## Polarization delivery in heterodyne interferometry |

Optics Express, Vol. 21, Issue 22, pp. 27119-27126 (2013)

http://dx.doi.org/10.1364/OE.21.027119

Acrobat PDF (1193 KB)

### Abstract

Optically heterodyned laser interferometry, as applied to measuring linear displacements, requires different optical frequencies to be encoded onto unique polarization states. To eliminate non-linear contributions to the interferometer signal, the frequency difference must be introduced after beam splitting and the interfering beams must be recombined via spatially separated paths. The polarization jitter of the frequency-shifted beams still originates a noise in the beat-signal phase. A formula is given expressing the noise amplitude in terms of the illuminating beam’s extinction ratio.

© 2013 OSA

## 1. Introduction

1. K.-N. Joo, J. D. Ellis, J. W. Spronck, P. J. M. van Kan, and R. H. M. Schmidt, “Simple heterodyne laser interferometer with subnanometer periodic errors,” Opt. Lett. **34**, 386–388 (2009). [CrossRef] [PubMed]

4. C. Weichert, P. Koechert, R. Koening, J. Fluegge, B. Andreas, U. Kuetgens, and A. Yacoot, “A heterodyne interferometer with periodic nonlinearities smaller than ±10 pm,” Meas. Sci. Technol. **23**, 094005 (2012). [CrossRef]

5. A. Bergamin, G. Cavagnero, and G. Mana, “Phase holonomy in optical interferometry,” J. Mod. Opt. **39**, 2053–2074 (1992). [CrossRef]

6. C. M. Wu, J. Lawall, and R. D. Deslattes, “Heterodyne interferometer with subatomic periodic nonlinearity,” Appl. Opt. **38**, 4089–4094 (1999). [CrossRef]

7. J. Lawall and E. Kessler, “Michelson interferometry with 10 pm accuracy,” Rev. Sci. Instrum. **71**, 2669–2676 (2000). [CrossRef]

8. T. L. Schmitz and J. F. Beckwith, “Acousto-optic displacement-measuring interferometer: a new heterodyne interferometer with Ångstrom-level periodic error,” J. Mod. Opt. **49**, 2105–2114 (2002). [CrossRef]

*γ*-ray diffractometry [9]. In our interferometer, though the two low- and high-frequency beams are independently generated and kept spatially separated up to the beat-signal detectors, polarization division was still used to avoid back-reflections and light recycling and to minimize losses of optical power [10

10. G. Cavagnero, G. Mana, and E. Massa, “Effect of recycled light in two-beam interferometry,” Rev. Sci. Instrum. **76**, 053106 (2005). [CrossRef]

11. C. Weichert, J. Flügge, R. Köning, H. Bosse, and R. Tutsch, “Aspects of design and the characterization of a high resolution heterodyne displacement interferometer,” in: *Fringe 2009, 6th International Workshop on Advanced Optical Metrology*, W. Osten and M. Kujawinska, eds. (SpringerBerlin Heidelberg, 2009) 263–268.

## 2. Interferometer concept

*γ*-ray diffractometry requires high resolution angle measurements, which are traced back to laser-interferometry measurements of the displacement of two retro-reflectors pairs mounted at the ends of two beams embedded in the diffractometer spindles. A detailed description of the diffractometer and angle interferometer can be found in [9]. In our Laue-Laue setup, depicted in Fig. 2, the relevant measurement quantity is the angle between the two diffractometer crystals. Two acousto-optic modulators generate two beams, linearly polarized and shifted in frequency, from a stabilized single frequency laser source. These beams are delivered to the interferometer by two single-mode and polarization-preserving fibers (not shown in Fig. 2). They enter the interferometer linearly polarized along a direction forming a 45° angle with the interferometer reflection-plane, are delivered to the diffractometer retro-reflectors according to horizontal and vertical polarizations, are back-reflected by two fixed reference mirrors, and are combined coming out of the interferometer. Since the measurand is the angle between the two crystals, both the interferometer arms can serve as the reference or the measurement purpose. Auxiliary monolithic arms, ending in the RRP reference roof-prism, are added for redundancy and testing.

## 3. Interferometer model

### 3.1. Jones’ formalism

5. A. Bergamin, G. Cavagnero, and G. Mana, “Phase holonomy in optical interferometry,” J. Mod. Opt. **39**, 2053–2074 (1992). [CrossRef]

*δ*exp(i

_{i}*α*)]

_{i}*are two-components spinor describing the polarization of the fields emerging from the fibers,*

^{T}*ϕ*is the retardation difference at the interferometer entrance, Ω is the heterodyne frequency, and the matrix rotates the polarizations by

*π*/4 rad. The [1,

*δ*exp(i

_{i}*α*)]

_{i}*spinors are represented by using orthogonal (horizontal and vertical) linear polarization states as basis. The beams entering the interferometer are quasi linearly polarized; hence*

^{T}*δ*

^{2}≪ 1, where

*δ*

^{2}is the extinction ratio. The phases

*α*

_{1,2}are arbitrary, they vary randomly between zero and 2

*π*, e.g., they depend on the thermal and mechanical stress of the fibers.

*π*/4, horizontal, and vertical linear polarization states and the optical-path difference between the two reference arms has been assumed zero. Since only the first order terms will be retained, small aberrations of

*P*

_{∠},

*P*

_{||}, and

*P*

_{⊥}contribute only through terms independent of the extinction ratios. Therefore, fixing Eqs. (4a–c) at the specified values, do not jeopardize the model generality.

*x*is the retardation between the two interferometer arms. The interfering beams are where |∠〉 is a linear polarization at 45° with respect to the reflection plane and only the leading terms have been retained. Hence, the normalized beat-signals are Some Lissajous curves of the measurement beat-signal

*vs.*the reference one – Eqs. (7a) and (7b), respectively – are shown in Fig. 5. If the field entering the interferometer is linearly polarized, that is, if

*δ*

_{1}=

*δ*

_{2}= 0, the beat signals are as expected. It is worth noting that the differential phase noise

*ϕ*(

*t*) in beam delivering shifts the heterodyne frequency from Ω to Ω +

*∂*. The phase difference between the beat signals (7a) and (7b) is

_{t}ϕ*α*

_{1}and

*α*

_{2}phases walk randomly in the [0, 2

*π*] interval, by assuming

*δ*

_{1}≈

*δ*

_{2}≈

*δ*and by observing that Var[sin(

*α*

_{1})] = Var[sin(

*α*

_{2})] = 1/2, the root-mean-square noise of the phase-difference measurement is Therefore, the extinction ratio of the polarized beams entering the interferometer must be better than

*σ*= 2

_{x}*π*10

^{−4}rad, the extinction ratio of the beams illuminating the interferometer must be at least

*δ*

^{2}≈ 10

^{−7}. It is hard to have polarizers with such a high extinction ratio. This problem can be relieved by a careful alignment of the (linear) polarization of the delivered beams to the slow axis of the polarization preserving fibers to make to make the extinction ratio at the fiber output as small as possible and by using multiple polarizers. Whenever it is possible, polarization division must be avoided.

### 3.2. Berry’s geometrical-phase

14. M. V. Berry, “Interpreting the anholonomy of coiled light,” Nature **326**, 277–278 (1987). [CrossRef]

5. A. Bergamin, G. Cavagnero, and G. Mana, “Phase holonomy in optical interferometry,” J. Mod. Opt. **39**, 2053–2074 (1992). [CrossRef]

15. T. van Dijk, H. F. Schouten, W. Ubachs, and T. D. Visser, “The Pancharatnam-Berry phase for non-cyclic polarization changes,” Opt. Express **18**, 10796–10804 (2010). [CrossRef] [PubMed]

*u*

_{1}〉 and |

*u*

_{2}〉 – are split and recombined according to (4a–c). The |||〉 and |⊥〉 split states are the north and south poles; the intersection of the zero meridian with the equator, |∠〉, is the detection state of the beat signals. To calculate the geometric phase – the angle subtended at the origin by the path of the polarization states – let us observe that the longitudes of the input states are 2

*δ*

_{1}sin(

*α*

_{1}) and 2

*δ*

_{2}sin(

*α*

_{2}). Hence, with a unit radius and up the first order in

*δ*

_{1}and

*δ*

_{2}, the area of the spherical wedge shown in Fig. 6 is 4[

*δ*

_{1}sin(

*α*

_{1}) +

*δ*

_{2}sin(

*α*

_{2})]. This shows that the phase noise in Eq. (9) originates from the transport of the polarization states and it is caused by the jitter of the |

*u*

_{1}〉 and |

*u*

_{2}〉 location on the sphere.

*P*

_{||},

*P*

_{⊥}, and

*P*

_{∠}projectors in Eqs. (3) and (5). Figure 6 (right) shows the circuit of the polarization states when beam splitting and recombination is slightly perturbed. It is worth noting that, with a first order calculation, the circuit area is still 4[

*δ*

_{1}sin(

*α*

_{1}) +

*δ*

_{2}sin(

*α*

_{2})]. This explains why Eq. (9) does not lose generality when the Eqs. (4a–c) are fixed at the specified values.

## 4. Conclusions

## Acknowledgment

## References and links

1. | K.-N. Joo, J. D. Ellis, J. W. Spronck, P. J. M. van Kan, and R. H. M. Schmidt, “Simple heterodyne laser interferometer with subnanometer periodic errors,” Opt. Lett. |

2. | K.-N. Joo, J. D. Ellis, E. S. Buice, J. W. Spronck, and R. H. M. Schmidt, “High resolution heterodyne interferometer without detectable periodic nonlinearity,” Opt. Express |

3. | M. Pisani, A. Yacoot, P. Balling, N. Bancone, C. Birlikseven, M. Çelik, J. Fluegge, R. Hamid, P. Koechert, P. Kren, U. Kuetgens, A. Lassila, G. B. Picotto, E. Sahin, J. Seppä, M. Tedaldi, and C. Weichert, “Comparison of the performance of the next generation of optical interferometers,” Metrologia |

4. | C. Weichert, P. Koechert, R. Koening, J. Fluegge, B. Andreas, U. Kuetgens, and A. Yacoot, “A heterodyne interferometer with periodic nonlinearities smaller than ±10 pm,” Meas. Sci. Technol. |

5. | A. Bergamin, G. Cavagnero, and G. Mana, “Phase holonomy in optical interferometry,” J. Mod. Opt. |

6. | C. M. Wu, J. Lawall, and R. D. Deslattes, “Heterodyne interferometer with subatomic periodic nonlinearity,” Appl. Opt. |

7. | J. Lawall and E. Kessler, “Michelson interferometry with 10 pm accuracy,” Rev. Sci. Instrum. |

8. | T. L. Schmitz and J. F. Beckwith, “Acousto-optic displacement-measuring interferometer: a new heterodyne interferometer with Ångstrom-level periodic error,” J. Mod. Opt. |

9. | J. Krempel, |

10. | G. Cavagnero, G. Mana, and E. Massa, “Effect of recycled light in two-beam interferometry,” Rev. Sci. Instrum. |

11. | C. Weichert, J. Flügge, R. Köning, H. Bosse, and R. Tutsch, “Aspects of design and the characterization of a high resolution heterodyne displacement interferometer,” in: |

12. | S. Cosijns, |

13. | S. Pancharatnam, “Generalized theory of interference and its applications,” Proc. Indian Acad. Sci. A |

14. | M. V. Berry, “Interpreting the anholonomy of coiled light,” Nature |

15. | T. van Dijk, H. F. Schouten, W. Ubachs, and T. D. Visser, “The Pancharatnam-Berry phase for non-cyclic polarization changes,” Opt. Express |

**OCIS Codes**

(040.2840) Detectors : Heterodyne

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

(120.3180) Instrumentation, measurement, and metrology : Interferometry

(130.5440) Integrated optics : Polarization-selective devices

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: July 2, 2013

Revised Manuscript: September 26, 2013

Manuscript Accepted: September 29, 2013

Published: November 1, 2013

**Citation**

E. Massa, G. Mana, J. Krempel, and M. Jentschel, "Polarization delivery in heterodyne interferometry," Opt. Express **21**, 27119-27126 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-22-27119

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### References

- K.-N. Joo, J. D. Ellis, J. W. Spronck, P. J. M. van Kan, and R. H. M. Schmidt, “Simple heterodyne laser interferometer with subnanometer periodic errors,” Opt. Lett.34, 386–388 (2009). [CrossRef] [PubMed]
- K.-N. Joo, J. D. Ellis, E. S. Buice, J. W. Spronck, and R. H. M. Schmidt, “High resolution heterodyne interferometer without detectable periodic nonlinearity,” Opt. Express18, 1159–1165 (2010). [CrossRef] [PubMed]
- M. Pisani, A. Yacoot, P. Balling, N. Bancone, C. Birlikseven, M. Çelik, J. Fluegge, R. Hamid, P. Koechert, P. Kren, U. Kuetgens, A. Lassila, G. B. Picotto, E. Sahin, J. Seppä, M. Tedaldi, and C. Weichert, “Comparison of the performance of the next generation of optical interferometers,” Metrologia49, 155–167 (2012). [CrossRef]
- C. Weichert, P. Koechert, R. Koening, J. Fluegge, B. Andreas, U. Kuetgens, and A. Yacoot, “A heterodyne interferometer with periodic nonlinearities smaller than ±10 pm,” Meas. Sci. Technol.23, 094005 (2012). [CrossRef]
- A. Bergamin, G. Cavagnero, and G. Mana, “Phase holonomy in optical interferometry,” J. Mod. Opt.39, 2053–2074 (1992). [CrossRef]
- C. M. Wu, J. Lawall, and R. D. Deslattes, “Heterodyne interferometer with subatomic periodic nonlinearity,” Appl. Opt.38, 4089–4094 (1999). [CrossRef]
- J. Lawall and E. Kessler, “Michelson interferometry with 10 pm accuracy,” Rev. Sci. Instrum.71, 2669–2676 (2000). [CrossRef]
- T. L. Schmitz and J. F. Beckwith, “Acousto-optic displacement-measuring interferometer: a new heterodyne interferometer with Ångstrom-level periodic error,” J. Mod. Opt.49, 2105–2114 (2002). [CrossRef]
- J. Krempel, A new spectrometer to measure the molar Planck constant (Ludwig-Maximilians Universität München, Fakultät für Physik, 2011).
- G. Cavagnero, G. Mana, and E. Massa, “Effect of recycled light in two-beam interferometry,” Rev. Sci. Instrum.76, 053106 (2005). [CrossRef]
- C. Weichert, J. Flügge, R. Köning, H. Bosse, and R. Tutsch, “Aspects of design and the characterization of a high resolution heterodyne displacement interferometer,” in: Fringe 2009, 6th International Workshop on Advanced Optical Metrology, W. Osten and M. Kujawinska, eds. (SpringerBerlin Heidelberg, 2009) 263–268.
- S. Cosijns, Displacement laser interferometry with sub-nanometer uncertainty (Technische Universiteit Eindhoven, 2004).
- S. Pancharatnam, “Generalized theory of interference and its applications,” Proc. Indian Acad. Sci. A44, 247–262 (1956); reprinted in: Collected Works of S. Pancharatnam, G. W. Series ED. (Oxford University, 1975)
- M. V. Berry, “Interpreting the anholonomy of coiled light,” Nature326, 277–278 (1987). [CrossRef]
- T. van Dijk, H. F. Schouten, W. Ubachs, and T. D. Visser, “The Pancharatnam-Berry phase for non-cyclic polarization changes,” Opt. Express18, 10796–10804 (2010). [CrossRef] [PubMed]

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