AFIBER Bragg grating is well known to exhibit strong

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1 1892 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 19, NO. 12, DECEMBER 2001 Dispersion of Optical Fibers With Far Off-Resonance Gratings J. E. Sipe, Fellow, OSA C. Martijn de Sterke, Member, OSA Abstract We derive analytic expressions for the quadratic cubic dispersion of optical fiber Bragg gratings, at frequencies far from the Bragg condition, where the usual coupled-mode theory (CMT) fails. We use these to design fibers that have no quadratic dispersion at a particular wavelength, but find that the cubic dispersion then increases. We also show that it is possible to design a fiber link with vanishing average quadratic cubic dispersion by combining a length of stard fiber a length of dispersion compensating fiber with an off-resonance grating. Index Terms Gratings, optical fiber dispersion, optical fiber theory. I. INTRODUCTION AFIBER Bragg grating is well known to exhibit strong reflection of an incident light pulse, provided the pulse frequency matches the grating s Bragg frequency [1], [2]. Yet, even if the reflection is small due to a mismatch between these frequencies, the grating still affects the light propagation. For example, light can travel at a group velocity that is much smaller than that in uniform fiber, even though the reflectivity is negligible [3]. However, this effect is appreciable only over a small bwidth, which, for typical fiber gratings, is less than a nanometer. Another effect at frequencies away from the Bragg frequency, a potentially more relevant one, is that gratings lead to strong dispersion of the transmitted light with negligible reflection [4], [5]. This behavior is illustrated in Fig. 1. It was shown by Eggleton et al. that the dispersion of a grating can be used to compensate for the dispersion of a long fiber link [5]. This experimental work was followed by a theoretical analysis to optimize the parameters of this scheme [6]. Here, we extend the previous theoretical work to frequencies that are far from the Bragg frequency of the grating. Previous work on grating dispersion made use of coupled-mode theory (CMT) [6]. The assumptions in this theory are that the grating is weak that the frequency of the light is close to the Bragg condition [7]. The former assumption is always justified in a fiber grating, but the latter is justified only for a narrow frequency range. However, for the application to dispersion compensation considered here, we are mostly interested in frequencies far from the Bragg condition, where Manuscript received September 6, 2000; revised August 20, This work was supported in part by the Australian Research Council by the National Science Engineering Research Council of Canada. J. E. Sipe is with the Department of Physics, University of Toronto, Toronto, ON M5S 1A7, Canada. C. M. de Sterke is with the School of Physics, University of Sydney, Sydney, NSW 2006, Australia. Publisher Item Identifier S (01) Fig. 1. (a) If the incident pulse s spectrum is close to grating s Bragg resonance, there is substantial reflection, both the reflected transmitted pulses are broadened due to grating dispersion. (b) Far from the Bragg resonance, the reflection is negligible, yet the transmitted pulse experiences strong dispersion. CMT, thus, fails. In fact, we show in Section II that results for the quadratic dispersion following form CMT can be off by a factor of more than two. Applications of this work not only include dispersion compensation, but also fiber lasers, in which the magnitude sign of the quadratic dispersion is crucial in determining the properties of the emitted pulses [8], [9]. Because the usual CMT fails for the problem of interest, we first give, in Section II, an approximate analytic description of off-resonance light propagation through a grating. This leads to an expression for the wavenumber of the light as a function of frequency, which is somewhat reminiscent of the formula for the frequency response of a Lorentz atom. However, the usual denominators resonate at the first- higher order Bragg frequencies of the grating, for which the wavenumber satisfies where is the period of the grating is a positive integer. This corresponds to the well known relation, where is the Bragg wavelength of the grating. Once the wavenumber of the light in the grating is known, it is, in principle, easy to extract the quadratic higher order dispersion by differentiation with respect to frequency (Section III). When these are evaluated, we find that a large range of quadratic dispersion can be obtained, but that the cubic dispersion is essentially always positive, thus limiting the situations in which dispersion compensation can be achieved effectively (Section IV). II. THEORY We consider a one-dimensional (1-D) model for the propagation of light in a fiber mode, deferring considerations that arise because of the true three-dimensional (3-D) nature of the fiber (1) /01$ IEEE

2 SIPE AND DE STERKE: OPTICAL FIBERS WITH FAR OFF-RESONANCE GRATINGS 1893 to Section V. At a given frequency, the electric field can be taken to satisfy (2) with period [cf. (1)], that the amplitude of the oscillation varies, at most, at the level. That is, we write (8) All quantities, such as the effective index of refraction of the mode, can also depend on frequency. We now extract a constant reference refractive index from by writing this equation as where. Now, the Green function of (3), with outgoing waves as, which is the solution to, is given by [10] A formal solution of (3) can, thus, be written as (3) (4) where the designation indicates all positive negative integers, zero. If the were constants, then would be a purely periodic function the grating would, thus, be uniform. By making the depend on the slow parameter, is almost periodic, with slowly varying, but otherwise arbitrary, phase amplitude modulation; this can be used, for example, to describe chirped or apodized gratings. The prefactor 2 in (8) is introduced for later convenience; all of the are assumed to be of order unity or less. We take to be real, so that, we can, thus, write for nonnegative, where are real,. Using these expressions in (8) gives (9) (5) where (10) (6) where the are arbitrary constants. Differentiating (6) once, with respect to, gives the differential equations We now assume that there are no resonances; that is, is of the order of for all. This means that, as discussed in Section I in contrast with stard CMT [7], the light spectrum is well away from all the grating s Bragg resonances [see (1)]. Using expression (8) for in (7), we find the pair of equations where (5) for was used. Thus, (7), in combination with (5), formally solves (2). Although (7) can easily be solved numerically, this does not give any insight into the nature of the solutions. In the following subsections, therefore, we solve (7) analytically, using an approximate scheme that makes use of the fact that is close to unity, that the refractive index distribution is approximately periodic, that the field spectrum is well away from any Bragg resonance. The solution we are after, then, is predominantly forward-propagating because the grating reflection is small (see Fig. 1). Our main result is that this dominant field contribution approximately propagates as a plane wave, but with a wavenumber that differs from that in bare fiber, due to the presence of the grating [(29) (30)]. A. Multiple-Scales Analysis We now assume that is a small quantity of order. We do a multiple-scales analysis [11] in this parameter, introducing in the usual way. Further, we assume that the quantity has a periodicity at the level (7) (11) We solve these equations asymptotically using the assumed forms (12) According to these, the field is predominantly forward-propagating, this field component has a wavenumber that is close to [(5)]. The other, smaller contributions to the field are all allowed to vary rapidly. Putting these forms in (11), we can begin to collect equations to successively higher orders in. B. First-Order Equations Their Solutions To order, (11) gives (13)

3 1894 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 19, NO. 12, DECEMBER 2001 We solve the first of these equations by taking (14) where, in the second line, we used (14) for. Collecting all of these expressions, we can now write (18) as (15) Equation (15) can be immediately integrated to give (16) solving for in terms of, which still must be determined from (14) the results to follow. Similarly, the second of (13) can be integrated to so determine (22) To solve this, we let the term take care of all the nonsecular contributions, while takes care of the secular terms. We do not write down the former; the latter gives C. Second-Order Equations Their Solutions To order, (11) gives (17) (23) because. The first term, here, vanishes when summed over, because each contribution with positive cancels the corresponding term with negative. From the second term, we get (24) (18) (19) Using (16) (17) for, respectively, we find that, after some rearranging, the right-h side of (19) can be written as D. Results The terms for give a phase that oscillates in space; it is the term that acquires a steadily increasing phase as the light propagates through the medium, which is of interest here. Because, from (12) we have (25) (26) Furthermore, using (16) for, we can write (20) where (14) (24) were used. Now, we simplify our notation by setting (27) Then, (8) for becomes (21) (28)

4 SIPE AND DE STERKE: OPTICAL FIBERS WITH FAR OFF-RESONANCE GRATINGS 1895 Our expression (26), for the phase accumulation in now be written as, can (29) where the local wave number is given by (30) (5) was used. Equations (29) (30) are the main results of this section. They show that the dominant field contribution approximately satisfies the simple differential (29). However, the wavenumber of the resulting plane wave solution is not, associated with the reference refractive index, but is changed due to the off-resonant grating. This shift is given by (30). The second third terms in (30) are associated with the direct current value of the grating s refractive index are not surprising; they also do not depend on frequency. The new result in (30) is the contribution due to the grating s various nonzero Fourier coefficients, a contribution that depends on frequency through. However, if the grating affects the wavenumber in a way that depends on frequency, then it must also change its various derivatives with respect to frequency, i.e., the dispersion. We extract, in Section III, from (30), the dispersion introduced by a grating compare it to exact calculations to results from CMT. III. GRATING DISPERSION Here, we calculate, from (30), the quadratic dispersion the cubic dispersion (dispersion slope) introduced by the grating, which corresponds to taking various derivatives with respect to. For the examples considered below, only the term is of importance, therefore, we drop reference to all higher order resonances. Also, ignoring the DC contribution to the grating s refractive index profile, we then find that the wavenumber shift for a grating with unit strength is given by (31) defining parameter, which is convenient to use in the analysis below. It is straightforward to show that (32) corresponding to the change in the quadratic dispersion due to the presence of an off-resonance grating with unit strength. Here, the dimensionless constant is defined as (33) Fig. 2. Quadratic dispersion versus frequency of a Bragg grating described in the text. Shown are a numerical result (solid line) that is indistinguishable from (32). The dashed line follows from CMT. Finally, the change in the cubic dispersion can be found by differentiating with respect to one more time where was defined in (33) Now, let us estimate the magnitudes of the parameters. For a stard fiber at m, we have km ps/km (34) (35) ps km ps km (36) where we took the effective mode index to be 1.46 for both the phase group velocities, ps/nm/km ps/nm /km [12]. With these numbers, we find that (37), thus,. Although the results, here, were derived for stard fibers, we find that even for special-purpose fibers, such a dispersion-compensating fiber, are still quite small. In Section IV, we show that the effects of these terms are, in fact, negligible,, therefore, we ignore them. We now first evaluate the expressions derived previously compare them to numerical results. We model a fiber grating as a thin-film stack, consisting of 4000 layers of equal thickness with alternating refractive indices that are taken to be dispersionless. According to stard CMT, this grating has a strength. To reduce out-of-b reflection, we actually simulate a tapered grating, in that the refractive index of the high-index layers increases according to a sine-shaped profile in the 40 periods at both ends. In Fig. 2, we compare three different calculations of the quadratic dispersion of this grating. The figure shows, versus ; this combination of parameters allows one to use a single universal curve. The figure shows that the dispersion decreases far away from the Bragg resonance, which is at in the units used here. The dashed line

5 1896 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 19, NO. 12, DECEMBER 2001 is the stard result from CMT. The other two results, an exact calculation based on the transfer matrix method that from (32), are indistinguishable on the scale of figure, confirming that our expressions are correct. The coupled-mode results are quite good close to the Bragg resonance, but fail far from the Bragg frequency at low frequencies, as expected. IV. ANALYSIS AND APPLICATION Here, we analyze the results from Section III, ultimately ignoring the small parameters [(33) (35)]. The general idea is that, because gratings change the wavenumber for light propagating in the fiber,, thus, the various orders of dispersion, they can, in principle, be used to change the dispersion in fiber links. After some general considerations in Section IV-A, we consider two different cases. The first case is the compensation of quadratic dispersion in a stard in dispersion-compensating fiber (Sections IV-B -C). The second refers to a link consisting of these two types of fiber in which the quadratic dispersion is perfectly compensated, but in which the grating is used to compensate the residual dispersion slope (Section II-D). A. General Considerations Let us fix consider different values of.as, we have long period gratings with respect to the wavelength of interest, as, we have short period gratings. As ranges from 0 to, we go from very long period gratings to the Bragg resonance; in this range,, defined in (31), varies from 1to.As ranges from to, we go from Bragg resonance to very short period gratings; in this range, varies from 0 to. Hence, the range of physical values of is. Now, from (34), for the cubic dispersion, we find that if we ignore the small terms, it is strictly positive in this regime, because the quartic polynomial has roots at,. Under this approximation, the cubic dispersion is, thus, always positive for finite frequencies. Let us briefly consider how this conclusion is affected when are included. This is important because the quartic polynomial in (34) changes sign at the edges of the physically relevant interval at. Now, it is clear from (34) that remains a root for any, does not shift. It is also easy to show that remains a root, irrespective of. The constant does affect the position of this root; to the lowest order, it shifts to. Because, we conclude that the cubic dispersion of a fiber with an off-resonance grating is, essentially, always positive. Although the same conclusion was drawn earlier [6], this was based on CMT was, thus, not reliable for large detunings from the Bragg frequencies. B. Dispersion Compensation in Stard Fiber We now first consider a grating that is used to compensate for the dispersion of a stard single-mode fiber, the parameters of which, for m, were given in (36). We start with (30), in which we only include the dominant terms. Setting, we find from (30), (31), (32) the condition (38) where the first term is the dispersion in bare fiber, the second is due to the presence of the grating. Using definition (37) for, this can be written as (39) As an example, we take a grating with, corresponding by (8) to a modulation depth of roughly so that, with the first of (37), applicable at m, the polynomial on the left-h side (LHS) of (39) must have the value 591. This equation has a single root,or. It is easy to see that, thus, we require that m, where is the lowest order Bragg wavelength of the grating. We can now evalute the cubic dispersion at this wavelength using (30) (34). This leads to (40) where the first term in the brackets corresponds to the dispersion of the bare fiber, whereas the other terms are due to the grating. For, this leads to (41) the cubic dispersion, thus, increases by a factor of 4.2 to about 0.53 ps /km. Thus, even though the quadratic dispersion vanishes at the designated wavelength of 1.55 m, it reaches its original value of ps /km at a detuning of about 50 nm from the design wavelength. C. Dispersion-Compensating Fiber Here, we consider a grating in dispersion compensating fiber, for which ps/nm/km ps/nm /km [12], so leading to ps km ps km (42) (43) so that even for this fiber, is substantially below unity. Using the same procedure as in Section IV-B, using the same grating strength, we find that now the LHS of (39) should have the value , so that,, thus, m. Following the same method as in Section IV-B, we find for the cubic dispersion that (44) Thus, even though, in this case, the cubic dispersion introduced by the grating that of the fiber have opposite signs, the former dominates, the magnitude of the total cubic dispersion increases by a factor of three. This conclusion is not affected by or. D. Dispersion Compensation in a Fiber Link Here, we consider a -km fiber link, in which the stard fiber from Section IV-B the dispersion-compensating fiber from Section IV-C are used in combination with a

6 SIPE AND DE STERKE: OPTICAL FIBERS WITH FAR OFF-RESONANCE GRATINGS 1897 grating, which has vanishing quadratic cubic dispersion. Ignoring the quadratic dispersion of the grating for now, we find that we need to take the length of stard fiber km, while the length of dispersion-shifted fiber km, to have vanishing quadratic dispersion, leading to a residual cubic dispersion of ps. This negative cubic dispersion can be compensated for by the grating. It is noted that these are examples to illustrate the way in which gratings can affect the dispersion. We now design a grating of 1 m length with that compensates for this cubic dispersion. From (30), (31), (34), we find or. Taking the former, because it corresponds to a Bragg wavelength that is smaller than that of the signal,, thus, does not lead to cladding-mode losses (see Section V), we obtain that nm. Now, calculating the quadratic dispersion due to the grating using (32), we find a small residual quadratic dispersion ps. This, in turn, can be compensated by taking km km, which leads to negligible changes in the grating parameters. We note that the inclusion of also leads to negligible changes in the grating parameters. V. DISCUSSION AND CONCLUSION We have analyzed the dispersion introduced by a grating at frequencies far from the Bragg frequency, applied to this to the design of optical fibers without (quadratic) dispersion. We find that the quadratic dispersion can effectively be made to vanish at a designated wavelength, here m. However, in both cases, the magnitude of the cubic dispersion increases because the cubic dispersion of the grating dominates that of the fiber itself. We note that, in the stard fiber, the grating s Bragg wavelength of 1.37 m is smaller than the design wavelength; thus, cladding-mode coupling, which occurs on the short-wavelength side of the Bragg resonance, does not occur [13]. In contrast, in the dispersion-compensating fiber, the Bragg wavelength is 1.66 m; thus, cladding mode coupling occurs at 1.55 m unless precautions are taken. We also used our formalism for the design of a 10-km fiber link in which the quadratic cubic dispersion both vanish, on average. The Bragg wavelength must now be closer to the operating wavelength because the grating must compensate the dispersion of a length of fiber that is longer than the grating. It is noted that these are examples to illustrate the way in which gratings can affect the dispersion; they are not proposals for novel types of devices. In Section IV, we demonstrate that the cubic dispersion of a grating is, essentially, always positive. This is unfortunate because it precludes designs in which the cubic dispersion of different gratings cancel each other. Although untreated fiber can have cubic dispersion of either sign, the cubic dispersion of gratings tends to be much larger [4], a cancellation is, thus, not likely, as illustrated by the examples in Section IV. As mentioned, the analysis we have described here is 1-D; thus, any effect of the transverse dimensions is assumed to reside in the function, introduced in (2). Although many effects can be included in this way, in its present form, it does not account for dispersion of the grating s refractive index. These could, for example, be due to variations of the overlap of the mode that we are considering with the grating, or simply due to differential dispersion of the constituent materials of the fiber core cladding. Although the inclusion of these effects is beyond the scope of the present paper, one can argue quite simply that their effect is likely to be small. To see this, note that the inclusion of the dispersion of the grating strength would give an additional term in the numerator of (30). However, the largest contributions to the dispersion in (32) (34) result from the leading terms in, which originate from taking successive derivatives of the denominator of. In conclusion, we have derived simple analytic expressions for the dispersion of light in gratings at wavelengths far from the Bragg resonance where stard CMT fails. We find that, although the quadratic dispersion can have either sign can, thus, compensate the dispersion of any fiber, the cubic dispersion dominates that of the fiber, is always positive. We also present a procedure to design a grating that compensates the quadratic dispersion of a fiber (or that can give any nonzero value), as well as a 10-km fiber link with vanishing average quadratic cubic dispersion. ACKNOWLEDGMENT The authors wish to thank B. Eggleton for helpful discussions. REFERENCES [1] R. 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Martijn de Sterke, photograph biography not available at the time of publication.