Precise evaluation of polarization mode
`dispersion by separation of even- and
`odd-order effects in quantum
`interferometry
`
`A. Fraine1, D.S. Simon1,∗, O. Minaeva2, R. Egorov1, A.V. Sergienko1,3,4
`1 Dept. of Electrical and Computer Engineering, Boston University, 8 Saint Mary’s St.,
`Boston, MA 02215.
`2 Dept. of Biomedical Engineering, Boston University, 44 Cummington St., Boston, MA 02215.
`3 Photonics Center, Boston University, 8 Saint Mary’s St., Boston, MA 02215.
`4 Dept. of Physics, Boston University, 590 Commonwealth Ave., Boston, MA 02215.
`simond@bu.edu
`
`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 en-
`tangled 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 Optical Society of America
`OCIS codes: (260.2030) Dispersion; (120.3180) Interferometry; (270.0270) Quantum Optics.
`
`[2]
`
`[3]
`
`[4]
`
`References and links
`H. Kogelnik and R. Jopson, ”Polarization Mode Dispersion,” in Optical Fiber Telecommunications IVB System
`[1]
`and Impairments, I. Kaminow, T. Li, eds. (Academic, Press 2002), pp. 725-861.
`D. Andresciani, E. Curti, E. Matera, 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).
`B. Costa, D. Mazzoni, M. Puleo, E. Vezzoni, ”Phase shift technique for the measurement of chromatic dispersion
`in optical fibers using LED’s,” IEEE J. Quantum Elect. 18, 1509-1515 (1982).
`C.D. Poole, C.R. Giles, ”Polarization-dependent pulse compression and broadening due to polarization disper-
`sion in dispersion-shifted fiber,” Opt. Lett. 13, 155-157 (1987).
`C.D. Poole, ”Measurement of polarization-mode dispersion in single-mode fibers with random mode coupling,”
`Opt. Lett. 14 523-525 (1989).
`D. Derickson, Fiber Optic Test and Measurement (Prentice Hall, 1998).
`B. Bakhshi, J. Hansryd, P.A. Andrekson, J. Brentel, E. Kolltveit, B.K. Olsson, M. Karlsson, ”Measurement of
`the differential group delay in installed optical fibers using polarization multiplexed solitons,” IEEE Phot. Tech.
`Lett. 11, 593-595 (1999).
`S. Diddams, J. Diels, ”Dispersion measurements with white-light interferometry,” J. Opt. Soc. Am. B 13, 1120-
`1129 (1996).
`P. Williams, ”PMD measurement techniques and how to avoid the pitfalls,” J. Opt. Fiber. Commun. Rep. 1,
`84-105 (2004).
`D. Branning, A.L. Migdall, A.V. Sergienko, ”Simultaneous measurement of group and phase delay between two
`photons,” Phys. Rev. A 62, 063808 (2000).
`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).
`[12] M.H. Rubin, D.N. Klyshko, Y.H. Shih, A.V. Sergienko, ”Theory of two-photon entanglement in type-II optical
`parametric down-conversion,” Phys. Rev. A 50, 5122-5133 (1994).
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`0001
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`Capella 2022
`Fujitsu v. Capella
`IPR2015-00727
`
`

`
`[13]
`[14]
`
`D.N. Klyshko, Photons and Nonlinear Optics (Gordon and Breach, 1988).
`O. Minaeva, C. Bonato, B.E.A. Saleh, D.S. Simon, A.V. Sergienko, ”Odd- and Even-Order Dispersion Cancella-
`tion in Quantum Interferometry,” Phys. Rev. Lett. 102, 100504 (2009).
`
`Introduction: Dispersion Measurement - Classical versus Quantum
`1.
`As optical communication networks migrate towards higher 40 Gbps and 100 Gbps data rates,
`system impairments due to dispersion, especially polarization mode dispersion (PMD), become
`a primary issue. This includes not only fiber PMD, but also contributions from switches, ampli-
`fiers, and all other components in the optical path. The fiber PMD and component PMD tend to
`accumulate in different manners as the size of the network grows. In the long length regime, the
`√
`differential group delay (DGD) due to fiber PMD has a known dependance on length, growing
`L [1]. In a similar manner, contributions from chromatic dispersion increase linearly in L.
`as
`This known length dependence makes the dispersion of the optical fibers themselves relatively
`straightforward to measure and to take into account.
`In contrast, component PMD was until recently considered to be too small in comparison to
`fiber PMD to affect significant penalties at the system level. Since the introduction of recon-
`figurable add-drop multiplexers (ROADMs), the number of components that could potentially
`contribute to the PMD in a given system has increased significantly. Although the dispersive
`contribution of each separate component is relatively small, together they are capable of accu-
`mulating and of thereby making a significant contribution to the total system impairment. It is
`therefore important to be able to precisely and efficiently measure small values of DGD. How-
`ever, since only fiber PMD was important in the past, no measuring techniques were developed
`for efficient evaluation of small DGD values. With component PMD starting to play a signifi-
`cant role, developing high-resolution evaluation of small PMD values in a single optical switch
`or other small discrete optical component represents a new challenge to optical researchers that
`must be addressed by modern optical metrology. In this paper, we address the measurement of
`dispersive effects in such discrete elements.
`Polarization mode dispersion is the difference between wavenumbers of two orthogonal
`states of light at fixed wavelength, or equivalently, a polarization-dependent variation of a
`material’s index of refraction. A number of methods have been developed for measuring it
`[2, 3, 4, 5, 6, 7, 8, 9]. Many traditional techniques for measuring PMD rely on an interfero-
`metric approach for high-resolution measurements of absolute values of optical delays. This
`approach requires one to use a monochromatic laser source and to keep track of the number
`of interference fringes. Therefore, the accuracy of the approach is limited by the stability of
`the interferometer, by the signal-to-noise level of the detector, and by the wavelength of the
`monochromatic radiation, leading to significant limitations. For example, the use of monochro-
`matic classical polarized light does not allow one to measure the relative delay between two
`orthogonally polarized waves in a single measurement, so several measurements at different
`frequencies must be used to reconstruct the polarization dispersion properties of materials. The
`use of highly monochromatic laser sources creates the additional problem of multiple reflec-
`tions and strong irregular interference that may have detrimental effect on measuring polariza-
`tion dispersion.
`White-light or low-coherence interferometry [8] is another widely used approach. The ulti-
`mate resolution of such interferometric measurements will depend on the spectral bandwidth
`of the light source. Achieving sub-fs resolution in PMD measurement dictates the use of light
`sources with bandwidth in excess of 200 nm. Generating light of such a bandwidth with a
`smooth spectral profile is not an easy task in itself. Spectral modulations from existing sources
`with bumpy spectra produce ’ghost’ features during measurement, leading to complications in
`dispersion evaluation. In addition, the visibility of interference with such super-broadband light
`
`0002
`
`

`
`is diminished due to dispersion effects.
`Overall, while classical techniques can provide high-resolution measurement of polariza-
`tion mode dispersion they still have limitations in many areas that quantum-based techniques
`can address. For example, entangled photon states intrinsically provide an absolute value for
`polarization optical delay, in contrast to the conventional (classical) case, which is limited to
`determination of delay modulo an integer number of cycles of the light. This is mainly due to
`the fact that quantum interferometry exploits both phase and group velocity effects in the same
`measurement [10, 11], a feat not possible in classical optics.
`The current practical resolution of conventional dispersion evaluation techniques is limited
`to a few femtoseconds (fs). The primary goal here is to use an interferometric setup with an
`entangled photon source to measure the component PMD of a small, discrete optical element
`to sub-femtosecond precision. Ideally, it would be desirable to measure chromatic dispersion
`with the same device, while allowing for the polarization and chromatic effects to be easily
`separable. We will show that this is indeed possible. Due to the frequency-anticorrelation in
`the entangled downconversion source used as illumination, we may independently determine
`the even-order and odd-order parts of the PMD’s frequency dependence. Due to the reliance on
`the frequency anticorrelations within pairs of photons, the separation method is intrinsically a
`two-photon quantum effect, and is not present in the classical interferometer.
`Classical attempts to simulate this even-odd separation effect by symmetrical chirping and
`anti-chirping of femtosecond laser pulses are constrained to a very narrow size of wavepacket
`thus making it not very practical. The availability of such a separation is useful in a number
`of circumstances. One example is when there is enough pulse broadening (second-order dis-
`persion) to make accurate measurement of group velocity (first order dispersion) difficult. In a
`fiber, group velocity and broadening effects can be separated to some extent by simply taking
`a sufficiently long length of fiber as sample; the longer the fiber, the more accurately each can
`be measured. When dealing with switching elements or other small discrete optical elements,
`this option is not available. Another means must be found to prevent accurate measurement of
`the first-order group delay from being obscured by second-order broadening effects. That is
`what is accomplished here: the location of a dip in the coincidence rate may be used to find
`the group velocity, and this location is unaffected by the second-order broadening as a result
`of the even-order dispersion cancellation. Conversely, although the amount of broadening in
`a single small component may seem negligible, the total broadening from many such compo-
`nents present in a large network may be significant; thus high-accuracy measurements of these
`very small second-order dispersive contributions is important. Separating them off from the
`generally larger first-order contributions makes accurate measurements much easier.
`We note that since the component being analyzed is assumed to be relatively small, the prin-
`cipal polarization axes may be assumed to remain constant over the longitudinal length of the
`object and to be independent of frequency, with the dispersive contributions of the two polar-
`ization components remaining independent of each other. This greatly simplifies the analysis.
`After a review of background and notation in section 2, three measurement methods will be
`discussed in sections 3-5. The apparatus of section 3 uses a single detector to make a classical
`measurement; the system is illuminated with a broadband classical light source. In contrast,
`quantum measurements are made using two detectors connected in coincidence with illumina-
`tion provided by a source of entangled photon pairs (spontaneous parametric downconversion,
`(SPDC)). We will examine two quantum measurement setups in sections 4 and 5. In addition,
`in section 5 we give a qualitative analysis that allows the positions of dips (or peaks) indepen-
`dently of the mathematical formalism.
`The two quantum configurations will be distinguished from each other by referring to them
`as type A or type B. They differ only in the presence or absence of a final beam splitter before
`
`0003
`
`

`
`detection, so they may both be implemented in a single apparatus by allowing a beam splitter
`to be switched in or out of the optical path. Similarly, by adding an additional polarizer and
`counting the singles rate at one detector instead of coincidence events, the classical setup may
`also be implemented in the same device. Thus, a single apparatus could be made which is
`capable of performing any of the three types of measurements to be discussed.
`This paper builds on two previous lines of work. The apparatus used for the type A setup was
`introduced previously [10, 11], where it was shown that quantum interferometry can achieve
`higher resolution than classical methods in measurements of PMD. Separately, the segregation
`of even- and odd-order chromatic dispersion effects was demonstrated in [14]. Here, we bring
`the two strands together in a single device (type B), showing that we can separate even- and
`odd-order effects in PMD, as well as in chromatic dispersion, and that we can do so with the
`resolution available to the type A device.
`As a further benefit of the quantum devices over classical methods, note that for the quantum
`cases there is no need to know in advance the principal axis directions of the device or ob-
`ject being measured. Although the incoming photons are aligned along particular axes that are
`linked to a birefringent crystal orientation, their projections onto any rotated pair of orthogonal
`axes (including the principle axes of the sample) will remain equally entangled, allowing the
`method to work without any need to align the axes of the source and the device under test.
`
`2. Chromatic Dispersion and Polarization Mode Dispersion
`First consider a material for which the index of refraction is independent of polarization. The
`frequency dependence of the wavenumber k = 2πn(λ)
`is given by a dispersion relation, which
`λ
`can be written near some central frequency Ω0 as
`k(Ω0 ± ω) = k0 ± αω+ βω2 ± γω3 + . . .
`(1)
`for |ω| << Ω0. The coefficients α, β, . . . characterize the chromatic dispersion or variation of
`the refractive index with frequency. Explicitly,
`
`k0 = k(Ω0),
`d2k(ω(cid:4))
`dω(cid:4)2
`
`β = 1
`2!
`
`(cid:2)(cid:2)(cid:2)(cid:2)
`
`α= dk(ω(cid:4))
`dω(cid:4)
`γ= 1
`3!
`
`,
`
`,
`ω(cid:4)=Ω0
`d3k(ω(cid:4))
`dω(cid:4)3
`
`(cid:2)(cid:2)(cid:2)(cid:2)
`
`ω(cid:4)=Ω0
`ω(cid:4)=Ω0
`Rather than looking at the individual terms in the expansion (1), we may also collect together
`all terms containing even powers of ω and all terms containing odd powers to arrive at an
`expansion containing only two terms:
`k(Ω0 + ω) = keven(ω) + kodd(ω),
`
`(cid:2)(cid:2)(cid:2)(cid:2)
`
`, . . .
`
`(2)
`
`(3)
`
`(4)
`
`where
`
`keven(ω) = k0 + βω2 + O(ω4),
`
`(5)
`
`and
`
`kodd(ω) = αω+ γω3 + O(ω5).
`(6)
`In the case of nonzero 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:
`
`kH (Ω0 ± ω) = kH0 ± αHω+ βHω2 + . . .
`
`(7)
`
`0004
`
`

`
`(8)
`(9)
`(10)
`
`= kH,even(ω) + kH,odd(ω)
`kV (Ω0 ± ω) = kV0 ± αV ω+ βVω2 + . . .
`= kV,even(ω) + kV,odd(ω),
`where H,V denote horizontal and vertical polarization.
`To describe the PMD, we must define quantities that measure the differences between the
`two polarization states:
`Δβ = βV − βH.
`Δα= αV − αH,
`Δk0 = kV0 − kH0,
`These parameters are defined per unit length. For the case of primary interest to us, discrete
`fixed-size objects, the formulas should really be written in terms of the relevant lumped quan-
`tities
`ΔA ≡ lΔα,
`ΔB ≡ lΔβ,
`Δφ≡ lΔk0,
`(12)
`where l is the axial thickness of the device under study. However, we will continue to use the
`α, β, and Δk0 parameters of eq. 11, both because they are more commonly used, and because
`they allow easy comparison to the formulas used in fiber optics.
`Note that Δk0 = Ω0Δn(Ω0)
`is a measure of the difference in phase velocity between the two
`c
`polarization modes, while Δα and Δβ are related to the difference in group velocity. Also, it
`should be pointed out that the PMD and the chromatic dispersion are not entirely independent
`effects; in particular, the PMD coefficients themselves (Δk0, Δα, Δβ) are frequency dependent.
`In the quantum cases, it is convenient to also define τ− = DL, where L is the thickness of the
`0 − u−1
`nonlinear downconversion crystal and D = u−1
`is the difference of the group velocities
`e
`of the two polarizations inside the crystal. We will restrict ourself to the simplest case of a
`bulk crystal, so the spectral distribution of the downconverted pairs is described by the function
`(cid:4)
`(cid:3)
`[12, 13]
`τ−ω
`
`(11)
`
`(13)
`
`,
`
`12
`
`Φ(ω) = sinc
`
`where the sinc function is defined by sinc(x) = sin(x)
`. Photons are emitted from the downconver-
`sion process in frequency- anticorrelated pairs: the frequencies Ω0 ± ωin each pair are shifted
`x
`equally, but in opposite directions, from the central frequency Ω0 = ωpump/2, with the distribu-
`tion of frequency shifts ω being given by Φ(ω) of eq. 13. The downconversion time scale,
`τ−, is inversely proportional to the spectral width of the source, and therefore determines the
`precision of the resulting measurements. The spectrum may be made wider by using a thinner
`nonlinear crystal, but this occurs at the expense of reducing the intensity of the downconverted
`light. High intensity and large bandwidth may be obtained simultaneously by use of a chirped
`crystal, although some of the details of the following analysis will then be changed.
`
`3. Classical PMD Measurement
`An apparatus equivalent to that shown schematically in fig. 1 [8] is commonly used to measure
`polarization mode dispersion. The illumination may be provided by any sufficiently broadband
`light source. For easier comparison with the later sections, we will assume the illumination is
`provided by type II parametric downconversion, but this is not necessary; since we use a single
`detector, the entanglement of the downconverted photons will play no role.
`Assume an arbitrary amount of H and V polarization out of the downconversion crystal, so
`(cid:5) (cid:3)
`(cid:4)
`that the incident field in Jones vector notation is proportional to
`AH (ω)
`AV (ω)
`
`dω,
`
`(14)
`
`0005
`
`

`
`Crystal
`
`Half-wave
`plate (45°)
`
`NPBS
`
`Pump
`
`Horizontal
`Polarizer
`
`d
`
`1
`
`d
`
`2
`
`τ
`
`Non-
`birefringent
`delay
`
`Polarizer
`at 45°
`
`Object
`
`D
`
`NPBS
`
`l
`
`Fig. 1. Classical (single-detector) white-light setup for finding total PMD.
`
`where AH and AV 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
`(cid:4)
`(cid:5) (cid:3)
`think of it as a classical broadband source of horizontally polarized light,
`AH(ω)
`0
`
`dω.
`
`(15)
`
`dω=
`
`dω.
`
`(16)
`
`For path 1 (lower), the horizontally polarized light accumulates a phase corresponding to the
`path length d1. For path 2 (upper), the horizontally polarized light passes through a λ
`2 wave
`plate with fast axis 45◦ from the horizontal, converting it into vertically polarized light,
`(cid:4)(cid:5) (cid:3)
`(cid:5) (cid:3)
`(cid:4)
`(cid:3)
`(cid:4)
`0 −1
`AH(ω)
`0
`AH (ω)
`1
`0
`0
`In addition, the vertically polarized light in path 2 experiences a phase corresponding to the
`path length d2 and an adjustable delay δ = cτ2.
`(cid:4)
`(cid:3)
`At the second beam splitter, the two components form a superposition of the form
`(cid:5)
`eik(ω)d1
`eik(ω)(d2+δ)
`with k(ω) = ω
`c (assuming the paths are in free space). In the absence of any sample after the
`second beam splitter, this superposition will pass through a linear polarizer at 45◦, resulting in
`(cid:4)
`(cid:3)
`(cid:5)
`AH (ω)√
`eik(ω)d1 + eik(ω)(d2+δ)
`J(cid:4)
`0 =
`eik(ω)d1 + eik(ω)(d2+δ)
`2
`The intensity at the detector is then given by
`(cid:5)
`I = |J(cid:4)
`0|2 =
`|AH (ω)|2 [1 + cos(k(ω) (Δd − δ))] dω.
`Here, Δd = d1 − d2 is the path length difference between the two arms.
`If a birefringent sample of length l is introduced between the last beam splitter and the final
`(cid:4)
`(cid:3)
`polarizer, an additional polarization-dependent phase shift is added to the vector in eq. 17:
`(cid:5)
`ei[k(ω)d1+kH (ω)l]
`e[ik(ω)(d2+δ)+kV (ω)l]
`
`(17)
`
`(18)
`
`(19)
`
`(20)
`
`J0 =
`
`AH(ω)
`
`dω,
`
`dω,
`
`J0 =
`
`AH (ω)
`
`dω.
`
`0006
`
`

`
`Fig. 2. Interferograms produced by apparatus of fig. 1 for samples of different thicknesses.
`For a fixed thickness, the size of the shift may be used as a measure of the difference in
`phase velocities of the two polarizations.
`
`(cid:7)ω
`
`(cid:8)(cid:9)
`
`dω.
`
`(21)
`
`(cid:6)
`The resulting intensity at the detector is then:
`(cid:5)
`I = |J(cid:4)
`(Δd − δ)− Δk(ω)l
`0|2 =
`|AH (ω)|2
`1 + cos
`c
`(cid:11)
`(cid:10)
`For Type II downconversion, the AH (ω) and AV (ω) are both proportional to Φ(ω) =
`τ−ω
`. Plotting eq. 21 as a function of birefringent delay δ leads to interferograms such
`sinc
`as those shown in fig. 2. Each interferogram will be phase shifted (moving the positions of
`the peaks and troughs within the envelope) due to the zeroth order difference in dispersion
`Δk0, while the envelope as a whole will be shifted horizontally due to the first order difference
`in dispersion Δα and broadened due to the second order difference Δβ. 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
`λ2
`0Δλ = 12 μm.
`xc =
`4. Type A Quantum Measurement
`The goal now is to extract the polarization mode dispersion of an object with a higher precision
`than is possible with the classical apparatus of the previous section. In addition, we would like
`to be able to measure the even and odd orders of chromatic dispersion for each polarization.
`The setup [10, 11] is shown in fig. 3. The downconversion is type II so that the two photons
`have opposite polarization (H and V). The photons have frequencies Ω0 ± ω, where 2Ω0 is
`the pump frequency. Controllable birefringent time delays τ1 and τ2 are inserted before and
`after the beam splitter. Objects of lengths l1 and l2 may be placed before and after the beam
`splitter, respectively. Polarizers at angles θ1 and θ2 from the horizontal are placed before the
`two detectors. In the following, we will take θ1 = θ2 = π
`4 and assume that the beam splitter is
`50/50. Information about which polarization state travels in which branch of the apparatus will
`therefore be erased, allowing interference to occur with maximum visibility.
`Rather than the Jones matrix formalism used in the previous section, it will be more conve-
`nient here to use creation and annihilation operators for horizontally and vertically polarized
`photons. The portion of the output from the downconversion process that is relevant to our
`purposes is the biphoton state
`(cid:5)
`|Ψ(cid:7) =
`
`
`
`dωΦ(ω) ˆa†H (Ω0 + ω) ˆa†V (Ω0 − ω)|0(cid:7),
`
`
`
`(22)
`
`0007
`
`12
`
`

`
`ˆE (+)
`1
`ˆE (+)
`2
`
`(cid:2)(cid:2)(cid:2)2
`
`G(2)(t1,t2) =
`
`Rc(τ1,τ2) =
`
`dt1dt2G(2)(t1,t2).
`
`=
`
`e−iω1t1
`
`(23)
`
`(24)
`
`(25)
`
`(26)
`
`(27)
`dω|Φ(ω)|2 = 2π
`τ− .
`
`(cid:13)
`
`(28)
`
`which will serve as the incident state of our setup. The positive-frequency parts of the fields at
`detectors D1 and D2, respectively, can be written in the forms
`(cid:12)
`(cid:13)
`(cid:5)
`(t1) = 1
`ˆaH (ω1)eikH (ω1)l1 + ˆaV (ω1)ei[kV (ω1)l1+ω1τ1]
`dω
`(cid:12)
`(cid:5)
`2
`(t2) = 1
`ˆaH (ω2)eikH (ω2)(l1+l2)
`dω
`(cid:13)
`2
`e−iω2(t2+τ).
`+ ˆaV (ω2)ei[kV (ω2)(l1+l2)+ω2(τ1+τ2)]
`The coincidence rate is then computed by integrating the correlation function
`(t2)|Ψ(cid:7)|2
`
`(cid:2)(cid:2)(cid:2)(cid:8)0|E (+)
`(t1)E (+)
`1
`2
`over the characteristic time scale T of the detectors:
`(cid:5) T /2
`−T /2
`Since T is generally much larger than the downconversion time τ−, we may safely simplify by
`taking T → ∞.
`Using eqs. 22-26, the coincidence rate may be written in the general form ([12])
`Rc(τ1,τ2) = R0{1 +CM(τ1,τ2)} ,
`(cid:14)
`where R0 is a constant (delay-independent) background term and C−1 =
`The dependence on the time delays is contained in the modulation term
`(cid:5)
`M(τ1,τ2) = 1
`dω|Φ(ω)|2 e−i[Δk(ω)−Δk(−ω)]l1−2iωτ1
`(cid:12)
`2
`×
`eiΔk(−ω)l2+i(Ω0−ω)τ2 + e−iΔk(ω)l2−i(Ω0+ω)τ2
`(cid:5)
`dω|Φ(ω)|2
`(29)
`× cos{[Δk(ω)− Δk(−ω)]l1 + Δk(ω)l2 + 2ωτ1 + (Ω0 + ω)τ2} ,
`where the second form follows by changing the sign of the integration variable in the first term
`of the previous line. It can be seen that even-order PMD terms arising from the pre-beam splitter
`object cancel. Thus, measurements made with the object before the beam splitter will give us
`the odd-order PMD, and measurements made with the object after the beams splitter give the
`total PMD; making both measurements and then taking the difference will provide the even-
`order PMD. We can see the roles of the even and odd parts more clearly by splitting Δk into its
`even and odd parts, then using the identity cos(A + B) = cosAcosB− sinAsinB. The result is:
`(cid:5)
`
`M(τ1,τ2) =
`
`dω|Φ(ω)|2{cos [Δkodd(ω)(2l1 + l2) + ω(2τ1 + τ2)]cos [Δkeven(ω)l2 + Ω0τ2]
`−sin [Δkodd(ω)(2l1 + l2) + ω(τ1 + τ2)]sin [Δkeven(ω)l2 + Ω0τ2]} .
`(30)
`Note that the integrand in the second term is odd in ω, so the integral over that term vanishes.
`Therefore, this simplifies to
`(cid:5)
`
`M(τ1,τ2) =
`
`dω|Φ(ω)|2 cos [Δkodd(ω)(2l1 + l2) + ω(2τ1 + τ2)]
`× cos [Δkeven(ω)l2 + Ω0τ2] .
`
`(31)
`
`0008
`
`

`
`Controllable birefringent delays
`
`Crystal
`
`Object
`
`τ
`1
`
`1l
`
`Object
`
`l2
`
`Pump
`
`Polarizer
`
`D
`2
`
`τ
`
`2
`
`Polarizer
`
`D
`1
`
`Coincidence
`counter
`
`Fig. 3. Type A setup for measuring PMD parameters Δα≡ αV − αH and Δβ≡ βV − βH.
`
`We see that the even- and odd-order terms have separated into different cosine terms.
`In the special case that Δβ and all higher order terms vanish, the integral of the previous line
`can be done explicitly:
`(cid:16)
`(cid:15)
`M(τ1,τ2) = 2π
`τ− cos [Δk0l2 + Ω0τ2]Λ
`In the last line we have used the result
`(cid:5)
`
`Δα(2l1 + l2) + (2τ1 + τ2)
`τ−
`(cid:7) τ
`(cid:8)
`
`,
`
`2a
`
`π a
`
`Λ
`
`dωsinc2(aω)cos (ωτ) =
`(cid:17)
`
`.
`
`(32)
`
`(33)
`
`(34)
`
`(35)
`
`where
`
`Λ(x) =
`
`1−|x|,
`0,
`
`|x| ≤ 1
`|x| > 1
`
`is the unit triangle function.
`(cid:17)
`The coincidence rate is then
`
`Rc(τ1,τ2) = R0
`
`1 + cos[Δk0l2 + Ω0τ2]Λ
`
`(cid:15)
`
`Δα(2l1 + l2) + (2τ1 + τ2)
`τ−
`
`(cid:16)(cid:18)
`
`.
`
`This result is consistent with equation A31 of [10], with the caveat that an extra time delay
`τ1 has been added here. We now have two possibilities: we can scan over τ1 while holding τ2
`fixed, or vice-versa. If we scan over τ1 with τ2 = 0, we find a triangular dip similar to the HOM
`dip, as shown in fig. 4. The first order term in the PMD, Δα shifts the triangular envelope left
`or right, so that the bottom of the dip is at τ1 = − Δα
`(2l1 + l2); thus Δαmay be determined by
`2
`measuring the location of the minimum. The absolute value of the factor cos(Δk0l2) in front of
`the triangle function gives the visibility of the dip; so measuring the depth of the dip allows Δk0
`to be determined. Note that (depending on the sign of cos(Δk0l2)) the ”dip” may actually be a
`peak.
`Alternatively, we may scan over τ2 while holding τ1 = 0. This leads to an oscillating inter-
`ference fringe pattern within the triangular envelope, similar to those of fig. 2. The shift of the
`triangular envelope allows Δα, the first order term in the PMD, to be determined as before. In
`this case, rather than determining visibility, the zeroth order term Δk0 horizontally shifts the
`fringe pattern by distance τ2 = Δk0l2Ω0
`within the envelope, allowing determination of Δk0 from
`the size of this shift. To see clearly the effects of each order of dispersion, fig. 5 shows examples
`
`0009
`
`

`
`Coincidence rate
`
`
`
`_.'A(I(L.)ll
`2
`
`0
`
`T1
`
`Fig. 4. Scanning over 1:; while keqring 1:2 = 0. The horizontal shifi‘ ofthe minimum away
`fiom the origin determines Aa, while the depth of the dip determines
`Ihe triangle
`function mayleadeithertoadip (asshown)ortoapealr, dqrendingon thesign ofthe
`cosine.
`
`(8)
`
`(D)
`
`Fig. 5. Scanning over 1:2 while beping 1:1 = 0. In (a), a nonzem Au shifm the envelope
`from itspositionfor At! = 0 in part
`The size ofthe shift can be measured with accuracy
`on the order of0.1
`In part (c), a nonzero Ah) shifts the locations of the peak within
`the unshified envelope. The size ofthe shift can be measured with accuracy on the order of
`.001 fs = l as.
`
`0010
`
`

`
`of such scans in the presence of zeroth-order and first-order dispersion separately. The fringes
`within the envelope as τ2 is scanned allow evaluation of the phase delays (the Δk0 term) to an
`accuracy on the order of attoseconds (10−18 s) [10]. Group delays from the Δα term down to
`the order of 0.1 fs.
`Note that only the differences of the horizontal and vertical polarization parameters (Δα, Δβ,
`etc.) appear in the formulas above. The resulting interferogram is independent of the values of
`the parameters for fixed polarization (αH, αV , etc.) and so are insensitive to non-polarization-
`dependent dispersive effects.
`In principle, Fourier transforming experimental data for the coincidence rate and then fit-
`ting parameterized curves to it will allow the determination of higher order PMD parameters.
`However, this requires a large quantity of data to be obtained at high precision. By adding an
`additional beam splitter to the apparatus in the next section, we will arrive at a better method,
`which allows us to extract additional information; namely, it will also give us information about
`the H and V polarizations separately, not just their difference.
`
`5. Type B Quantum Measurement
`The goal here is to see if additional information may be obtained with a variant of the previous
`apparatus that mixes the final beams via an additional beam splitter. This variation is inspired
`by the setup of ref. [14], in which even and odd portions of the chromatic dispersion were
`separated into different parts of an interferogram, allowing them to be studied independently of
`each other.
`Consider the setup in fig. 6. This differs from the setup of the previous section (fig. 3) only
`by the addition of an extra beam splitter before the detectors and an additional nonbirefringent
`delay τin one arm, after the first beam splitter. Two birefringent samples of lengths l1 and l2 are
`placed before and after the first beam splitter. Birefringent delays τ1 and τ2 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 τ1 and τ2 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 2Ω0, while the signal and idler frequencies will be written as
`Ω0 ± ω. We will make use of the fact that the downconversion spectral function is symmetric,
`Φ(ω) = Φ(−ω).
`(36)
`
`We will identify the e and o polarizations with V and H respectively.
`It should be emphasized that in the notation used here, τ is an absolute delay, so it must be
`positive. However, τ1 and τ2 are relative delays of the vertically polarized photon compared to
`the horizontal, and so τ1 and τ2 may be positive or negative.
`We will find below that the effects of the even and odd orders separate and play different
`roles: the location of each dip in the interferogram (represented mathematically by a triangle
`function in the coincidence rate) is determined by the odd part, while the relative depths of
`the dips are controlled by the even part. We may predict the number and location of each of
`these dips by identifying the ways in which it becomes impossible from the relative timing of
`detection events in the two detectors to identify which photon took which path. To do so, first
`note that the delay between the V and H photons arising before the first beam splitter is
`Δτpre ≡ τV − τH = Δαl1 + τ1.
`(37)
`(This is the delay due to the object and τ1 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
`
`0011
`
`

`
`Controllable birefringent delays
`
`Crystal
`
`Object
`
`τ
`1
`
`1l
`
`Object
`
`l2
`
`Pump
`
`Non birefringent
`delay
`
`τ
`
`2
`
`τ
`
`D 2
`
`Polarizers
`
`D
`1
`
`Coincidence counter
`
`Fig. 6. Type B setup for finding even- and odd-order PMD
`
`delay, leaving a total delay of zero between the two photons. These are enumerated in the
`table of figure 7, which gives the total post-beam splitter delay Δτpost for each case in the final
`column. Setting
`Δτpre + Δτpost = 0
`(38)
`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.
`One additional dip (represented by the second triangle function in eq. 46 below) arises in a
`slightly different fashion. Here the time delay between the two photons is nonzero, but has a
`value tha