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Degree of Polarization in the Lyot Depolarizer JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. LT-1, NO. 3, SEPTEMBER 1983 415 G. Zeidler, J. Appl. Phys., vol. 42, no. 884, 1971. [ 181 F. P. Schaefer, Ed., Dye Lasers, 2nd ed., (Topics in Applied Phys- E. P. Ippen and C. V. Shank, Appl. Phys. Lett., vol. 21, no. 301, ics Series). New York: Springer Verlag, 1977. 1972. [ 191 E. Jahar and D. Treves, IEEE J. Quantum Electron., vol. QE-13, H. Kogelnik and C. V. Shank, Appl. Phys. Lett., vol. 18, no. 152, ' p. 962, Dec. 1977. 1971. [20] N. M. Lawandy,Appl. Opt., vol. 18, p. 189, 1977. J. E. Bjorkholm and C. V. Shank, IEEE J. Quantum Electron., VOI. QE-8, p. 833, NOV.1972. 9 J. E. Biorkholm and C. V. Shank, Appl... Phys. Lett., vol. 20, p. 306, i972. P. Zory, presented at OSA Topical Meeting on Integrated Optics, W. Hodel, photograph andbiography not available atthe time of Las Vegas, NV, 1972. publication. B. P. Schinke et el., Appl. Phys. Lett., vol. 21, no. 494, 1972. K. 0. Hill and A. Watanabe, Opt. Commun., vol. 5, p. 389,1972. R. L. Fork and 2. Kaplan, Appl: Phys. Lett., vol. 20, p. 472, I 1972. C. J. Koester, IEEE J. Quantum Electron., vol. QE-2,p. 580, Sept. 1966. P. Anliker, photograph andbiography not available atthe time of N. Periasamy and F. P. Schaefer, Appl. Phys., vol. 24, p. 201, publication. 1981. H. Injeyan et al., Appl. Opt., vol. 21, p. 1928, 1982. 9 K. 0. Hill et al., Appl. Opt., vol. 11, p. 1952, 1972. K. 0. Hill et al., J. Opt. SOC.Amer., vol. 64, p. 26 3,1974. A. Watanabe et al., Can. J. Phys., vol. 51, p. 761,1973. H. P. Weber ("72), photograph and biography not available at the time D. Gloge, Appl. Opt., vol. 10, p. 2252, 1971. of publication. Degree of Polarization in the Lyot Depolarizer Abstract-Analytic output equations for the degree of polarization of plate axes is madeso the depolarizer will be operative indepen- light exiting a Lyot depolarizer are derived from a coherency matrix dent of the input polarization state. If depolarization is not representation. Design miteria areobtained for sources of different achieved by the first plate, it will be achieved by the second. spectral shape. Billings [Z] analyzed the Lyot depolarizer by assuming suf- ficient birefringence to provide 2n of phase retardation over I. INTRODUCTION the source bandwidth in the first plate. His model for a rec- YOT DEPOLARZERS [ 11, [2] have recently attracted re- tangular-shaped source suggested that the plates should have Lnewed attention due to the successfuluse of one in reduc- an integer thickness ratio not equal to one. Loeber has treated ing polarization noise in a fiber-optic gyroscope[3]. Recently, the problem for a blackbody [5]. Recent approaches have fo- such a device has been reported [4] which was constructed cused on frequency decomposition of the output polarization frombirefringent single-mode fibers rather than the usual sjates on the Poincare sphere 1161, attempting to achieve a de- birefringent crystal plates. sign that provides a uniform distribution of such output states The Lyot deploarizer classically consists of two birefringent [7], [8]. However, these approaches are not capable of pro- crystal plates (X or Y cut), whose crystal axes are oriented at viding quantitative design rules for Lyot depolarizers. 45" with respect to each other, and whose thicknesses are in Rashleigh and Ulrich [9] havepointed outthat polariza- the ratio 1:2. The device is intended to be used with broad- tion-modedispersion in single-mode fibers leads to a group bandlight and may be viewed as performinga polarization delay difference whichcauses the depolarization of broad-band conversion on the spectral components of polarized input light.light. The role of polarization dispersion as opposed to'bire- At the output, each spectral component appears with a differ- fringence in depolarization has also been emphasized theoret- ent polarization state, and upon averaging over bandwidth, the ically by Sakai et al. [IO]. In this paper, we follow this ap- output is depolarized. The choice of a 45" angle between the proach to calculate analytic output equations for the degree of polarization of a Lyot depolarizer, as a function of the input Manuscript received April 18,1983; revised June 6, 1983. polarization state, source bandwidth and band shape, andfiber The author is with The Naval Research Laboratory, Washington, DC 20375. or crystal group delay differ%ce. This result will aid in the de- US. Government work not protected byU.S. copyright 416 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. LT-1, NO. 3, SEPTEMBER 1983 sign of depolarization devices in applications. Our model has beenpresented qualitatively byEpworth [ll]. We ignore mode coupIingbetween polarization modes which has been treated in the paper byBohm et al. [4]. 11. THEORY We assume a configuration forthe Lyot depolarizer as shown in Fig. 1. The birefringent sections may be crystal plates or sections of briefringent fiber, although we will use fiber ter- dL' minology. We assumefiber-mode propagation constants 0, Fig, 1. Model for Lyot depolarizer formed with birefringent fibers. and fly, which are functions of optical frequency w. The fiber birefringent axes are denoted byX and Y in the first section of the coherency matrix J length L1, and X' and Y' in thesecond section of length L2(L22 L1). The second section is rotated relative to the first by 45". We assume an optical source with spectral intensity I (4) lv(w)I2and center frequency wo. The optical inputto the de- polarizer is assumed to be linearly polarized with input azimuth at an angle 0, from the mode axis X. For unity power input, where t signifies the Hermitian transpose. To compute (4), we the time-dependent electric-field input to the fiber may then requiretime-varying electric-field outputs. We express time- be represented varying electric fields in terms of a complex analytic signal in the usual way [ 101 by defining e(t)in (1a) as where where ~(w)is the (complex) amplitude spectrum ofthe source. The output time-varying fields from the depolarizer are then given by and ( ) signifies a time average. At the joint between the two 1 sections, polarization-mode coupling will occur. The transfer ~,t(t) = E {cos e, e i[wot-@xo(L1 +'2)J e(t - P:,(L, + L,)) matrix for such couplingis given by +sin 0, ei(wot-flxoL2-PyoL1) . e(t - /3:0L2 - P;oLl)) (6a) 1 ~~,(t)=-{-cos ,gp ei(wOt-PxoL1-PyoL2) v5 . e(t - pioL1 - 0' L + sin ,gp e'Iwot-Pyo(L1 +LdJ Propagation in the fibersections is governed bythe fre- YO 2 quency-dependentpropagation constants f px(o) and /3, . e(t - P;O(Ll + L?,))} (6b) py -p,(w). Employing the transfermatrix in (2), we can write the frequency-dependent output fields at the end ofsec- where px0 px(oo),py0 E /3,,(w0) and p' denotes differentia- tion L2 as tion of /3 by w. To derive (6), we have used (3) and papers by Sakai et al. [ 101 and Burns et al. [ 131. Using (6), we calculate the components of the coherency matrix by performing the necessary time averages JXtxt = 1 [So + sin28, S1(L,) cos A/3,L1] (74 Jy,yt = 1 [So - sin20, Sl(L1)cos ApOLl] (7b) ~~t~f= eiAPoL2 (-c~~2e, s,(L~) (3) + 3 sin20, eiA@dl[S~(L, +L,) where EoX(w)and Eoy(w)are the input fields at thefrequency - e-izaPoL1 s,(L,- L,)J} (7b) component w. Jy;' = (JXfY')* (74 The degree of polarization is defined as the fraction of opti- cal power that is polarized [lo], [12]. It is calculated from where 48, = py(wo)- pX(w0)and So and S1(z) are defined by BURNS: DEGREE OF POLARIZATION IN THE LYOT DEPOLARIZER 477 Sl(z) = 8rJm Iv(w)12 cos [(a- o0)Srgz] dw. (8b) 0 To derive (7) and (8), we have assumed the source spectrum is symmetrical with respect to oo.In (8b), z is a length variable Q E 6w 87 z and ST, is the polarization-modedispersion or group delay Fig. 2. Magnitude of the degree of coherence for sources with various difference band shapes. 6 w is the spectral half-width at half-power, ~7~the fiber group delay difference, andz the fiber length. We first look at (1 1) for special cases of 0,. For 8, = 0 the The degreeof polarization is obtainedfrom the coherency input is polarized parallel to theX-axis of the first section (L1) matrix as [ 121 and no depolarization occurs in this section. The input to the second section (L2) is at 45" to its birefringent axes and we have P(0) = ly(Lz)l. For 8, = 45' depolarization will occur in the first section and the second section becomes unnecessary. whichyields P(0,) asa function of the input polarization We take 0, = 45" and L2 = 0 and obtain P(45') = )y(L1>I. azimuth. The result is These results agree with thecalculation for a single fiber length in the paper by Sakaiet al. [IO]. In general, we want P(0,) = 0 p(ep)'=(cos2 28, y2(~2) for any 6,. We assume L2 2 L1 and a monotonically decreas- + 3 sin228, {y2(~,)+ [Y~(L~) ing y(z). From inspection of (1 l), we require - y(L, +L2)Y(L2 - Llll cos 2APoL1 Y(L1) = 0 ( 14a) + p [Y2(Ll + L2) + Y2GZ - L111) and - sin28, COS 28,y(L2) [y(L1+Lz) Y(L2) = 0 (14b) - y(L2 - Ll)] COS APoL1)"' (11) which then implies y(L, t L2)= 0. We also require where y(z) S,(z)/S, is the degree of coherence [12].
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