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.

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 sufficient 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 platesshould 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- sucha device hasbeen 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 toa 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 depolarizer 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 optical 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]. r(L2 - L1) = 0 (1 4c) 111. DISCUSSION which is satisfied if (14a) is satisfied and if L2 - L1 2 L1 or Equation (1 1) gives the output degree of polarization forthe L2 2 2 L1. The entire condition can then be stated as Lyot depolarizer. It depends on the degree of coherence y(z) m [ Iv(w)l' cos [(a- W0)STgZ] dw Y(L1) = 0 (154 Lz 22L1 5b) (1

to ensure P(6,) = 0 for all Bo with Lorentzian- or Gaussian- shapedsourc& This yields'a mathematical explanation for which depends On the Source parameters wo and Iv(0>12 and the original condition ofL2 = 2 L1in a Lyot depolarizer. For the fiber parameter sTg. We note that y(z) is a even func- a rectangularshaped source,.-which was assumed by Billings tion of z. For band shapes withspectral width 2Sw, Sakai [2],the situation is morecomplicated because y doesnot et has 7 for the shapes monotonically decrease with z. Following the argumentabove, we arrive at the condition -sin Q Rectangular: Y" (13a) a Y(L1) = 0 Q Gaussian: y = exp [-( t/in2 )'1 (13b) L2 = n L1, n integer # 1 It is clear from Fig. 2 that precision in L1 and L2 is only re- Lorentzian: y = exp (- a) (13') quired with a rectangular-shaped source. where Q = 606~~~.These functions are compared in Fig. 2. The condition for OW output degree of polarization is that For the Gaussian and Lorentzian cases, y is a monotonically the quantity 6wSrgzbe large (2.). We may express the group decreasing function of a. Experimental results have been given delay difference Srg in terms of the effective-index difference for rectangular- [9] and Gaussian- [ 141 shaped sources. Aneff between the modes 191. Since Ab = Aneff o/c,where c 478 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. LT-1, NO. 3, SEPTEMBER 1983

Equations (1 1j and (15 j predict that the degree of polarization with a given bandwidth source may be made as small as desired by using sufficient lengths of fiber with a large group delay difference. For a Gaussian source with half-width 6h 7 nm and a fiber with 67, E 1 ns/km, a fiber length (I,,) of 0.5m should be sufficient [14]. Of course thetheory presented here will be limited by mode coupling between polarization modes, whichwas treated in the paper by Bohm et al. [4]. Experimentally,a value P - wasachieved ina Lyotde- polarizer [4], anda value P - ina single section of high- birefringence fiber [16].

IV. CONCLUSIONS We have provided analytic output equations for the degree Fig. 3. Model for depolarization in a Lyot depolarizer of broad-band of polarization in Lyot depolarizers which can be applied di- lightwith coherence length LC. AL1 and AL2 are the group delay differences experienced in the first and second sections, respectively. rectly to experimentalarrangements. This derivation clearly For depolarization wercquire LC << AL1 and ALz - 2AL1. De- points out the importance of source band shape and of group polarization is then independent of input azimuth. delaydifference in the birefringentelements, and will allow quantitative device design. Our approach has been put in con- is the light velocity, we have textwith previousdescriptions of the device. This analysis will be useful to device designers who will employ this impor- dA0 - Aneff 0 d(Aneff) tant device in fiber gyroscopes. -- f- (17) 67g=- dm C c dcd ACKNOWLEDGMENT so that the dispersion of An,,, when it is not negligible, con- tributes to depolarization as well as the index difference itself. Theauthor appreciates helpful conversationswith R. B. This is the case, for example, in elliptically cored fibers. Smith on this subject. Epworth [I 11 has given auseful physical picture of the REFERENCES mathematics presented here, following the original model of Rashleigh and Ulrich [9]. Broad-band light has a finite region [ 11 B. Lyot, Annales de l’observatoire d’astronomie de Paris (Mendon), vol. VIII, p. 102, 1929. of coherence [ 121. Depolarization then occurs when a polar- [2] B. H. Billings, “A monochromatic depolarizer,”J. Opt. Soc. Amer., ized input is evenly divided between two polarization modes, VO~.41, pp. 966-975, 1951. and the polarization-mode dispersion is sufficiently large that [3] K. Bohm, P. Marten, K. Peterman, E. Weidel, and R. Ulrich, “Low- drift fibre gyro using a superluminescent diode,” Electron. Lett., the resultinggroup delay is greater thanthe length of the VO~.17, pp. 352-353,1981. coherent region. Using two fiber lengths with a relative axis [4] K. Bohm et al., “Performanceof Lyot-depolarizers with bire- rotation of 45” insures that this conditionis met in one of the fringent single-mode fibers,” J. Lightwave Technol., vol. 1, pp. 71-74, 1983. two sections for all linear inputs. An arbitrary input polariza- [5] A. P. Loeber, “Depolarization of white light by a birefringent tion state is then depolarized by superposition. The 1 : 2 length crystal. I1 The Lyot depolarizer,”J. Opt. SOC.Amer., vol. 72, pp. ratio of the sections avoids overlap of the coherent regions in 650-656,1982. [6] R. Ulrich, “Polarization and depolarization in the fiber-opticgyro- the second section. We adapt this picture in Fig. 3, assuming scope,” in Fiber OpticRotation Sensors, S. Ezekiel and H. J. a nonrectangular source. Arditty, Eds. Berlin, Germany: Springer-Verlag, 1982,pp. 52-77. Finally, wecan compareour resultshere withthe model [7]R. B. Smith, C. H. Anderson,and G. L. Mitchell, “Numerical modeling of dual polarization interferometric gyros and sensors,” used to analyze Lyot depolarizers by performing a spectral de- in Fiber Optic RotationSensors, S. Ezekiel and H. J. Arditty,Eds. compositionof output polarization states on the Poincare Berlin, Germany: Springer-Verlag, 1982, pp. 93-101. sphere [6]-[&I. The approach developed here points out the [8] R. B. Smith, G. L. Mitchell, and C. D. Anderson, “Fiber depolar- izcrs using polarization preserving fibers,” in Proc. OFC’83, New importance of spectral band shape and dispersion of An,ff to Orleans, 1983, papcr TuG9. depolarization, factors which are not readily included in the 191 S. C. Rashleigh and R. Ulrich, “Polarization mode dispersion in Poincare sphere representation. The theoretical significance of single-mode fibers,” Opt. Lett., vol. 3, p. 60, 1978. [lo] J. I. Sakai, S. Machida, and T. Kimura, “Degree of polarization in a uniform distribution of output polarization states (spectrally anisotropic single mode optical fibers: Theory,” J. Quantum Elec- decomposed) on the Poincare sphere, has not been rigorously tron. vol. QE-18, p. 488, 1982. defined. In constrast, depolarized light has been rigorously de- [ 111 R. E. Epworth, “The temporal coherence of various semiconduc- tor lightsources used inoptical fiber sensors,” in Fiber Optic fined as light whose polarization state, viewed in a sequence Rotation Sensors, S. Ezekiel and H. J. Arditty, Eds. Berlin, of sufficiently short time intervals, does uniformly sample the Germany: Springer-Verlag, 1982, pp, 237-244. surface of the Poincare sphere [15]; or as light whose Stokes [12] M. Born and E. Wolf, Principles of Optics, 3rd ed. New York: Pergamon, 1965, pp. 316-323,491-506,544-554. parameterswhich describe polarization are zero [12], [ 151. [13] W. K. Burns, R. P. Moeller, and C. L. Chen, “Depolarization in Thesecriterion are directly related to thecoherency matrix a single-mode optical fiber,’’ J. Lightwave Technol., vol. 1, pp. representation used above,and so must also besatisfied for 44-50,1983. [14] W. K. Burns and R. P. Moeller, “Measurement of polarization any arrangement which gives zero degree of polarization as de- mode dispersion in high-birefringence fibers,” Opt. Lett., vol. 8, fined by (1 l). pp. 195-197,1983. JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. LT-1, NO. 3, SEPTEMBER 1983 479

[IS] R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized During 1971he wasstaff a memberArthurat Light. New York:North Holland, 1979,pp. 52-59. D. Little, Inc., Cambridge, MA, wherecon- he [16] R. P. Moeller and W. K. Burns,“Depolarized broadband source,” tributed to variouslaser-related studiesand Electron. Lett., vol. 19, pp. 187-188,1983. surveys. Sincehas he 1972 been a Research Physicist atthe Naval Research Laboratory, * Washington, DC. He is presently Head of the Optical Waveguide Section of the Optical Tech- William K. Burns (”80) was born in Philadelphia, PA in ’June 1943. niques Branch. His research interestsinclude He received the B.S. degree in engineering physics from Cornel1 Univer- , integratedoptics, single-modefiber optics,and sity,Ithaca, NY, 1965,in and the M.S. and Ph.D. degrees in applied ;5ss the applicationof single-mode technology to physics from Harvard University,Cambridge, MA,1967 in and1971, communications, sensors, and signal processing. respectively. His thesisresearch was in nonlinear optics. Dr. Burns is a member of the OpticalSociety of America.

Low-Loss Coupling of Multiple Fiber Arrays to Single-Mode Waveguides

EDMOND J. MURPHY AND TRUDIE C. RICE

Abstract-A methodfor obtaining permanent low-loss coupling would be ideal. In principle, the tongue and groove could be between arrays of single-mode fibers andTi:LiNbOs waveguides is formed byetching. Unfortunately,the submicrometer accu- described. The technique, based on the use of silicon chip V-grooves, racy required here may preclude this possibility especially for simplifies the coupling problem by simultaneously aligning the entire array and by providing a large surface area for a higher integrity adhe- chemicallyinert materials like lithium niobate. Theother sive bond. At h = 1.3 pm, we measure an average 1.9dB coupling loss obvious and, in fact, more tractable approach involves the use (exclusive of propagation loss) for the assembled array. The average of etched V-grooves in silicon. The etching of these grooves is excess loss due to the fiber array is0.8 dB. well understood [3] and the grooves can be accurately placed We present ananalysis of the effect ofvarious types of array misalign- by using standardphotolithographic processes. Moreover, ment on coupling efficiency. Angular alignment and array periodicity are found to be critical. If the fiber and waveguide periodicities ate thesechips are commonly used forother fiberalignment matched exactly, the fibers need only be placed within * 1.3 pm of applications [4]. their optimum position to maintain coupling efficiencies greater than Several otherauthors have reportedon permanent fiber- 90 percent. waveguide fixturing. The earliest work involvedthe “flip-chip’’ approach [5]. Ramer et al. reported on a 2 X 2 switch at X = HE EFFICIENT COUPLING of light from a single-mode 0.83 pm with attached fiber pigtails [6] . The excess lossdue to T fiber into a single-mode waveguide (in glass or in an elec- fiber coupling was approximately 1.5 dB per fiber per interface. troopticmaterial) is animportant and formidable problem. In the work by Kondo et al., a pigtailed 1 X 4 switch array at Recentresults [l], [2] demonstratethat through a careful h = 1.3 pm is reported. Inthis case, a four-element Si chip choice of waveguide fabricationparameters, high efficiency array was used at the switch output with an excess loss of at can be attained. In this research, micropositioners were used least 1.O dB per fiber. to align one input and one output fiber tosingle a waveguide, in The approach we utilize here is to fabricate arrays of fibers what amounts to a tedious and time-consuming manual opera- in Si chips and to align them to appropriately spacedwave- tion. However, two important problems were not addressed. guides in LiNb03. If the periodicities of the fibers and wave- No attempt was madeto simultaneouslycouple intomore guides are equal, alignment of any two fibers will result in the than one waveguide and no attempt was made to permanently alignmentof all of the fibers. The Si chips used hereare attach the fibers. In this paper, we report our initial results on standard Western Electric twelve-groove chips used for multi- a multiple fiber coupling and calculate the influence of various mode-fiberribbon connectors [4]. Thenominal groove misalignments on coupling efficiency. spacing is 228.5 ym. We use a standard fiber for X = 1.3 pm There are several approaches to the solution of these prob- with a mode size (half width at l/e amplitude) of 4.0 ym and a lems that deserve consideration. Clearly, a technology which coreeccentricity of 0.7 pm.The fiberarray (see Fig. 1) is allows fora simpletongue-in-groove automatic alignment formed by stripping a small length of thefiber’s plastic cladding and epoxying the fibers in a Si chip “sandwich.” The chip is Manuscript received April 18, 1983; revised June 16, 1983. Theauthors are with Bell Telephone Laboratories, Allentown, PA then cut and polished to an smooth 18103. experiment, this For waveguidesfabricated were on a z-cut

0733-8724/83/0900-0479$01.OO 0 1983 IEEE