## Precise evaluation of polarization mode dispersion by separation of even- and odd-order effects in quantum interferometry |

Optics Express, Vol. 19, Issue 23, pp. 22820-22836 (2011)

http://dx.doi.org/10.1364/OE.19.022820

Acrobat PDF (1016 KB)

### Abstract

The use of quantum correlations between photons to separate measure even- and odd-order components of polarization mode dispersion (PMD) and chromatic dispersion in discrete optical elements is investigated. Two types of apparatus are discussed which use coincidence counting of entangled photon pairs to allow sub-femtosecond resolution for measurement of both PMD and chromatic dispersion. Group delays can be measured with a resolution of order 0.1 fs, whereas attosecond resolution can be achieved for phase delays.

© 2011 OSA

## 1. Introduction: dispersion measurement - classical versus quantum

*L*. This known length dependence makes the dispersion of the optical fibers themselves relatively straightforward to measure and to take into account.

2. D. Andresciani, E. Curti, E. Matera, and B. Daino, “Measurement of the group-delay difference between the principal states of polarization on a low-birefringence terrestrial fiber cable,” Opt. Lett. **12**, 844–846 (1987). [CrossRef] [PubMed]

9. P. Williams, “PMD measurement techniques and how to avoid the pitfalls,” J. Opt. Fiber Commun. Rep. **1**, 84–105 (2004). [CrossRef]

8. S. Diddams and J. Diels, “Dispersion measurements with white-light interferometry,” J. Opt. Soc. Am. B **13**, 1120–1129 (1996). [CrossRef]

10. D. Branning, A. L. Migdall, and A. V. Sergienko, “Simultaneous measurement of group and phase delay between two photons,” Phys. Rev. A **62**, 063808 (2000). [CrossRef]

10. D. Branning, A. L. Migdall, and A. V. Sergienko, “Simultaneous measurement of group and phase delay between two photons,” Phys. Rev. A **62**, 063808 (2000). [CrossRef]

14. O. Minaeva, C. Bonato, B. E. A. Saleh, D. S. Simon, and A. V. Sergienko, “Odd- and even-order dispersion cancellation in quantum interferometry,” Phys. Rev. Lett. **102**, 100504 (2009). [CrossRef] [PubMed]

## 2. Chromatic dispersion and polarization mode dispersion

_{0}as for |

*ω*| << Ω

_{0}. The coefficients

*α*,

*β*,... characterize the

*chromatic dispersion*or variation of the refractive index with frequency. Explicitly,

*ω*and all terms containing odd powers to arrive at an expansion containing only two terms: where and

*polarization mode dispersion*(PMD), the index of refraction varies with polarization. We now have two copies of the dispersion relation, one for each independent polarization state: where

*H*,

*V*denote horizontal and vertical polarization.

*l*is the axial thickness of the device under study. However, we will continue to use the

*α*,

*β*, and Δ

*k*

_{0}parameters of Eq. (11), both because they are more commonly used, and because they allow easy comparison to the formulas used in fiber optics.

## 3. Classical PMD measurement

8. S. Diddams and J. Diels, “Dispersion measurements with white-light interferometry,” J. Opt. Soc. Am. B **13**, 1120–1129 (1996). [CrossRef]

*A*and

_{H}*A*are the incoming amplitudes of the horizontal and vertical components. After a horizontal polarizer, we destroy the quantum state and just pick off one component. We can think of it as a classical broadband source of horizontally polarized light,

_{V}*d*

_{1}. For path 2 (upper), the horizontally polarized light passes through a

*d*

_{2}and an adjustable delay δ =

*cτ*

_{2}.

*d*=

*d*

_{1}−

*d*

_{2}is the path length difference between the two arms.

*l*is introduced between the last beam splitter and the final polarizer, an additional polarization-dependent phase shift is added to the vector in Eq. (17): The resulting intensity at the detector is then:

*A*(

_{H}*ω*) and

*A*(

_{V}*ω*) are both proportional to

*within*the envelope) due to the zeroth order difference in dispersion Δ

*k*

_{0}, while the envelope as a whole will be shifted horizontally due to the first order difference in dispersion Δ

*α*and broadened due to the second order difference Δ

*β*. The interferograms shown in Fig. 2 are shifted by different amounts due to the use of different sample thicknesses. In this plot, a 200 nm bandwidth centered at 1550 nm was assumed, with a coherence length of

## 4. Type A quantum measurement

10. D. Branning, A. L. Migdall, and A. V. Sergienko, “Simultaneous measurement of group and phase delay between two photons,” Phys. Rev. A **62**, 063808 (2000). [CrossRef]

*H*and

*V*). The photons have frequencies Ω

_{0}±

*ω*, where 2Ω

_{0}is the pump frequency. Controllable birefringent time delays

*τ*

_{1}and

*τ*

_{2}are inserted before and after the beam splitter. Objects of lengths

*l*

_{1}and

*l*

_{2}may be placed before and after the beam splitter, respectively. Polarizers at angles

*θ*

_{1}and

*θ*

_{2}from the horizontal are placed before the two detectors. In the following, we will take

*D*

_{1}and

*D*

_{2}, respectively, can be written in the forms The coincidence rate is then computed by integrating the correlation function over the characteristic time scale

*T*of the detectors: Since

*T*is generally much larger than the downconversion time

*τ*

_{−}, we may safely simplify by taking

*T*→ ∞.

12. M. H. Rubin, D. N. Klyshko, Y. H. Shih, and A. V. Sergienko, “Theory of two-photon entanglement in type-II optical parametric down-conversion,” Phys. Rev. A **50**, 5122–5133 (1994). [CrossRef] [PubMed]

*R*

_{0}is a constant (delay-independent) background term and

*k*into its even and odd parts, then using the identity cos(

*A*+

*B*) = cos

*A*cos

*B*− sin

*A*sin

*B*. The result is:

*ω*, so the integral over that term vanishes. Therefore, this simplifies to We see that the even- and odd-order terms have separated into different cosine terms.

*β*and all higher order terms vanish, the integral of the previous line can be done explicitly: In the last line we have used the result where is the unit triangle function.

**62**, 063808 (2000). [CrossRef]

*τ*

_{1}has been added here. We now have two possibilities: we can scan over

*τ*

_{1}while holding

*τ*

_{2}fixed, or vice-versa. If we scan over

*τ*

_{1}with

*τ*

_{2}= 0, we find a triangular dip similar to the HOM dip, as shown in Fig. 4. The first order term in the PMD, Δ

*α*shifts the triangular envelope left or right, so that the bottom of the dip is at

*α*may be determined by measuring the location of the minimum. The absolute value of the factor cos (Δ

*k*

_{0}

*l*

_{2}) in front of the triangle function gives the visibility of the dip; so measuring the depth of the dip allows Δ

*k*

_{0}to be determined. Note that (depending on the sign of cos (Δ

*k*

_{0}

*l*

_{2})) the “dip” may actually be a peak.

*τ*

_{2}while holding

*τ*

_{1}= 0. This leads to an oscillating interference fringe pattern within the triangular envelope, similar to those of Fig. 2. The shift of the triangular envelope allows Δ

*α*, the first order term in the PMD, to be determined as before. In this case, rather than determining visibility, the zeroth order term Δ

*k*

_{0}horizontally shifts the fringe pattern by distance

*k*

_{0}from the size of this shift. To see clearly the effects of each order of dispersion, Fig. 5 shows examples of such scans in the presence of zeroth-order and first-order dispersion separately. The fringes within the envelope as

*τ*

_{2}is scanned allow evaluation of the phase delays (the Δ

*k*

_{0}term) to an accuracy on the order of attoseconds (10

^{−18}s) [10

**62**, 063808 (2000). [CrossRef]

*α*term down to the order of 0.1 fs.

*differences*of the horizontal and vertical polarization parameters (Δ

*α*, Δ

*β*, etc.) appear in the formulas above. The resulting interferogram is independent of the values of the parameters for fixed polarization (

*α*,

_{H}*α*, etc.) and so are insensitive to non-polarization-dependent dispersive effects.

_{V}*H*and

*V*polarizations separately, not just their difference.

## 5. Type B quantum measurement

14. O. Minaeva, C. Bonato, B. E. A. Saleh, D. S. Simon, and A. V. Sergienko, “Odd- and even-order dispersion cancellation in quantum interferometry,” Phys. Rev. Lett. **102**, 100504 (2009). [CrossRef] [PubMed]

*τ*in one arm, after the first beam splitter. Two birefringent samples of lengths

*l*

_{1}and

*l*

_{2}are placed before and after the first beam splitter. Birefringent delays

*τ*

_{1}and

*τ*

_{2}are present before and after the beam splitter as well, and a nonbirefringent delay

*τ*is added to one of the two arms after. For the sake of definiteness, assume that

*τ*

_{1}and

*τ*

_{2}delay the vertical (V) polarization and leave the horizontal (H) unaffected. The system is illuminated with type II downconversion beams. The pump frequency is at 2Ω

_{0}, while the signal and idler frequencies will be written as Ω

_{0}±

*ω*. We will make use of the fact that the downconversion spectral function is symmetric, We will identify the

*e*and

*o*polarizations with

*V*and

*H*respectively.

*τ*is an

*absolute*delay, so it must be positive. However,

*τ*

_{1}and

*τ*

_{2}are

*relative*delays of the vertically polarized photon compared to the horizontal, and so

*τ*

_{1}and

*τ*

_{2}may be positive or negative.

*V*and

*H*photons arising

*before*the first beam splitter is (This is the delay due to the object and

*τ*

_{1}alone; it is assumed that the intrinsic delay introduced by the known birefringence of the crystal itself has been compensated.) There are four possible ways in which the delay

*after*the first beam splitter may compensate this pre-beam splitter delay, leaving a total delay of zero between the two photons. These are enumerated in the table of Fig. 7, which gives the total post-beam splitter delay Δ

*τ*for each case in the final column. Setting for these four possibilities predicts four dips in the coincidence rate at delay values for which the difference in the final column vanish; at these values, there is no path information available because the two photons arrive at the detector simultaneously, allowing for complete destructive interference between paths.

_{post}*τ*= −2Δ

_{post}*τ*)as in ref. [10

_{pre}**62**, 063808 (2000). [CrossRef]

*α*(2

*l*

_{1}+

*l*

_{2}) + (2

*τ*

_{1}+

*τ*

_{2}) = 0. This leads to Δ

*τ*= −Δ

_{total}*τ*. Because the photons arrive at different times and with different phases we see that in this case interference can occur, leading to the possibility that sines or cosines may modulate this term. These expectations will be explicitly verified below for the linearized case.

_{pre}### 5.1. Linearized dispersion

*τ*

_{1}, while holding

*τ*and

*τ*

_{2}fixed. Then each of the Λ factors above gives a triangular spike (of width 2

*τ*

_{−}) in the coincidence rate centered at the value of

*τ*

_{1}for which the argument of Λ vanishes. We can then easily read off the locations of these spikes from Eq. (46). Explicitly, the various terms of Eq. (46) indicate that there should be triangular spikes centered at the values So suppose we have a sample only after the beam splitter (

*l*

_{1}= 0) and then we do three scans over

*τ*

_{1}, each with different values of

*τ*and

*τ*

_{2}:

- Take |
*τ*_{2}| large, with*τ*= 0. Then the*τ*_{2}-dependent peaks move far from the origin, off the edge of the plot. We will be left with peaks at*τ*_{1}= 0 and at*τ*_{1}=*β*_{H}l_{2}; from the locations of the latter we can read off*α*._{H} - Take |
*τ*_{2}| and |*τ*| both large, but with |*τ*_{2}+*τ*| small. Then we will be left with peaks at*τ*_{1}= 0 and*τ*_{1}=*α*_{V}l_{2}, so we can read off*α*._{V} - Take |
*τ*| large, with*τ*_{2}= 0. We will be left with peaks at*τ*_{1}= 0, and Δ*α l*_{2}, so we can read off Δ*α*.

*k*

_{0}, so that measuring the heights of these spikes relative to the others will allow Δ

*k*

_{0}to be determined as well.

### 5.2. Adding in quadratic dispersion

*β*) term is added back in, analytic expressions can no longer be obtained and numerical simulations must be done. An example is shown for one pair of triangular peaks in Fig. 8. In the figure, unrealistically large values of Δ

*β*were used to make the effect clearly visible. For Δ

*β*= 0 (red curve), the peaks have the same triangular form predicted earlier. As Δ

*β*increased for fixed Δ

*α*and Δ

*k*

_{0}the top of the triangle flattens and gains small oscillatory features; the triangle also broadens slightly.

*k*

_{0}and Δ

*β*, the alteration of the peak’s height by Δ

*β*is negligible, so that the height of the peak can still be used to measure Δ

*k*

_{0}. The most straightforward method to separate the value of Δ

*β*from Δ

*k*

_{0}is to fit a parameterized curve to the data and look for the values of the parameters Δ

*k*

_{0}and Δ

*β*that give the best fit.

### 5.3. Example: postponed delay only

*l*

_{1}=

*τ*

_{1}= 0. (In reality, the downconversion crystal itself acts as a birefringent sample before the beam splitter, but a fixed

*τ*

_{1}may be inserted to cancel it, so that without loss of generality, we can still take the combination Δ

*α*

_{crystal}*l*

_{1}+

*τ*

_{1}= 0.)

*τ*

_{2}fixed and scanning over the nonbirefringent delay

*τ*, there should be dips at as in Fig. 10. So, running two scans over

*τ*using two different values of

*τ*

_{2}, the location of the dip that remains at the same position in both scans gives us the value of

*α*. The other dip moves between the scans; measuring its location during either scan will then give the value of

_{H}*β*. Δ

_{V}*α*is then given by the difference between the two measured values.

*τ*

_{2}large enough to satisfy

*τ*

_{2}= Δ

*α l*

_{2}, then the term with the sines will be large, in which case we may also be able to extract

*k*

_{0}

*and*

_{H}*k*

_{0}

*by scanning*

_{V}*τ*over a range of values for which the other triangle functions vanish and fitting the resulting data curve to the function sin[

*k*

_{0}

_{V}*l*

_{2}+ Ω

_{0}(

*τ*+ Δ

*α l*

_{2})]sin[

*k*

_{0}

_{H}l_{2}− Ω

_{0}

*τ*)]. Alternatively, if only Δ

*k*

_{0}is needed (not

*k*

_{0}

*and*

_{H}*k*

_{0}

*separately), it may be simpler to remove the final beam splitter (turning the type B apparatus back into type A), then scanning over*

_{V}*τ*

_{2}and find Δ

*k*

_{0}from the shift in oscillation fringes via Eq. (35).

## 6. Conclusions

*τ*

_{−}in the quantum case (typically tenths of a femtosecond).

## Acknowledgments

## References and links

1. | H. Kogelnik and R. Jopson, “Polarization mode dispersion,” in |

2. | D. Andresciani, E. Curti, E. Matera, and B. Daino, “Measurement of the group-delay difference between the principal states of polarization on a low-birefringence terrestrial fiber cable,” Opt. Lett. |

3. | B. Costa, D. Mazzoni, M. Puleo, and E. Vezzoni, “Phase shift technique for the measurement of chromatic dispersion in optical fibers using LED’s,” IEEE J. Quantum Electron. |

4. | C. D. Poole and C. R. Giles, “Polarization-dependent pulse compression and broadening due to polarization dispersion in dispersion-shifted fiber,” Opt. Lett. |

5. | C. D. Poole, “Measurement of polarization-mode dispersion in single-mode fibers with random mode coupling,” Opt. Lett. |

6. | D. Derickson, |

7. | B. Bakhshi, J. Hansryd, P. A. Andrekson, J. Brentel, E. Kolltveit, B. K. Olsson, and M. Karlsson, “Measurement of the differential group delay in installed optical fibers using polarization multiplexed solitons,” IEEE Photon. Technol. Lett. |

8. | S. Diddams and J. Diels, “Dispersion measurements with white-light interferometry,” J. Opt. Soc. Am. B |

9. | P. Williams, “PMD measurement techniques and how to avoid the pitfalls,” J. Opt. Fiber Commun. Rep. |

10. | D. Branning, A. L. Migdall, and A. V. Sergienko, “Simultaneous measurement of group and phase delay between two photons,” Phys. Rev. A |

11. | E. Dauler, G. Jaeger, A. Muller, and A. Migdall, “Tests of a two-photon technique for measuring polarization mode dispersion with subfemtosecond precision,” J. Res. Natl. Inst. Stand. Technol. |

12. | M. H. Rubin, D. N. Klyshko, Y. H. Shih, and A. V. Sergienko, “Theory of two-photon entanglement in type-II optical parametric down-conversion,” Phys. Rev. A |

13. | D. N. Klyshko, |

14. | O. Minaeva, C. Bonato, B. E. A. Saleh, D. S. Simon, and A. V. Sergienko, “Odd- and even-order dispersion cancellation in quantum interferometry,” Phys. Rev. Lett. |

**OCIS Codes**

(120.3180) Instrumentation, measurement, and metrology : Interferometry

(260.2030) Physical optics : Dispersion

(270.0270) Quantum optics : Quantum optics

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: March 28, 2011

Revised Manuscript: June 8, 2011

Manuscript Accepted: July 16, 2011

Published: October 27, 2011

**Citation**

A. Fraine, D. S. Simon, O. Minaeva, R. Egorov, and A. V. Sergienko, "Precise evaluation of polarization mode dispersion by separation of even- and odd-order effects in quantum interferometry," Opt. Express **19**, 22820-22836 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-23-22820

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### References

- H. Kogelnik and R. Jopson, “Polarization mode dispersion,” in Optical Fiber Telecommunications IVB: System and Impairments , I. Kaminow and T. Li, eds. (Academic Press, 2002), pp. 725–861.
- D. Andresciani, E. Curti, E. Matera, and B. Daino, “Measurement of the group-delay difference between the principal states of polarization on a low-birefringence terrestrial fiber cable,” Opt. Lett. 12, 844–846 (1987). [CrossRef] [PubMed]
- B. Costa, D. Mazzoni, M. Puleo, and E. Vezzoni, “Phase shift technique for the measurement of chromatic dispersion in optical fibers using LED’s,” IEEE J. Quantum Electron. 18, 1509–1515 (1982). [CrossRef]
- C. D. Poole and C. R. Giles, “Polarization-dependent pulse compression and broadening due to polarization dispersion in dispersion-shifted fiber,” Opt. Lett. 13, 155–157 (1987). [CrossRef]
- C. D. Poole, “Measurement of polarization-mode dispersion in single-mode fibers with random mode coupling,” Opt. Lett. 14, 523–525 (1989). [CrossRef] [PubMed]
- D. Derickson, Fiber Optic Test and Measurement (Prentice Hall, 1998).
- B. Bakhshi, J. Hansryd, P. A. Andrekson, J. Brentel, E. Kolltveit, B. K. Olsson, and M. Karlsson, “Measurement of the differential group delay in installed optical fibers using polarization multiplexed solitons,” IEEE Photon. Technol. Lett. 11, 593–595 (1999). [CrossRef]
- S. Diddams and J. Diels, “Dispersion measurements with white-light interferometry,” J. Opt. Soc. Am. B 13, 1120–1129 (1996). [CrossRef]
- P. Williams, “PMD measurement techniques and how to avoid the pitfalls,” J. Opt. Fiber Commun. Rep. 1, 84–105 (2004). [CrossRef]
- D. Branning, A. L. Migdall, and A. V. Sergienko, “Simultaneous measurement of group and phase delay between two photons,” Phys. Rev. A 62, 063808 (2000). [CrossRef]
- E. Dauler, G. Jaeger, A. Muller, and A. Migdall, “Tests of a two-photon technique for measuring polarization mode dispersion with subfemtosecond precision,” J. Res. Natl. Inst. Stand. Technol. 104, 1–10 (1999).
- M. H. Rubin, D. N. Klyshko, Y. H. Shih, and A. V. Sergienko, “Theory of two-photon entanglement in type-II optical parametric down-conversion,” Phys. Rev. A 50, 5122–5133 (1994). [CrossRef] [PubMed]
- D. N. Klyshko, Photons and Nonlinear Optics (Gordon and Breach, 1988).
- O. Minaeva, C. Bonato, B. E. A. Saleh, D. S. Simon, and A. V. Sergienko, “Odd- and even-order dispersion cancellation in quantum interferometry,” Phys. Rev. Lett. 102, 100504 (2009). [CrossRef] [PubMed]

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