Tilted fiber gratings have frequently been used to se-lectively couple light out of the fiber, and polarization-dependent scattering (PDS) is an important issuefor those gratings. For fiber grating spectrometersused in metrology and in dense wavelength-divisionmultiplexing channel monitoring,1,2 PDS makes themeasurement inaccurate and needs to be minimized.For other devices that actually utilize this polariza-tion dependence, such as in-fiber polarimeters3 andpolarization-dependent loss compensators,4,5 PDS alsoneeds to be optimized to improve device performance.The volume-current method6 (VCM) has been shownto provide accurate predictions of scattering from fibergratings.7 As in many other perturbation analyses,8,9
the physical model described in Ref. 6 is based on ascalar wave approach in which the incident light isassumed to be linearly polarized. This treatmentis usually considered valid, and it has already beenshown within coupled-mode theory that the inf luenceof Ez in Bragg ref lection is negligible.10
However, in our recent measurement we found thatPDS decreases much faster with longer wavelengthsthan predicted by the VCM in Ref. 6. We are able toaccount for this discrepancy by including the longitudi-nal E-f ield component of the LP01 fiber mode, showingthat Ez is not negligible even though the grating waswritten in a standard single-mode fiber that satisfiesthe condition of weak guidance. Applying the VCM tothe complete LP01 mode field, we show that the scat-tered f ields from Et and Ez can overlap [see Fig. 1(a)]and that this overlap is much stronger than the scat-tering from Ez alone. Moreover, the dependence on
0146-9592/04/070691-03$15.00/0
wavelength of Ez scattering is different from that ofEt; thus it can be signif icant when the scattering fromEt is small. For the 10±-tilted f iber grating that weused, our VCM simulation that includes both Et andEz has an excellent match with the measured data.
Fig. 1. Polarization-dependent scattering from fibergratings and experimental setup for PDS measurement.(a) Dependence of the polarization of the scattering fieldon the polarization of a guided wave. Note the behavior ofthe longitudinal mode component. (b) PDS measurementsetup for the Mueller matrix method. SM, single mode;TLS, tunable laser source.
Our PDS measurement setup is shown in Fig. 1(b).A tilted fiber grating was written in the core of astandard single-mode fiber by UV inscription with aphase mask. The sample was then index matchedto a silica block, allowing light to escape from thefiber. The phase mask had a pitch of 1.09525 mmand was tilted by 6.7± relative to the fiber duringwriting. Cylindrical lensing by the f iber claddingincreased the tilt angle to 10± inside the core. TheUV-induced index modulation was approximately3.0 3 1024. The overall PDS was measured with anaccuracy of 60.05 dB by the Mueller–Stokes method11
for every wavelength setting, and special care wastaken to avoid polarization-dependent loss through theoutput face of the block so the data would emphasizegrating-induced PDS.
The measurement data at 1530–1585 nm wereshown in Fig. 2(a) together with the predictions fromthe VCM based on a scalar-wave model. The VCMpredicts that, without the longitudinal field, the PDSwill decrease from 0.73 to 0.33 dB for wavelengthsfrom 1530 to 1565 nm (C band for dense wavelength-division multiplexing), a slope of 1.14 3 1022 dB�nm.However, in the actual measurement the PDS de-creased from 0.82 to 0.27 dB over the same range,with a slope of 1.57 3 1022 dB�nm, which is 40% fasterthan predicted. Moreover, the PDS measurementreached zero crossing between 1575 and 1580 nmand then increased for longer wavelengths. Usingthe longitudinal phase-matching formula from Ref. 6,we found that, for 1575–1580 nm, scatter angle a0
[the angle between the scattering direction and thenegative fiber axis, as shown in Fig. 1(b)] is still �12±,indicating that this minimum PDS was not occurringat Bragg resonance. The scalar VCM model could notaccount for these differences.
To explain these discrepancies we found it necessaryto consider the complete vector mode field of the guidedwave. We model our standard single-mode fiber as astep-index fiber with 8.3-mm core diameter and indexstep Dn�n � 0.4%. For weakly guided optical fiber,longitudinal f ield Ez may be written as12
Ez�r,f, z� �iuk0n
E0J1�ur�cos�f 2 d�exp�2ibz� , (1)
where u is the transverse wave number of the LP01mode, E0 is the amplitude of transverse field Et, d isthe polarization angle of the transverse field (d � 0±corresponds to S polarization and d � 90± correspondsto P polarization), J1 is a Bessel function of thefirst order, and the time dependence omitted hereis exp�1ivt�. For standard single-mode fiber theamplitude of Ez is usually 5% or less than that of Et.Following treatments similar to those described inRef. 6, the induced field from Ez can also be obtained:
Ein, z �2ktAz
ivm0e�Dr 2 ktz� . (2)
In Eq. (2), Az is the corresponding scatter potentialfor Ez and D and kt are the longitudinal and transverse
wave numbers for the scattered light, respectively, de-fined in the same way as in Ref. 6. The overall powerf low density for the scattered light will be
S �1
2vm02e
�jAtj2�D2 1 kt
2 sin2�d 2 f��
2 �AtAz� 1 AzAt
��ktD cos�d 2 f� 1 jAzj2kt
2�
3 �ktr 1 Dz� . (3)
Besides the original contribution from Et (the termproportional to jAtj
2), now we also have scatteringfrom Ez (the term proportional to jAzj
2) as well asthe overlap of the two (the term proportional toAtAz
� 1 AzAt�). Each term in Eq. (3) has a different
polarization dependence, and the wavelength depen-dences of At and Az are also different.
To illustrate the underlying optical interactionswe look at azimuthal direction f � 90±, which isthe preferred direction of scattering according to thegrating tilt. Equation (3.19) of Ref. 6 describesthe induced E field from Et. For S polarization,d � 0±, d 2 f � 290±, the formula reduces toEin, t �2k0
2n02At�ivm0e�f, which is perpendicular
to Ein, z, as given in Eq. (2). As a result, the inten-sity correction from Ez is small. However, for Ppolarization, d � 90±, d 2 f � 0±, the same formulareduces to Ein, t �DAt�ivm0e� �Dr 2 ktz�, which ispolarized in the same direction as Ein, z [see Fig. 1(a)];therefore Ein, z and Ein, t overlap. This overlap forP polarization is the basic reason that Ez cannot beneglected in scattering. We show below that Ein, zand Ein, t also have different wavelength dependences,a fact that also contributes to the change in PDS.
Fig. 2. PDS measurement for a 10±-tilted f iber grating,and VCM predictions. (a) Measurement compared withprediction from a scalar-wave model. Note the cleardiscrepancy in slope and zero crossing. (b) Measurementcompared with a vector model that includes the longi-tudinal field. A simulation with nanowatt-level noiseis also shown, as is the first-order approximation fromrelation (6).
In this preferred direction (f � 90±), PDS of the grat-ing can be obtained from Eq. (3):
PDS 10 log10
ÇSS
SP
� 220 log10
Çcos a0 2
Rz
Rtsin a0
Ç.
(4)
Angle a0 is the scatter angle defined earlier; it satisfieslongitudinal phase-matching condition k0n0 cos a0 �Kg 2 b, where Kg represents the longitudinal wavenumber of the grating and b is the propagationconstant for the LP01 mode. We use a0 here to dis-tinguish it from angle a defined in Ref. 6; in fact,a0 � 180± 2 a. Rz and Rt represent the strengthof Ez scattering and Et scattering, respectively; atf � 90± they can be written as
Rz �uk0n
Z a
0J1�ur0�J1�Ktr0 1 ktr0�r0dr0,
Rt �Z a
0J0�ur0�J0�Ktr0 1 ktr0�r0dr0, (5)
where a is the f iber core radius and Kt is the trans-verse wave number of the grating. Without Ez, thePDS is a simple function of a0, as predicted in the origi-nal VCM and in other perturbation analyses. WithEz included, PDS now depends on the strength ratiobetween Rz and Rt. At the peak resonance the trans-verse phase-matching condition is satisfied, Kt 1 kt �0. As a result, Rt reaches maximum and Rz is zero,Ez has no inf luence on the PDS. When the wave-length is shifted away from peak resonance, Rt willdecrease and eventually go through a zero crossing.Near this region, P polarization dominates the scatter-ing, so it is not only possible to have a PDS of 0 dB butalso likely to have large negative PDS values. FromFig. 2(b) we can see that the measured PDS changedin the same way as predicted here, especially after thedetector noise near the zero crossing point was takeninto account.
Using the f irst-order approximation of Rt, Rz, anda0 near peak resonance, we obtained a formula for theargument of PDS in Eq. (4) inside the resonant scat-tering band:
cos a0 2Rz
Rtsin a0 cos amax
0
1
∑1 1
uaJ2�ua�2J1�ua�
∏l 2 lmax
n0LZ
. (6)
Here LZ is the longitudinal grating period, lmax isthe peak resonance wavelength, and amax
0 is the scat-tering angle at l � lmax, which is approximately twicethe grating tilt angle. This formula should be com-pared with the result when only transverse field Et isconsidered:
cos a0 cos amax0 1
l 2 lmax
n0LZ
. (7)
For the single-mode fiber that we used and a wave-length near 1550 nm, the slope correction factor in re-lation (6) is �35%, which is close to what we observedin the experiment. The prediction from relation (6) isalso shown in Fig. 2(b); we can see that it works reason-ably well for wavelengths inside the resonance band of1520–1570 nm. For wavelengths near the zero cross-ing point, this formula is no longer accurate and theexact solution has to be used.
In summary, we have demonstrated that the lon-gitudinal E-f ield component of the waveguide modeplays an important role in polarization-dependentscattering from tilted f iber gratings, even in standardsingle-mode fiber that exhibits weak guidance. Asimple first-order formula for the wavelength depen-dence of PDS was also obtained and was shown to beaccurate for most wavelengths inside the resonanceband of scattering.
Y. Li (e-mail [email protected]) acknowledgespartial support from the National Science Foundationunder grant ECS-9816251.
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