. RESEARCH PAPERS .

SCIENCE CHINAInformation Sciences

February 2010 Vol. 53 No. 2: 390–397

doi: 10.1007/s11432-010-0017-9

c© Science China Press and Springer-Verlag Berlin Heidelberg 2010 info.scichina.com www.springerlink.com

Theoretical analysis of polarization properties fortilted fiber Bragg gratings

XU Ou1∗, LU ShaoHua2 & JIAN ShuiSheng2

1Faculty of Information Engineering, Guangdong University of Technology, Guangzhou 510006, China;2Institute of Lightwave Technology, Beijing Jiaotong University, Beijing 100044, China

Received July 22, 2008; accepted January 4, 2009; published online January 28, 2010

Abstract The polarization properties for tilted fiber Bragg gratings (TFBGs) are investigated theoretically

based on coupled-mode theory and Mueller matrix method. The expression of wavelength-related polarization-

dependent loss (PDL) for TFBGs with different tilt angles is derived and calculated. Simulation results are

compared, and the results indicate that the polarization capability of TFBGs with 45◦ angle is stronger than

other TFBGs with smaller angles. The degree of polarization for unpolarized light passing TFBGs is also

simulated to further evaluate the polarization properties for various tilt angles. In addition, the relationship

between physical parameters of a TFBG and its polarization capability is discussed.

Keywords coupled-mode theory, degree of polarization (DOP), Mueller matrix method, polarization depen-

dent loss (PDL), tilted fiber Bragg gratings

Citation Xu O, Lu S H, Jian S S. Theoretical analysis of polarization properties for tilted fiber Bragg gratings.

Sci China Inf Sci, 2010, 53: 390–397, doi: 10.1007/s11432-010-0017-9

1 Introduction

Tilted fiber Bragg gratings (TFBG), also called blazed, slanted, and side-tap gratings [1], as a specificsubset of grating components, have attracted increasing attention in recent years due to their uniqueabilities. TFBGs can realize efficient coupling of core mode to cladding and radiation modes, and thecoupling can be highly polarization-dependent at large tilt angles [2], thus suggesting the potential ofTFBGs for polarization-related applications. Taking advantages of this characteristic, TFBGs has beenimplemented as in-fiber polarizers [3], polarization-dependent loss (PDL) equalizers [4], PDL compen-sators [5] and in-line polarimeters [6, 7]. A theoretical analysis of the polarization properties for TFBGsis thus desirable to present evaluations of the polarization-dependent quantities such as PDL and thedegree of polarization (DOP) of transmitted light through a TFBG. However, as far as we know, inprevious papers the polarization properties of TFBGs has not been analyzed quantitatively. Thus theoptimization and design of such devices have been somewhat affected.

In this paper, based on the coupled-mode theory and Mueller matrix method, the polarization prop-erties of TFBGs with different tilt angles are studied and described. Through the Mueller matrix, anexplicit expression of wavelength-dependent PDL for TFBGs is derived. The polarization capability of

∗Corresponding author (email: aliceseaer@163.com; xuou@gdut.edu.cn)

XU Ou, et al. Sci China Inf Sci February 2010 Vol. 53 No. 2 391

Figure 1 Schematic diagram of a tilted fiber grating.

TFBGs with different tilt angles are compared by simulating the corresponding PDL and DOP of an un-polarized light after it passes through TFBGs. And the influences of specific grating physical parameterson the polarization properties are investigated so as to provide some rough guidelines for the design ofTFBG related devices.

2 Theory

2.1 Transmission coefficient for s-polarized or p-polarized light in TFBG

Figure 1 shows an x-tilted grating written in a fiber whose index-modulation planes are tilted aboutthe y axis by an angle θ from the x axis, and the z axis coincides with the fiber axis. Here only thecoupling between the bound core mode and the radiation modes in a fiber with an infinite cladding isconsidered, which can be realized in experiments by immersing the fiber in a medium with a refractiveindex equal to or higher than that of the cladding. According to the coupled-mode theory [2], thegrating-induced coupling between the forward-propagating bound core mode LP01 and a whole set ofbackward-propagating radiation LP modes can be described by

du

dz= β[iγ − κ2As(p)]u(z) + iβgbκυ(z),

dυ

dz= −β[iγ − κ2As(p)]υ(z) − iβgbκu(z),

(1)

where u(z) and υ(z) are forward-going and backward-going core-mode amplitude related variables; β

is the reference wavelength number, γ is a general “dc” self-coupling coefficient [8]; γ = gfσ + δ for agrating without chirp; gf and gb are coupling constants mainly determined by the fiber characteristicsand the grating pitch along the fiber axis; σ and 2κ describe the grating index amplitude which areconstants for uniform grating. This paper focuses on perfectly modulated uniform gratings, i.e. σ =2κ. δ = K−1(β01 − K), β01 is the propagation constant for LP01 mode and K represents the wavenumber of the grating along the fiber axis (Figure 1). As(p) is called the effective extinction coefficient,As(p) =

∑q A

s(p)q , q = 1, 2, 3, . . ., and A

s(p)q represents the coupling efficiency between the s-polarized

(p-polarized) LP01 core mode and s-polarized (p-polarized) LPq radiation mode. Because the couplingstrength between two modes of the same polarization state is by several orders of magnitude strongerthan the coupling between modes of different polarization states, or depolarized modes, we only considerthe s → s or p → p coupling.

Assuming u(L/2) = 1 and υ(L/2) = 0 for a uniform tilted grating of length L, we can get thetransmission coefficient of s-polarized or p-polarized light for the grating as

ts(p) =u(L/2)

u(−L/2)=

1

cosh[βLγs(p)] + κ2As(p)−i(gf σ+δ)

γs(p) sinh[βLγs(p)], (2)

where γs(p) =√

(gbκ)2 − (gfσ + δ + iκ2As(p))2, and the power transmission coefficient Ts(p) = |ts(p)|2.

392 XU Ou, et al. Sci China Inf Sci February 2010 Vol. 53 No. 2

2.2 Description of the transmission properties of the TFBG by a Mueller matrix

Considering a Cartesian coordinates system such that the x(y) axis corresponds to the p(s) polarization,the incident polarized light can be represented by the following Jones vector:

[Ex

Ey

]

=

[Ep

Es

]

. (3)

Because the mode coupling is very weak between the different polarizations [2, 8] and can be ignored,the transmitted light can be simply represented by

[E′

x

E′y

]

=

[tpEp

tsEs

]

. (4)

The Stokes parameters can be deduced by the Jones vectors. For the incident light characterized bythe Stokes vector Sin, the four Stokes parameters can be obtained from the Jones vector in (3) as

S0 = E∗pEp + E∗

sEs, (5)

S1 = E∗pEp − E∗

sEs, (6)

S2 = E∗pEs + E∗

sEp, (7)

S3 = i(E∗s Ep − E∗

pEs), (8)

Similarly, the Stokes parameters of Sout for the transmitted light are

S′0 = t∗ptpE

∗pEp + t∗stsE

∗sEs, (9)

S′1 = t∗ptpE

∗pEp − t∗stsE

∗sEs, (10)

S′2 = t∗ptsE

∗pEs + t∗stpE

∗sEp, (11)

S′3 = i(t∗stpE

∗sEp − t∗ptsE

∗pEs). (12)

With known Sin and Sout, the Mueller matrix for a TFBG can be deduced from Sout = M · Sin as

M =

⎡

⎢⎢⎢⎢⎣

m11 m12 0 0

m21 m22 0 0

0 0 m33 m34

0 0 m43 m44

⎤

⎥⎥⎥⎥⎦

, (13)

m11 = m22 = tpt∗p + tst

∗s, m12 = m21 = tpt

∗p − tst

∗s ,

m33 = m44 = tpt∗s + tst

∗p, m34 = −m43 = i(tst∗p − tpt

∗s).

From the Mueller matrix, the polarization dependent loss of the TFBG can be calculated [9, 10]

PDL = 10 log(

m11 + |m12|m11 − |m12|

)

. (14)

Substituting ts(p) into the matrix, the polarization characteristics of the TFBG under different condi-tions can be obtained quantitatively.

3 Simulations and discussion

3.1 Spectral characteristics of the TFBG with different tilt angles

Since the matrix elements ts(p) are wavelength-dependent transmission coefficients and strongly impactedby the tilt angle of the TFBG, first the transmission spectra of the light after passing the TFBG with

XU Ou, et al. Sci China Inf Sci February 2010 Vol. 53 No. 2 393

different tilt angles are simulated by eq. (2). In the simulations the core size is set at 5.25 µm, core indexn1 = 1.445, grating length L=5 cm, σ = 5× 10−4. Figure 2 shows the calculated transmission spectra ofs-polarized and p-polarized light for selected values of θ. In order to get a clear comparison and eliminatethe influence of dispersion on the TFBG, the simulations are performed while the centre wavelength ofthe transmission loss spectra is fixed and the grating period is varied.

For different θs, the wavelength range in the corresponding figure is adjusted to show the transmissionloss spectra completely. Note that, the width of the loss spectra are much wider than the transmissionspectra of untilted fiber Bragg gratings. That is because, for a tilted grating, the coupling of the boundcore mode to radiation modes with all odd azimuthal quantum numbers and all even numbers is allowed.But, for an untilted grating, the bound mode can only be coupled to radiation modes with azimuthalquantum numbers 0 and 2 [2]. From the figures, the difference between s-polarized curve and p-polarizedcurve becomes remarkably obvious with the increase of the tilt angle. In order to verify this trend, thesimulations for more tilt angles are calculated, and the minimum power transmission coefficients of twopolarizations T

s(p)min for each angle are compared and plotted in Figure 3. It is clear that the values of

Ts(p)min for two polarizations become more and more different with increasing tilt angle. When θ is 45◦, the

curve of T p, i.e. the transmission spectra of p-polarized light, is almost like a wave line and T pmin is not as

obvious as other tilt angles. Through simulations, we see that T pmin−T s

min is more than 0.444 for θ = 45◦.It should be noted that the wavelengths where the minimum transmissions appear are not the same fortwo polarizations with large tilt angles. In Figure 2(b), the lowest point in the p-polarization curve movestowards longer wavelengths about 25 nm, compared with the lowest point in the s-polarization curve fixedat 1550 nm.

3.2 Polarization properties of TFBGs with different tilt angles

The transmission spectra analysis shows that the transmission loss of a TFBG is significantly polarization-dependent, and the polarization sensitivity is different for different tilt angles. When θ = 45◦, all thep-polarized light almost passes through the TFBG, whereas the loss for s-polarized light is large. When

Figure 2 The calculated transmission spectra of s-polarized and p-polarized light for different tilt angles.

394 XU Ou, et al. Sci China Inf Sci February 2010 Vol. 53 No. 2

Figure 3 The minimum power transmission coefficients of two polarizations and the difference between two polarizations

versus tilt angles.

θ is small, e.g. 5◦, the transmissions for two polarizations differ little. The physical meaning of thedifferent polarization-dependences of the minimum power transmission coefficients with different tiltangles can be explained by Fresnel equations and Brewster’s law, as is detailed in [11]. Ref. [11] simplymade clear why the difference in the minimum power transmission coefficients between two polarizationsreached maximum when θ = 45◦. In the following, the polarization properties of TFBGs are furtherdiscussed quantitatively.

As is well known, polarization dependent loss (PDL) is defined as the maximum change in the powertransmitted by an optical device as the input state of polarization (SOP) is varied over all possiblepolarization states. To verify the deduced expression of the wavelength-dependent PDL of TFBGs (eq.(14)), we calculate PDL in the range of 100 nm around 1550 nm according to the parameters in [4]: tiltangle 38.2◦ (achieved with eq. (2) in [4]), grating period about 1 µm along the fiber axis, grating lengthof 2 cm, and index change 3.5 × 10−3. Simulation results are plotted in Figure 4, which shows that therange and variation tendency of the calculated PDL basically agree with the measured PDL in [4]. Thesmall deviation between the calculated results shown in Figure 4 and experimental results presented in[4] could result from the difference in detailed parameters of fiber structure in the simulations in [4]. It isnoted that there is no ripple in the PDL curve in Figure 4, but a ripple exists in the PDL measurement

Figure 4 Calculated PDL versus wavelength using pa-

rameters in [4].Figure 5 Calculated wavelength-dependent PDL for dif-

ferent tilt angles.

XU Ou, et al. Sci China Inf Sci February 2010 Vol. 53 No. 2 395

in [4]. This is because we simulate the coupling behavior in an infinite-cladding fiber which is coatedwith index matching oil, but in [4] the PDL was measured when the blazed grating was just in the air.

By eq. (14), the wavelength dependent PDL is calculated for different tilt angles (Figure 5). Theparameters used in the calculations are the same as those in Figure 2, and for convenient comparison,the smallest wavelength range of the transmission loss spectra is chosen for three angles (Figure 2).

Simulation results show that the PDL of the TFBG with θ = 45◦ is the largest among the three θs,and the difference between the maximum and minimum PDL values for 45◦ is less than 0.8 dB in thewavelength range of 200 nm. The flat curve of PDL indicates that the TFBG with 45◦ angle can beimplemented as a PDL equalizer, PDL compensator or a polarizer. In Figure 5, the maximum PDL valueis about 2.6 dB which is not large enough to meet the practical needs. For a TFBG with a certain periodand tilt angle, grating length L and index modulation amplitude σ are two other adjustable physicalparameters. Figure 6(a) shows the wavelength-dependent PDL for the TFBG of 45◦ angle with differentσs and Figure 6(b) shows different grating lengths. The values of PDL in the whole wavelength rangeincrease with increasing L and σ.

In order to examine the influence of these two parameters, we calculated the values of PDL at 1550nm with different grating lengths and index modulation amplitudes for three θs. The results are plottedin Figure 7. In Figure 7(a), the calculations show that PDL increases linearly with increasing gratinglength, when σ = 5 × 10−4. The slope of 45◦ curve is about 0.52 dB/cm for the TFBG structure usedhere, and the 15◦ curve is only 0.086 dB/cm. When the grating length is fixed at 5 cm, the PDL variesnonlinearly with index modulation depth (Figure 7(b)). When σ is 2 × 10−3, which can be realized inthe practical fabrications [12], the PDL for 45◦ can reach as high as 41.72 dB, while the growing speedsof the other two θs are slow. Thus increasing the grating length and strengthening the index modulationdepth can help to achieve a large PDL. And it is evident that, the increasing speed of PDL with gratinglength and index modulation depth is the fastest for 45◦ angle among all smaller tilt angles.

Figure 6 The wavelength-dependent PDL for the TFBG of 45◦ angle with different (a) σ and (b) grating length.

Figure 7 The PDL at 1550 nm versus (a) grating lengths and (b) index modulation amplitudes for three θs.

396 XU Ou, et al. Sci China Inf Sci February 2010 Vol. 53 No. 2

Furthermore, the polarizing capability of the TFBG with different tilt angles is evaluated by calculatingthe degree of polarization (DOP) of unpolarized light after it passes through the TFBG. The Stokes vectorof the unpolarized light is [1 0 0 0]T. The transmitted light vector can be deduced from the productof the Mueller matrix of the TFBG and [1 0 0 0]T as

Sout =

⎡

⎢⎢⎢⎢⎣

S0

S1

S2

S3

⎤

⎥⎥⎥⎥⎦

=

⎡

⎢⎢⎢⎢⎣

m11 m12 0 0

m21 m22 0 0

0 0 m33 m34

0 0 m43 m44

⎤

⎥⎥⎥⎥⎦

⎡

⎢⎢⎢⎢⎣

1

0

0

0

⎤

⎥⎥⎥⎥⎦

=

⎡

⎢⎢⎢⎢⎣

m11

m21

0

0

⎤

⎥⎥⎥⎥⎦

. (15)

The DOP is defined as the ratio between the intensity of the totally polarized component to the totalintensity of the light and can be calculated using Stokes parameters of the vector [13]. Here, we have

DOP =

√S2

1 + S22 + S2

3

S0=

|m21|m11

=|tpt∗p − tst

∗s|

tpt∗p + tst∗s. (16)

By combining eq. (16) with eq. (2), the wavelength-dependent DOP for TFBGs with different tilt anglescan be studied.

Figure 8 shows the DOP in a wavelength range of 200 nm for three angles. In simulations, the gratinglength is 5 cm and σ = 1 × 10−3. Comparing the three curves in Figure 8, we find that the DOP for

Figure 8 The wavelength-dependent DOP of unpolarized light after passing through the TFBGs with different tilt angles.

Figure 9 The DOP of unpolarized light after passing through the TFBG at 1550 nm versus (a) grating lengths and (b)

index modulation amplitudes for three θs.

XU Ou, et al. Sci China Inf Sci February 2010 Vol. 53 No. 2 397

θ = 45◦ is the largest and the maximum point in 45◦ curve can reach about 0.834. The calculationsfurther demonstrate that TFBGs exert strong polarization-dependent effects especially when the tiltangle reaches 45◦. The DOP can also be increased with the grating length or index modulation depth.The variations of DOP with grating length and index change amplitude σ for the three tilt angles aresimulated at 1550 nm and plotted in Figure 9. In Figure 9(a), σ is fixed at 5 × 10−4, and in Figure 9(b)grating length is 5 cm. Comparison of the curves of the three angles in Figure 9 shows that the DOP for45◦ grows fastest and first acquires the value of 1, which means total polarization.

4 Conclusions

A theoretical investigation of the polarization properties of TFBGs with different tilt angles is presented.The wavelength dependent expression of PDL is derived by the combination of the coupled-mode theoryand Mueller matrix of TFBGs. To test the expression, the PDL is calculated using the parameters in [4]and the results basically agree with the measured ones in [4]. Then the PDL for TFBGs with differenttilt angles are simulated and compared. Furthermore, the DOP of an unpolarized light after passingthrough TFBGs with different angles are calculated and compared. The results show that TFBGs with45◦ angle have strong polarization sensitivities. In addition, the effects of grating physical parameters onpolarization characteristic are discussed. Analysis shows that the TFBG of 45◦ with proper grating lengthand index modulation depth can be a candidate for polarization devices in a wide range of applications.

Acknowledgements

This work was jointly supported by the High-Tech Research & Development Program of China (Grant No.

2008AA01Z15), and the National Natural Science Foundation of China (Grant No. 60771008).

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