## A passive method to compensate nonlinearity in a homodyne interferometer

Optics Express, Vol. 17, Issue 25, pp. 23299-23308 (2009)

http://dx.doi.org/10.1364/OE.17.023299

Acrobat PDF (268 KB)

### Abstract

This study presents an analysis of the nonlinearity resulting from polarization crosstalk at a polarizing beam splitter (PBS) and a wave plate (WP) in a homodyne interferometer. From a theoretical approach, a new compensation method involving a realignment of the axes of WPs to some specific angles according to the characteristics of the PBS is introduced. This method suppresses the nonlinearity in a homodyne interferometer to 0.36 nm, which would be 3.75 nm with conventional alignment methods of WPs.

© 2009 OSA

## 1. Introduction

1. I. Misumi, S. Gonda, T. Kurosawa, and K. Takamasu, “Uncertainty in pitch measurements of one-dimensional grating standards using a nanometrological atomic force microscope,” Meas. Sci. Technol. **14**(4), 463–471 (
2003). [CrossRef]

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

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

6. T. Keem, S. Gonda, I. Misumi, Q. Huang, and T. Kurosawa, “Simple, real-time method for removing the cyclic error of a homodyne interferometer with a quadrature detector system,” Appl. Opt. **44**(17), 3492–3498 (
2005). [CrossRef] [PubMed]

7. Z. Li, K. Herrmann, and F. Pohlenz, “A neural network approach to correcting nonlinearity in optical interferometers,” Meas. Sci. Technol. **14**(3), 376–381 (
2003). [CrossRef]

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

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

7. Z. Li, K. Herrmann, and F. Pohlenz, “A neural network approach to correcting nonlinearity in optical interferometers,” Meas. Sci. Technol. **14**(3), 376–381 (
2003). [CrossRef]

6. T. Keem, S. Gonda, I. Misumi, Q. Huang, and T. Kurosawa, “Simple, real-time method for removing the cyclic error of a homodyne interferometer with a quadrature detector system,” Appl. Opt. **44**(17), 3492–3498 (
2005). [CrossRef] [PubMed]

8. O. P. Lay and S. Dubovitsky, “Polarization compensation: a passive approach to a reducing heterodyne interferometer nonlinearity,” Opt. Lett. **27**(10), 797–799 (
2002). [CrossRef] [PubMed]

6. T. Keem, S. Gonda, I. Misumi, Q. Huang, and T. Kurosawa, “Simple, real-time method for removing the cyclic error of a homodyne interferometer with a quadrature detector system,” Appl. Opt. **44**(17), 3492–3498 (
2005). [CrossRef] [PubMed]

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

## 2. Nonlinearity in the homodyne interferometer

^{◦}phase difference from each other and same amplitude as shown in Eq. (1). Therefore, the phase θ can be acquired by using the simple Eq. (2). On the contrary, in the real case they have not only different amplitude and offset but also lack of quadrature, δ as in Eq. (3) [3

3. 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]

## 3. Passive compensation method

### 3.1 Principle of the passive compensating method

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

^{◦}of lack of quadrature induces 1 nm of nonlinearity, therefore, the phase error δ should also be compensated for the sub-nanometer accuracy.

### 3.2 Jones matrix of optical elements

**T**and

**R**in Eq. (6) are the transmission and the reflection matrices, respectively. The values

*t*and

*r*denote the transmissivity and reflectivity, and the subscripts

*p*and

*s*represent the polarization state. Finally, the superscript

*c*represents the optical crosstalk.

*s*-polarized) or horizontally polarized (

*p*-polarized) beam can be produced. A polarizer, P1 which has high extinction ratio about 10000:1, is placed after the HWP so that the transmitted light is mostly polarized linearly, and after the PBS, the same kind of polarizers, P2 and P3 are placed just before the photo detector PD1 and PD2.

*p*-polarized beam, and make the transmission axis of P1 to coincide with the axis of the linearly polarized beam. This allows most of the beam to transmit the PBS with a high extinction ratio. After the PBS, by rotating the P2 or P3, the intensity of the

*s-*and

*p*-polarized beam can be measured. For the

*s*-polarized beam the same experiment is conducted to get the intensities of each polarization state. Through the measured intensity, the parameters of the Jones matrix can be determined, and this procedure is expressed through Eqs. (7) to (12). Equations (7) and (8) describe the electric fields at the points ‘a’, ‘b’ and ‘c’ shown in Fig. 3, and their relation. Because only intensity can be detected at the photo detector, Eqs. (7) and (8) need to be modified into intensity relations as Eqs. (9) to (12).

*p*-polarized at the point ‘a’, the amplitude of

*E*is much larger than that of

_{x}*E*. Therefore, the second and third terms in Eqs. (9) to (12) can be neglected, and from the simplified equation,

_{y}*s*-polarized beam, or the amplitude of

*E*is much larger than that of

_{y}*E*, the first and second term in Eqs. (9) to (12) vanish. Then,

_{x}^{◦}for QWP and 22.5

^{◦}for HWP. However, to minimize the nonlinearity caused by the optical crosstalk, the WPs should not be aligned at the conventional angles according to the suggested passive compensation method in this paper. Instead the WPs need to be aligned at the specific angle relying on the characteristic properties of the PBS. Therefore, an extended formula of the Jones matrix for WP aligned at arbitrary angle is required to approach the new compensation method numerically. The Jones matrix,

**W**, for a WP aligned at an arbitrary angle (

*ψ + Δθ*) can be expressed as in Eqs. (13) and (14), where

**W**is the Jones matrix of a WP with phase retardation tolerance

_{o}*ε*with its fast axis at 0

^{◦}and

**R**is a rotation matrix. The nominal value of rotation angle parameter

*ψ*is π/4 for QWP and π/8 for HWP, and

*Δθ*is the deviation angle from

*ψ*. Γ denotes the nominal phase retardation of the wave plate, which is π/2 for the QWP and π for the HWP.

### 3.3 Optical model of single pass homodyne interferometer

^{◦}. The

*p*-polarized beam transmits PBS

_{1}and QWP

_{1}, and then reflected from TM. As it passes through the QWP

_{1}twice, the

*p*-polarization changes to

*s*-polarization so that it is reflected at the PBS

_{1}and propagates to the detection part. Contrary to the

*p*-polarization, the

*s*-polarized beam reflects from the PBS

_{1}and passes through QWP

_{2}. After reflecting from RM and passing through QWP

_{2}again, the

*s*-polarization also changes to

*p*-polarization. Therefore, it can transmit the PBS

_{1}and proceed for the detection part. The propagation of

*s-*and

*p*-polarization elements of incident beam in the interferometer part can be described using the Jones matrix as in the following equations. In Eqs. (16) and (17),

**E**and

_{tar}**E**are the resultant electric field passed through the target arm and the reference arm respectively.

_{ref}**Q**is the Jones matrix for the j

_{ψj}_{th}QWP (j = 1,2,3) that is angularly aligned at ψ

_{j}, and

**T**and

_{j}**R**denote the transmission and reflection matrix of the j

_{j}_{th}PBS (j = 1,2,3). In addition,

**E**is the electric field vector of the incident ray, and

_{in}**R**is a reflection matrix of a target mirror and a reference mirror. φ is the relative phase change due to the target mirror movement. At the entrance of the detection part, there are two electric fields,

_{m}**E**and

_{tar}**E**, which have traveled different arms of the interferometer. The sum of these two fields is to be denoted as

_{ref}**E**. In the detection part,

_{int}**E**experiences several polarization optical parts, as expressed by Eqs. (18) to (21).

_{int}**H**is the Jones matrix for the HWP whose fast axis is at ψ,

_{ψ}*T*and

*R*denote the transmission and reflection coefficient of the NPBS.

**D**represents the resultant electric field at the j

_{j}_{th}photo detector (j = 1,…,4). As the intensity is proportional to |

**D**

_{j}|^{2}, the final interferometric signals and nonlinear error can be analyzed using the aforementioned equations.

### 3.4 Simulation and Analysis

**T**= [0.9413, 0.0127; 0.0203, 0.0430] and

_{1}**R**= [0.2629, 0.0278; 0.0137, 0.9788]. Both

_{1}**T**and

_{2}**T**are given by [0.9777, 0.0009; 0.0133, 0.0137]; both

_{3}**R**and

_{2}**R**are given by [0.0405, 0.0146; 0.0115, 0.9837]. Because PBS

_{3}_{1}is tilted about 5

^{◦}to prevent from the multi-interference by ghost reflections in the experimental setup, its property is a little different from that of PBS

_{2}and PBS

_{3}. The WPs were assumed to have a phase retardation error value

*ε*of π/250, the value of which was taken from the manufacture’s specifications.

^{◦}and 22.5

^{◦}. From an observation of the curves in Fig. 4, the alignment state of the HWP is most sensitive to the nonlinearity. The rotation angle of QWP

_{2}shows a small variation to the nonlinearity error. Moreover, the angular alignment states of respective WPs are coupled in determining the magnitude of the nonlinear error. Therefore, a set of angular alignment values can be optimized, which minimizes the nonlinear error.

^{◦}. For the optimization, the nonlinear error was initially defined as a function of the WP alignment angles of the QWPs and HWP using the Matlab optimization function. The step size of the angle variation was set to 0.1

^{◦}, which is the minimum resolution available from rotational mounts. Through the optimization process, the set of angles for alignment of WPs are determined as 51.3

^{◦}for QWP

_{1}, 18.9

^{◦}for QWP

_{2}, 50.5

^{◦}for QWP

_{3}, and 22.2

^{◦}for HWP. By setting the WPs to these angles in the simulation, the magnitude of nonlinear error was reduced to 0.057 nm, which is much smaller than the original magnitude of nonlinear error, and the lack of quadrature is 0.022

^{◦}. As expected, the passive method effectively suppressed the lack of quadrature which still remained after applying the gain and offset correction method.

## 4. Experiments

### 4.1 Setup

### 4.2 Result and discussion

^{◦}. The same procedure aforementioned in section 3.4 is used for an experiment to measure the nonlinearity at the various alignment conditions of WPs. Figure 6(a) to 6(d) shows the experimental result of the each WP comparing with the theoretically expected value. There are some difference between the theoretical expectation and experimental result in the magnitude and the alignment angle showing the minimum nonlinearity. The exact characteristic parameters are necessary to approach the nonlinearity theoretically, but this is not available in practice. For this reason, the experimental results show little differences compared to the theoretical expectation. However, both of them are representing similar tendency: HWP is most sensitive in determining the nonlinearity, and the curve for QWP

_{3}has two minimum points. The angle with minimum nonlinearity for QWP

_{1}coincides in the theoretical expectation and experimental result, and the curve for QWP

_{2}is almost flat as predicted.

_{3}is highly sensitive, therefore, by adjusting them first near the expected angles, the optimized set can be easily obtained as 65.0

^{◦}for QWP

_{1}, 48.0

^{◦}for QWP

_{2}, 55.5

^{◦}for QWP

_{3}, and 21.3

^{◦}for HWP. Finally, setting up WPs to this alignment angle set gives out 0.36 nm of nonlinearity as shown in Fig. 7 , and only 0.12

^{◦}of lack of quadrature is observed. Because, the lack of quadrature δ is almost removed as 0.12

^{◦}, the main reason of remained nonlinearity is due to the electrical noise and short term power instability of laser.

## 5. Conclusion

## Acknowledgements

## References and links

1. | I. Misumi, S. Gonda, T. Kurosawa, and K. Takamasu, “Uncertainty in pitch measurements of one-dimensional grating standards using a nanometrological atomic force microscope,” Meas. Sci. Technol. |

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

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

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

5. | 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), 017003.1–017003.5 (2009). |

6. | T. Keem, S. Gonda, I. Misumi, Q. Huang, and T. Kurosawa, “Simple, real-time method for removing the cyclic error of a homodyne interferometer with a quadrature detector system,” Appl. Opt. |

7. | Z. Li, K. Herrmann, and F. Pohlenz, “A neural network approach to correcting nonlinearity in optical interferometers,” Meas. Sci. Technol. |

8. | O. P. Lay and S. Dubovitsky, “Polarization compensation: a passive approach to a reducing heterodyne interferometer nonlinearity,” Opt. Lett. |

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

**OCIS Codes**

(120.0120) Instrumentation, measurement, and metrology : Instrumentation, measurement, and metrology

(120.3180) Instrumentation, measurement, and metrology : Interferometry

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: October 23, 2009

Revised Manuscript: November 19, 2009

Manuscript Accepted: November 19, 2009

Published: December 4, 2009

**Citation**

Jeongho Ahn, Jong-Ahn Kim, Chu-Shik Kang, Jae Wan Kim, and Soohyun Kim, "A passive method to compensate nonlinearity
in a homodyne interferometer," Opt. Express **17**, 23299-23308 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-25-23299

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### References

- I. Misumi, S. Gonda, T. Kurosawa, and K. Takamasu, “Uncertainty in pitch measurements of one-dimensional grating standards using a nanometrological atomic force microscope,” Meas. Sci. Technol. 14(4), 463–471 (2003). [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, 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. B. Eom, J. Y. 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), 017003.1–017003.5 (2009).
- T. Keem, S. Gonda, I. Misumi, Q. Huang, and T. Kurosawa, “Simple, real-time method for removing the cyclic error of a homodyne interferometer with a quadrature detector system,” Appl. Opt. 44(17), 3492–3498 (2005). [CrossRef] [PubMed]
- Z. Li, K. Herrmann, and F. Pohlenz, “A neural network approach to correcting nonlinearity in optical interferometers,” Meas. Sci. Technol. 14(3), 376–381 (2003). [CrossRef]
- O. P. Lay and S. Dubovitsky, “Polarization compensation: a passive approach to a reducing heterodyne interferometer nonlinearity,” Opt. Lett. 27(10), 797–799 (2002). [CrossRef] [PubMed]
- G. Dai, F. Pohlenz, H.-U. Danzebrink, K. Hasche, and G. Wilkening, “Improving the erformance of interferometers in metrological scanning probe microscopes,” Meas. Sci. Technol. 15(2), 444–450 (2004). [CrossRef] [PubMed]

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