respectively.

*α*in the interval between −90

*°*and 90

*°*. However, in our case the rotation of the OWP enabled measurements of the phase shift in the interval between −36

*°*and 144

*°*, because an additional phase shift of 54

*°*originates from the polarization-sensitive light reflections at the BS. This phase shift was measured by removing the OWP from the HQLI. Its origin was proved by the BS rotation of 180

*°*around the

*y*-axis (see Fig. 1). After the rotation, the additional phase shift changed the sign, indicating that the BS was the only source of this shift. Due to this effect, the measured phase-shift interval in Fig. 4(c) exceeds 90

*°*. When the phase shift is smaller than 10

*°*it can no longer be determined, because of the extreme distortion of the Lissajous curve.

*u*and

_{errB}*u*(the solid lines in Fig. 4(c)). To calculate the boundaries we need to find the extremes of Eq. (7) with respect to

_{errb}*δ*. Those that give an error which is further away from the zero error are labeled

*δ*, the others, which are closer to the zero error, are named

_{B}*δ*. Substituting the locations of the extremes back into Eq. (7) gives the two bordering lines:

_{b}6. L. M. Sanchez-Brea and T. Morlanes, “Metrological errors in optical encoders,” Meas. Sci. Technol. **19**(11), 115104 (2008). [CrossRef]

*λ*/(4

*π*) in Eq. (7) is replaced by

*p*/(2

*π*). Here,

*p*is an arbitrary position period that equals

*λ*/2 for the case of HQLI or

*λ*/

*n*for

*n*-pass realizations of similar interferometers [12

12. T. Usuda and T. Kurosawa, “Calibration methods for vibration transducers and their uncertainties,” Metrologia **36**(4), 375–383 (1999). [CrossRef]

19. M. Pisani, “Multiple reflection Michelson interferometer with picometer resolution,” Opt. Express **16**(26), 21558–21563 (2008). [CrossRef] [PubMed]

*α*= 90

*°*givesand indicates that the phase-shift displacement error is periodic with respect to

*δ*. This two-cycle period, i.e., a periodicity of two-cycles as the difference in the optical path length changes from 0 to 2

*π*, is

*p*/2 (e.g.,

*λ*/4 for HQLI). The two-cycle period is seen in Figs. 3(a), 4(a) and 4(b).

*u*−

_{m}*u*=

_{c}*u*is justified. The results presented in this analysis show: (i) the error region is independent of the amplitude, the frequency and the shape of the displacement. It depends only on the laser wavelength or, in general, on the position period of quadrature-detection systems; (ii) the error is systematic, periodic and asymmetrical around the nominal displacement value; (iii) the described phase-shift correction algorithm significantly improves the accuracy, even for large deviations from the ideal phase quadrature. However, when the acquired signals lack the quadrature, the sensitivity of the HQLI is no longer constant. It is, therefore, necessary to adjust the interferometer close to the optimal 90

_{err}*°*phase shift.

## 5. Conclusion

## References and links

1. | N. Bobroff, “Recent advances in displacement measuring interferometry,” Meas. Sci. Technol. |

2. | R. Reibold and W. Molkenstruck, “Laser interferometric measurement and computerized evaluation of ultrasonic displacements,” Acustica |

3. | V. Greco, G. Molesini, and F. Quercioli, “Accurate polarization interferometer,” Rev. Sci. Instrum. |

4. | T. Keem, S. Gonda, I. Misumi, Q. Huang, and T. Kurosawa, “Removing nonlinearity of a homodyne interferometer by adjusting the gains of its quadrature detector systems,” Appl. Opt. |

5. | M. Novak, J. Millerd, N. Brock, M. North-Morris, J. Hayes, and J. Wyant, “Analysis of a micropolarizer array-based simultaneous phase-shifting interferometer,” Appl. Opt. |

6. | L. M. Sanchez-Brea and T. Morlanes, “Metrological errors in optical encoders,” Meas. Sci. Technol. |

7. | T. Požar, R. Petkovšek, and J. Možina, “Dispersion of an optodynamic wave during its multiple transitions in a rod,” Appl. Phys. Lett. |

8. | G. L. Dai, F. Pohlenz, H. U. Danzebrink, K. Hasche, and G. Wilkening, “Improving the performance of interferometers in metrological scanning probe microscopes,” Meas. Sci. Technol. |

9. | N.-I. Toto-Arellano, G. Rodriguez-Zurita, C. Meneses-Fabian, and J. F. Vazquez-Castillo, “Phase shifts in the Fourier spectra of phase gratings and phase grids: an application for one-shot phase-shifting interferometry,” Opt. Express |

10. | P. L. M. Heydemann, “Determination and correction of quadrature fringe measurement errors in interferometers,” Appl. Opt. |

11. | C.-M. Wu and C.-S. Su, “Nonlinearity in measurements of length by optical interferometry,” Meas. Sci. Technol. |

12. | T. Usuda and T. Kurosawa, “Calibration methods for vibration transducers and their uncertainties,” Metrologia |

13. | K. P. Birch, “Optical fringe subdivision with nanometric accuracy,” Precis. Eng. |

14. | C.-M. Wu, C.-S. Su, and G.-S. Peng, “Correction of nonlinearity in one-frequency optical interferometry,” Meas. Sci. Technol. |

15. | T. Eom, J. Kim, and K. Jeong, “The dynamic compensation of nonlinearity in a homodyne laser interferometer,” Meas. Sci. Technol. |

16. | J.-A. Kim, J. W. Kim, C.-S. Kang, T. B. Eom, and J. Ahn, “A digital signal processing module for real-time compensation of nonlinearity in a homodyne interferometer using a field-programmable gate array,” Meas. Sci. Technol. |

17. | C. B. Scruby and L. E. Drain, Laser Ultrasonics: Techniques and Applications, (Adam Hilger, Bristol, 1990). |

18. | I. Dániel, “Advanced successive phase unwrapping algorithm for quadrature output Michelson interferometers,” Measurement |

19. | M. Pisani, “Multiple reflection Michelson interferometer with picometer resolution,” Opt. Express |

**OCIS Codes**

(120.3180) Instrumentation, measurement, and metrology : Interferometry

(120.3930) Instrumentation, measurement, and metrology : Metrological instrumentation

(120.5050) Instrumentation, measurement, and metrology : Phase measurement

(120.7280) Instrumentation, measurement, and metrology : Vibration analysis

(230.5440) Optical devices : Polarization-selective devices

(280.4788) Remote sensing and sensors : Optical sensing and sensors

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: June 11, 2009

Revised Manuscript: August 25, 2009

Manuscript Accepted: August 25, 2009

Published: August 28, 2009

**Citation**

Peter Gregorčič, Tomaž Požar, and Janez Možina, "Quadrature phase-shift error analysis using a homodyne laser interferometer," Opt. Express **17**, 16322-16331 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-18-16322

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### References

- N. Bobroff, “Recent advances in displacement measuring interferometry,” Meas. Sci. Technol. 4(9), 907–926 (1993). [CrossRef]
- R. Reibold and W. Molkenstruck, “Laser interferometric measurement and computerized evaluation of ultrasonic displacements,” Acustica 49, 205–211 (1981).
- V. Greco, G. Molesini, and F. Quercioli, “Accurate polarization interferometer,” Rev. Sci. Instrum. 66(7), 3729–3734 (1995). [CrossRef]
- T. Keem, S. Gonda, I. Misumi, Q. Huang, and T. Kurosawa, “Removing nonlinearity of a homodyne interferometer by adjusting the gains of its quadrature detector systems,” Appl. Opt. 43(12), 2443–2448 (2004). [CrossRef] [PubMed]
- M. Novak, J. Millerd, N. Brock, M. North-Morris, J. Hayes, and J. Wyant, “Analysis of a micropolarizer array-based simultaneous phase-shifting interferometer,” Appl. Opt. 44(32), 6861–6868 (2005). [CrossRef] [PubMed]
- L. M. Sanchez-Brea and T. Morlanes, “Metrological errors in optical encoders,” Meas. Sci. Technol. 19(11), 115104 (2008). [CrossRef]
- T. Požar, R. Petkovšek, and J. Možina, “Dispersion of an optodynamic wave during its multiple transitions in a rod,” Appl. Phys. Lett. 92(23), 234101–234103 (2008). [CrossRef]
- G. L. Dai, F. Pohlenz, H. U. Danzebrink, K. Hasche, and G. Wilkening, “Improving the performance of interferometers in metrological scanning probe microscopes,” Meas. Sci. Technol. 15(2), 444–450 (2004). [CrossRef]
- N.-I. Toto-Arellano, G. Rodriguez-Zurita, C. Meneses-Fabian, and J. F. Vazquez-Castillo, “Phase shifts in the Fourier spectra of phase gratings and phase grids: an application for one-shot phase-shifting interferometry,” Opt. Express 16(23), 19330–19341 (2008). [CrossRef]
- P. L. M. Heydemann, “Determination and correction of quadrature fringe measurement errors in interferometers,” Appl. Opt. 20(19), 3382–3384 (1981). [CrossRef] [PubMed]
- C.-M. Wu and C.-S. Su, “Nonlinearity in measurements of length by optical interferometry,” Meas. Sci. Technol. 7(1), 62–68 (1996). [CrossRef]
- T. Usuda and T. Kurosawa, “Calibration methods for vibration transducers and their uncertainties,” Metrologia 36(4), 375–383 (1999). [CrossRef]
- K. P. Birch, “Optical fringe subdivision with nanometric accuracy,” Precis. Eng. 12(4), 195–198 (1990). [CrossRef]
- C.-M. Wu, C.-S. Su, and G.-S. Peng, “Correction of nonlinearity in one-frequency optical interferometry,” Meas. Sci. Technol. 7(4), 520–524 (1996). [CrossRef]
- T. Eom, J. Kim, and K. Jeong, “The dynamic compensation of nonlinearity in a homodyne laser interferometer,” Meas. Sci. Technol. 12(10), 1734–1738 (2001). [CrossRef]
- J.-A. Kim, J. W. Kim, C.-S. Kang, T. B. Eom, and J. Ahn, “A digital signal processing module for real-time compensation of nonlinearity in a homodyne interferometer using a field-programmable gate array,” Meas. Sci. Technol. 20(1), 017003 (2009). [CrossRef]
- C. B. Scruby and L. E. Drain, Laser Ultrasonics: Techniques and Applications, (Adam Hilger, Bristol, 1990).
- I. Dániel, “Advanced successive phase unwrapping algorithm for quadrature output Michelson interferometers,” Measurement 37(2), 95–102 (2005). [CrossRef]
- M. Pisani, “Multiple reflection Michelson interferometer with picometer resolution,” Opt. Express 16(26), 21558–21563 (2008). [CrossRef] [PubMed]

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