## Measurement of femtosecond Polarization Mode Dispersion (PMD) using biased π-shifted low-coherence interferometry

Optics Express, Vol. 7, Issue 6, pp. 228-236 (2000)

http://dx.doi.org/10.1364/OE.7.000228

Acrobat PDF (209 KB)

### Abstract

Conventional low-coherence interferometry (LCI) can be employed in the measurement of polarization mode dispersion (PMD) of fiber-optic components and fibers. However, the smallest PMD, which can be measured using this technique, is limited by the coherence length of the source. We propose a biased π-shifted Michelson interferometer where a birefringent crystal is inserted in front of the interferometer to introduce a bias differential group delay (DGD) larger than the coherence time of the source. In this way, the limitation imposed by the source coherence time has been overcome and PMDs much smaller than the source coherence time, in the order of several femtoseconds, can be measured. Experimental results for the PMD have been shown and compared with Jones matrix eigen-analysis. The theoretical model confirms the experimental observations.

© Optical Society of America

## 1. Introduction

4. P. Oberson, K. Juilliard, N. Gisin, R. Passy, and J. P. von der Weid, “Interferometric Polarization Mode Dispersion Measurements with Femtosecond Sensitivity”, J. Lightwave Tech. , **15**, 10, 1852 (1997). [CrossRef]

5. P. Martin, G. LeBoudec, E. Tauflieb, and H. Lefevre, “Optimized Polarization Mode Dispersion Measurement with “π-shifted” White Light Interferometry,” Opt. Fiber Technology , **2**, 207–212 (1996). [CrossRef]

*π*-shifted Michelson interferometer where a birefringent crystal (biasing crystal) has been inserted in front of the interferometer to introduce a bias differential group delay (DGD) due to polarization larger than the coherence time of the source and thus overcome the limitation it imposes on the measurement sensitivity. Using this technique, PMD of devices smaller than the source coherence time, in the order of several fs, can be measured. The interferometer has a QWP inserted in the fixed arm, which introduces a

*π*-shift in the passing polarizations. Thus in the interference pattern, the unwanted central autocorrelation peak is eliminated and the performance of the interferometer is improved. The biasing crystal and

*π*-shift combination increases the resolution of the interferometer and lowers the minimum PMD that one can measure. Experimental results have been presented and compared with Jones matrix eigenanalysis. It is clearly seen that the experimental observations agree well with the theoretical model.

## 2. Representation of the biased π-shifted Michelson interferometer

*τ*, is simply determined by the scanned distance, 2

*L*, between the two side peaks divided by the speed of light.

*τ*

*≤τ*

_{DUT}*, and that it cannot be resolved by the interferometer. We consider a linearly birefringent fiber-optic DUT without polarization dependent loss (PDL). For a given input polarization state, the output polarization from the DUT depends generally on the optical angular frequency ω. Nevertheless, there are always two input polarization states such that the output polarization is, to a first order, independent of the wavelength [7]. These two principal states of polarization are orthogonal and correspond to the maximum and minimum propagation time through the medium. For the sake of simplicity, we also consider circular polarization entering the biasing crystal or the DUT. An angular parameter*

_{coh}*ρ*is introduced to represent the misalignment of one component’s PSPs or polarization axes relative to those of the other components in the interferometer. The results presented here can readily be generalized for a DUT with PDL and for an elliptical input polarization. Throughout the work, we use DGD due to polarization interchangeably with first-order PMD.

5. P. Martin, G. LeBoudec, E. Tauflieb, and H. Lefevre, “Optimized Polarization Mode Dispersion Measurement with “π-shifted” White Light Interferometry,” Opt. Fiber Technology , **2**, 207–212 (1996). [CrossRef]

*x*- or

*y*-cut, so that the light travels perpendicular to the optic axis and a specific DGD for the o- and e-rays is introduced. According to the Jones matrix formalism [6

6. C. D. Poole and R. E. Wagner, “Phenomenological Approach to Polarization Mode Dispersion in Long Singlemode Fibers,” Elect. Lett. , **22**, 19, 1029 (1986). [CrossRef]

*J*. in its coordinate system is given by

_{crys}*δ*is the optical phase difference (OPD) between the o- and e-rays propagating in the crystal.

*ρ*. The Jones matrix of the QWP is given by:

*E⃗*(

*t*) is split by the beamsplitter into two fields that pass through the scanning

*E⃗*

_{1}(

*t*) and the fixed

*E⃗*

_{2}(

*t*) arms of the interferometer. When the scanning arm is moved away from the beamsplitter by a distance

*L*/2, the field in that arm

*E⃗*

_{1}(

*t*-

*τ*) is delayed by

*τ*=

*L*/

*c*with respect to the field in the fixed arm

*E⃗*

_{2}(

*t*). When the mirror in the scanning arm is continuously translated, the optical detector detects the interference pattern resulting from the interference of the two fields in the following form:

*ϖ*is the average optical frequency of the radiation, and the asterix (*) denotes the complex conjugate of the amplitudes associated with the electric field vector. The last two terms in Eq. (3) contain the information about the interference pattern and have the form of first-order correlation. Using the fact that the Fourier transformation of the correlation of two signals is equal to the product of their Fourier transforms, the interference terms in Eq. (3) can also be written as

*Ẽ*(

_{in}*ω*) is the Fourier transform of the source signal,

*T*

_{1}(

*ω*) and

*T*

_{2}(

*ω*) are the transfer functions for the scanning and fixed arms respectively,

*e⃗*

_{1}and

*e⃗*

_{2}account for the polarization states of the two fields. The quantity 〈

*Ẽ*(

_{in}*ω*)

*Ẽ**(

_{in}*ω′*)〉=2

*πδ*(

*ω*-

*ω′*)

*ω*) where

*ω*) is the autocorrelation function of the source. For the case of LCI, the light source is emitting in a wavelength bandwidth of Δ

*ω*=-(2

*πc*/

*λ*

^{2})Δ

*λ*and the source autocorrelation function has a Gaussian shape determined by the spontaneous emission. Therefore, the final form of the interference term is multiplied by an envelope function in the form of exp⌊-(2Δ

*L*/

*L*)

_{coh}^{2}⌋=exp⌊-(2Δ

*τ*/

*τ*)

_{coh}^{2}⌋, where Δ

*L*refers to the OPD and Δ

*τ*to the DGD accumulated by the light radiation passing through the interferometer arms, while

*L*and

_{coh}*τ*refer to the source coherence length and time respectively.

_{coh}*τ*., between its “fast” and “slow” component along its polarization axes much larger than the source coherence time, i.e., Δ

_{crys}*τ*. ≻

_{crys}*τ*, the two waves exiting the crystal recombine incoherently. There is no coupling between them and they propagate through the arms of the interferometer independently as:

_{coh}*E*

_{1x}(

*t*-

*τ*) and

*E*

_{1y}(

*t*-

*τ*) in the scanning arm and as

*E*

_{2x}(

*t*) and

*E*

_{2y}(

*t*) in the fixed arm respectively. Therefore, the Jones vector (

*E*(

_{x}*t*),

*E*(

_{y}*t*)) can now be treated as two separate vectors (

*E*(

_{x}*t*), 0) and (0,

*E*(

_{y}*t*)) at the exit of the crystal. In both arms there is one reflection taking place, either from the moving or fixed mirror. Furthermore, in the stationary arm, the radiation passes through the QWP twice. At the exit of the interferometer in the vicinity of the detector surface, the propagating waves have the form:

*ρ*between the crystal axes and the QWP is zero, cos(2

*ρ*)=1;sin(2

*ρ*)=0, only the source autocorrelation peak is present in the interference pattern and the information about DGD of the biasing crystal is lost. When the QWP is aligned at

*ρ*=

*45*deg, cos(2

*ρ*)=0;sin(2

*ρ*)=1, there are two peaks corresponding to the interference from the “slow” and “fast” polarization modes and the central autocorrelation peak is extinguished. From Eq. (7), the DGD due to polarization (or PMD) introduced by the crystal (or any DUT in the weak-mode coupling regime) can be calculated as:

*2L*is the scanned distance between the maxima of the two peaks, and

*c*is the speed of the light. In the LCI, Eq. (8) is used to calculate the PMD in fiber-optic devices according to [3, 5

5. P. Martin, G. LeBoudec, E. Tauflieb, and H. Lefevre, “Optimized Polarization Mode Dispersion Measurement with “π-shifted” White Light Interferometry,” Opt. Fiber Technology , **2**, 207–212 (1996). [CrossRef]

*τ*≤

_{DUT}*τ*. Under this condition the two peaks in Eq. (7) produce only one broadened peak in the fringe pattern and therefore cannot be resolved. With our proposed biased

_{coh}*π*-shifted configuration, having aligned the crystal and the QWP at 45 deg relative to each other, a reference scan is performed to determine precisely the biasing DGD, Δ

*τ*. The precise alignment is assured by the full elimination of the central autocorrelation peak. Equation (8) is also used to calculate the DGD of the biasing crystal.

_{crys}*π*-shifted interferometer in front of the crystal. A pictorial representation of the polarized light waves passing through the DUT and the crystal, as well as the accumulated DGDs, is illustrated in Fig. 2. For the sake of clarity, circular polarization is assumed to enter the crystal and the QWP is not shown. The angular misalignment

*ρ*is shown for the crystal rather than the DUT. Since Δ

*τ*≤

_{DUT}*τ*has been assumed, the radiation exiting the DUT with mutually perpendicular PSPs recombines coherently. This changes the resultant output polarization to elliptical and broadens the resulting wave train. Upon entering the biasing crystal, the wave train splits along the crystal polarization axes. At the crystal exit, since Δ

_{coh}*τ*. ≻

_{crys}*τ*, the waves with perpendicular polarizations accumulate OPD much larger than the source coherence length. The two waves recombine incoherently and propagate through the interferometer without any phase relationship between them. The biasing crystal and QWP are assumed aligned at 45 deg and hence the central autocorrelation peak is suppressed. The DUT is considered as a linear retarder with an OPD, Δ, introduced between the two polarization modes with axes at an arbitrary angle,

_{coh}*ρ*, relative to the crystal axes. It can be represented by its Jones matrix as:

*ρ*=0 deg, i.e., cos

^{2}

*ρ*=1, the DUT’s polarization (PSPs) are aligned with those of the crystal and the DGDs introduced by the DUT and the crystal are summed up. When

*ρ*=90, i.e., sin

^{2}

*ρ*=1, the “fast” axis of the DUT is aligned with the “slow” axis of the biasing crystal and vice versa and the two delays get subtracted. In both cases, there are two distinct well-separated peaks in the interference pattern resulting from the large DGD of the crystal. Hence, the PMD of the DUT can be obtained by the relative shift in the peaks’ positions from their nominal positions in the reference scan rather than by the separation of peaks it introduces. In the case of

*ρ*=45 deg, the sum and the difference of the DGDs are both present resulting in broadening of the peaks, if they cannot be resolved.

*ρ*=0 or 90 deg. Having determined either of said orientations, results from the scans would yield the combined DGD from Eq. (8), Δ

*τ*

_{crys}_{±DUT}, while together with the scan carried out previously for just the crystal would allow the DGD associated with the DUT to be calculated as

## 3. Measurements

*=1530 nm and a bandwidth (FWHM) of Δ*λ ¯

*λ*=42 nm at -10 dB. The source coherence length was

*τ*≈

_{DUT}*τ*≈0.18 ps. With the biasing crystal, this limitation was overcome and smaller DGDs were measured by the relative shifts of the two peaks from their nominal positions when only the crystal was present (according to Eqs. (8), (12) and (13)). The biasing crystal was an y-cut LiNiO

_{coh}_{3}. When circular polarization entered the crystal, the accurate alignment between the crystal and the QWP was achieved when the central peak was extinguished (according Eq. (7)). The QWP was then kept aligned at 45 deg to the crystal axes all throughout the experiments. First, a reference scan was carried out with the biasing crystal aligned at 45 deg to the QWP. The crystal DGD was evaluated from the recorded interference pattern according to Eq. (8) to yield Δ

*τ*

_{crys}_{.}=2.706

*ps*. The measured DGD was verified with the Jones Matrix Eigenanalysis (JME) method where the averaged DGD was obtained over the wavelength range of the broadband source. The JME method gave a value of Δ

*τ*

_{crys}_{.}=2.688 ps. The agreement between the results measured by the two methods is 0.018 ps, i.e., within 1%. The wavelength scan of the DGD due to polarization for the bias crystal from the JME method is depicted in Fig. 3.

*ρ*=

*90*deg), that is the “fast” and “slow” PSPs of the DUT are aligned with the “slow” and “fast” axes of the crystal, respectively, and the DGDs of the DUT and the crystal subtract; the brown plot shows the scan for the DUT polarization axes at an arbitrary angle (

*ρ*~

*45*deg). For comparison, the red plot in the centre depicts the interference pattern resulting from the DUT only without the bias crystal. The DGD of the second crystal used as DUT was calculated from Fig. 4 with Eqs. (8), (12) and (13) as Δ

*τ*=0.179 ps. For comparison, Fig. 5 illustrates the wavelength scan for the DGD from the JME method, which gives an average DGD for the DUT of Δ

_{DUT}*τ*=0.177 ps, an excellent agreement with the interferometric method within 2 fs or approximately 1%.

_{DUT}## 4. Conclusions

## Acknowledgements

## References

1. | TIA/EIA Standard, FOTP-124, “Polarization Mode Dispersion Measurement for Single-Mode Optical Fibres by Interferometric Method,” Aug., 1996. |

2. | P. Hernday, “Fibre Optic Test and Measurement,” Ed.D. Derickson, (Prentice Hall, N. J.1998), Ch. 10 and 12. |

3. | Y. Namihira, K. Nakajima, and T. Kawazawa,, “Fully Automated Interferometric PMD Measurements for Active EDFA, Fibre-optic Components and Optical fibres,” Electron. Letters, 29, 18, 1649–1650 (993). |

4. | P. Oberson, K. Juilliard, N. Gisin, R. Passy, and J. P. von der Weid, “Interferometric Polarization Mode Dispersion Measurements with Femtosecond Sensitivity”, J. Lightwave Tech. , |

5. | P. Martin, G. LeBoudec, E. Tauflieb, and H. Lefevre, “Optimized Polarization Mode Dispersion Measurement with “π-shifted” White Light Interferometry,” Opt. Fiber Technology , |

6. | C. D. Poole and R. E. Wagner, “Phenomenological Approach to Polarization Mode Dispersion in Long Singlemode Fibers,” Elect. Lett. , |

7. | D. S. Kliger, J. W. Lewis, and C. E. Randall, “Polarized Light in Optics and Spectroscopy,” (Academic Press Inc., 1990), Ch. 4 and 5. |

8. | B. L. Heffner, “Automatic measurement of polarization mode dispersion using Jones Matrix Eigenanalysis,” IEEE Ph. Tech. Let. , |

**OCIS Codes**

(060.2300) Fiber optics and optical communications : Fiber measurements

(120.3180) Instrumentation, measurement, and metrology : Interferometry

**ToC Category:**

Research Papers

**History**

Original Manuscript: July 26, 2000

Published: September 11, 2000

**Citation**

Eli Simova, Ian Powell, and Chander Grover, "Measurement of femtosecond Polarization Mode
Dispersion (PMD) using biased p-shifted low-coherence interferometry," Opt. Express **7**, 228-236 (2000)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-7-6-228

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### References

- TIA/EIA Standard, FOTP-124, "Polarization Mode Dispersion Measurement for Single-Mode Optical Fibres by Interferometric Method," Aug., 1996.
- P. Hernday, "Fibre Optic Test and Measurement," Ed. D. Derickson, (Prentice Hall, N. J. 1998), Ch. 10 and 12.
- Y. Namihira, K. Nakajima and T. Kawazawa,, "Fully Automated Interferometric PMD Measurements for Active EDFA, Fibre-optic Components and Optical fibres," Electron. Letters 29, 18, 1649-1650 (1993).
- P. Oberson, K. Juilliard, N. Gisin, R. Passy and J. P. von der Weid, "Interferometric Polarization Mode Dispersion Measurements with Femtosecond Sensitivity", J. Lightwave Tech., 15, 10, 1852 (1997). [CrossRef]
- P. Martin, G. LeBoudec, E. Tauflieb, and H. Lefevre, "Optimized Polarization Mode Dispersion Measurement with "p-shifted" White Light Interferometry," Opt. Fiber Technology, 2, 207-212 (1996). [CrossRef]
- C. D. Poole and R. E. Wagner, "Phenomenological Approach to Polarization Mode Dispersion in Long Single-mode Fibers," Elect. Lett., 22, 19, 1029 (1986). [CrossRef]
- D. S. Kliger, J. W. Lewis, C. E. Randall, "Polarized Light in Optics and Spectroscopy," (Academic Press Inc., 1990), Ch. 4 and 5.
- B. L. Heffner, "Automatic measurement of polarization mode dispersion using Jones Matrix Eigenanalysis," IEEE Ph. Tech. Let., 4, 1066-1069 (1992). [CrossRef]

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