cladding-mode resonances in short- and longperiod

7/28/2019 Cladding-Mode Resonances in Short- And Longperiod

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Cladding-mode resonances in short- and long-period fiber grating filters

Turan Erdogan

The Institute of Optics, University of Rochester, Rochester, New York 14627

Received July 29, 1996; accepted December 26, 1996

The transmission of a mode guided by the core of an optical fiber through an ultraviolet-induced fiber gratingwhen substantial coupling to cladding modes occurs is analyzed both experimentally and theoretically. Astraightforward theory is presented that is based on the calculation of the modes of a three-layer step-indexfiber geometry and on multimode coupled-mode theory that accurately models the measured transmission ingratings that support both counterpropagating (short-period) and co-propagating (long-period) interactions.These cladding-mode resonance filters promise unique applications for spectral filtering and sensing. 1997Optical Society of America [S0740-3232(97)02908-6]

1. INTRODUCTION

Optical fiber phase gratings formed by ultravioletirradiation1 have developed rapidly in recent years. Nu-merous applications have been demonstrated that utilizefiber gratings as mirrors, in which a forward-propagatingmode guided by the fiber core couples to a backward-propagating mode of the same type,1 and as mode con-

verters, in which one type of guided core mode couples toa different type.2 Fiber gratings can also function as lossfilters by enabling the guided core mode to couple to ra-diation modes of the fiber,3 which are effectively extin-guished by leakage of light away from the fiber. Whenthe cladding is surrounded by a medium with a refractiveindex lower than that of the glass, such as air, the coremode may couple to fiber cladding modes.4 These modestoo are easily extinguished by scattering loss, by bendingloss, and ultimately by leakage loss when the claddingmode reaches fiber that is coated with a material of indexequal to or higher than that of the glass, at which pointtruly guided cladding modes no longer exist. Radiation-mode coupling and cladding-mode coupling may occur asboth counterpropagating and co-propagating interactions.

Of the fiber grating applications reported to date thatutilize radiation-mode and cladding-mode coupling, mostare based on spectral filtering,5 although a few experi-ments have been described that utilize cladding-modecoupling gratings for sensors.6 The advantages of thesedevices as filters relative to competing technologies, such

as bulk pig-tailed filters, include low insertion loss, lowbackreflection, and potentially low cost. In addition, thespectral characteristics of fiber-grating filters are quiteflexible, since practically one can vary numerous param-eters, including induced-index change, length of the grat-ing, apodization, tilt of the grating fringes, chirp of thegrating period, and whether the grating supports a coun-terpropagating or a copropagating interaction at the de-sired wavelength. Normalized filter bandwidths ( /)of 0.1 to 104 are achievable. These grating filters willlikely be employed in optical communications applica-tions, such as optical-amplifier gain-spectrum flattening,

spectral filtering in wavelength-division-multiplexed sys-

tems, and suppression of amplified spontaneous emission,and in other applications such as fiber laser componentsand sensor systems.

In this paper we consider the details of the interactionbetween a guided core mode and the cladding modes of atypical, step-index fiber. To clarify the regime in whichsuch an interaction occurs, consider the measured trans-mission spectra through an untilted fiber grating shownin Fig. 1. The grating is approximately 5 mm long, has apeak index change of approximately 2 103, and is de-signed to reflect the fundamental (core) mode of the fiberat 1540 nm. Figure 1(a) shows the spectrum when thebare (uncoated) section of fiber that contains the gratingis immersed in index-matching fluid to simulate an infi-

nite cladding. There is a smooth transmission profile,demonstrating loss that is due to radiation-mode couplingfor wavelengths shorter than 1540 nm. This behaviorhas been described in detail in a number of papers. 3,7 InFig. 1(b) the fiber is immersed in glycerin, which has a re-fractive index slightly greater than that of the cladding.The transmission spectrum now exhibits fringes that arecaused by FabryPerot-like effects associated with theimperfect reflection of the radiation modes off thecladdingglycerin interface. We defer an analysis of thiscase to a later publication. In Fig. 1(c) the bare fiber issurrounded by air, such that clear resonance dips associ-ated with coupling to distinct cladding modes appear.From the quality of the resonances it appears that the

cladding modes propagate with little loss, at least overthe length of the grating. Note that the spectrally inte-grated loss is the same for all three cases because mate-rial absorption is insignificant in these devices. The caseshown in Fig. 1(c) is the subject of this paper.

One can understand the corecladding-mode interac-tion in a fiber grating by treating the coupling among thecore mode and multiple cladding modes simultaneously ata particular wavelength, using coupled-mode theory. Insome cases the individual resonances are sufficiently nar-row and spectrally separated that coupling between thecore mode and a single cladding mode well describes the

1760 J. Opt. Soc. Am. A/ Vol. 14, No. 8 / August 1997 Turan Erdogan

0740-3232/97/0801760-14$10.00 1997 Optical Society of America


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transmission. In such cases simple two-mode coupled-mode theory that involves constant-coefficient differentialequations with slowly varying amplitudes can be em-ployed. Once the coupling coefficient for the particular

pair of modes is calculated, the analysis is identical tothat described in numerous references.8,9 For a uniform(nonapodized) grating, an analytical solution for the re-flectivity (counterpropagating) or transmission (co-propagating) is available. However, in many cases ofpractical interest the core-modecladding-mode reso-nances overlap one another and even the core-modecore-mode reflection resonance. In these cases all modes thatare nearly resonant at a particular wavelength must beincluded simultaneously in the theory. The analysis be-low demonstrates these calculations.

The remainder of the paper is constructed as follows.Section 2 describes the method used to calculate the modepropagation constants and mode field profiles of both the

fundamental core mode and the cladding modes in a typi-cal step-index optical fiber. The cladding mode fields areobtained by using an exact, vector-field treatment, sincethe interface between the cladding and its surroundmight have a large index difference. Section 3 describesthe calculation of the coupling coefficients associated withthe interactions of interest. The coupled-mode theoryformalism and its particular implementation for cladding-mode coupling are described in Section 4. Both counter-propagating (short-period gratings) and co-propagating(long-period gratings) interactions are considered. InSection 5 some numerical results are compared with ex-

perimental measurements for short- and long-period grat-ings to verify the analysis. In the final section we discussthe results and consider limitations associated with ap-proximations made in the analysis.

2. MODES AND FIELDS

The interactions considered in this paper occur mainly be-tween the fundamental core mode (LP01 or HE11) and thecladding modes of a step-index fiber. The coupled-modetheory and ensuing calculations (Sections 4 and 5) aregeneral and apply to any fiber, assuming that all the nec-essary propagation constants and coupling coefficientshave been calculated. To keep the analysis as clear aspossible and focused on the mode interactions rather thancalculations of the modes themselves, we assume here thesimple three-layer, step-index fiber geometry shown inFig. 2. With this assumption, we can readily calculatethe fields of this structure and derive explicit expressionsfor the dispersion relations, the field profiles and inten-sity distributions, and the coupling coefficients that areused in Sections 4 and 5.

Since we are interested mainly in low fibers, where (n 1 n 2)/n 1 is the normalized corecladding indexdifference, the linearly polarized (LP) approximation10

should be sufficient to describe a mode guided by the fibercore. We use this description to find the mode propaga-tion constant, since the LP-mode dispersion relation issimpler than the exact HE11 expression, but we write thefield in terms of radial and azimuthal vector componentsbecause we ultimately seek overlap integrals betweenthis field and the exact cladding-mode fields. In particu-lar, the dispersion relation that we solve to obtain theLP01 mode effective index is

V1 bJ1 V1 b

J0 V1

b

VbK1 Vb

K0 Vb

, (1)

where J is a Bessel function of the first kind, K is a modi-fied Bessel function of the second kind, V

(2/)a1n 12 n22 is the V number of the fiber at awavelength , b is the normalized effective index, given

Fig. 1. Measured transmission through a typical short-period fi-ber grating under investigation where (a) the uncoated fiber isimmersed in index-matching liquid to simulate an infinite clad-ding, (b) the fiber is immersed in glycerin, and (c) the bare fiber issurrounded by air and thus supports cladding modes.

Fig. 2. Diagram of a cross section of the fiber geometry consid-ered here, showing the coordinate system, the refractive indexes,and the radii of the core ( a 1) and of the cladding ( a2).

Turan Erdogan Vol. 14, No. 8 / August 1997 / J. Opt. Soc. Am. A 1761


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by b (neff2

n22)/(n1

2 n2

2), and the rest of the param-eters are as defined in Fig. 2.

For the field expressions that describe the core modewe can approximate the exact radial and azimuthal vec-tor components of the HE11 mode fields in the core regionof the fiber ( r a1) as

11

Erco iE 01

coJ0 V1 br/a 1

exp i exp i z t r a 1 , (2)

Eco E01

coJ0 V1 br/a 1

exp i exp i z t r a 1 , (3)

where the normalization constant E01co, based on a total

power of 1 W carried by the mode, is

E01co Z0b

n21 2b

1/21

a1J1 V1 b(4)

and where Z0 0/0 377 is the electromagneticimpedance in vacuum. The z axis is along the axis of thefiber, and is the propagation constant, given by (2/)n eff. Here the notation 01 is used to denote theLP01 core mode. In Eqs. (2) and (3) we have assumedthat the factors that multiply the Bessel functions in theexact expressions can be approximated by 1 n eff/n1 2 and 1 n eff/n1 0. The normalization constant inrelation (4) is derived directly from that for the LP01 modeas described in Ref. 12. Note that both the propagationconstant and the mode fields of the guided core mode areobtained in the infinite-cladding geometry ( a2 ), im-plying the assumption that this mode does not sense thepresence of the claddingsurround interface. Since Eqs.

(2) and (3) are the only field components that we requirefor calculation of the coupling coefficients, and becausethe LP01/HE11 fiber mode is so familiar, we do not list thetransverse magnetic fields or the longitudinal field com-ponents here.

The cladding modes are somewhat more complicatedthan the core modes for the geometry of Fig. 2, since wemay not neglect one of the interfaces. The exact modesfor such a three-layer fiber have been detailed in Ref. 11,for example. Our analysis follows that of Ref. 11, but weinclude an explicit listing of the dispersion relation andmode fields here in a clearer, ready-to-program form.

The dispersion relation for a cladding mode with azi-muthal dependence exp(il), for which by definition n 3 n eff n2 , is given by

0 0, (5)

where

The following definitions have been used in Eqs. (5)(7):

1 il neff/Z0 , (8)

2 iln effZ0 , (9)

u21 1

u 22

1

u12

, (10)

u32 1

w 32

1

u22

, (11)

where

uj2 2/ 2 nj

2 n eff

2 j 1, 2 , (12)

w 32 2/ 2 neff

2 n 3

2 , (13)

J Jl u 1a 1

u 1Jl u 1a 1, (14)

K Kl w 3a 2

w 3Kl w3a 2, (15)

p l r Jl u 2r Nl u2a 1 Jl u 2a 1 Nl u2 r , (16)

q l r Jl u 2r Nl u 2a 1 Jl u 2a 1 Nl u2 r , (17)

r l r Jl u2r Nl u 2a 1 Jl u 2a 1 Nl u2 r , (18)

s l r Jl u2r Nl u 2a 1 Jl u2a 1 Nl u 2r .(19)

In Eqs. (14)(19) the prime notation indicates differentia-tion with respect to the total argument and N is a Besselfunction of the second kind, or the Neumann function.The dispersion relation given by Eqs. (5)(19) is straight-forward to solve numerically. For a given azimuthalnumber l , there are typically several hundred claddingmodes at near-infrared wavelengths in a 125-m-diameter fiber.

As is evident from the complexity of the dispersion re-lation for the cladding modes, the field expressions arealso somewhat complex. Because we are interested

0 1

2

u 2 JK 12u 21u 32n 22a 1a2 p l a 2 Kq l a 2 Jr l a 2 1u 2 s l a 2u 2

u 32

n 22a2

J u 21

n 12a1

K p l a 2 u 32

n 12a2

q l a 2 u 21

n 12a1

r l a 2

, (6)

0 1

u 2u 32

a2J

n 32u 21

n 22a 1

K p l a 2 u 32

a 2q l a 2

u 21

a1r l a 2

u 2

n 32

n 22

JK 12u 21u 32

n 12a1a2

p l a 2 n 32

n 12

Kq l a 2 Jr l a 2 n2

2

n 12u 2

s l a 2

. (7)



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mainly in ultraviolet-induced fiber gratings in which theindex perturbation responsible for mode coupling coeffi-cients generally exists only in the core of the fiber, herewe list the cladding-mode fields only in the core. Forcompleteness, the fields in the cladding and in the sur-round are listed in Appendix A. Furthermore, if we limitthe analysis to untilted gratings or, more generally, togratings that consist of a circularly symmetric index per-turbation in any transverse plane of the fiber, the only

nonzero coupling coefficients between the core mode andthe cladding modes involve cladding modes of azimuthalorder l 1 [Eq. (35)]. Therefore for the remainder of theanalysis we are concerned only with l 1 claddingmodes. The vector components of the electric field for thecladding modes in the fiber core ( r a1) are given by

Ercl

iE 1cl

u 1

2 J2 u 1r J0 u 1r 20

n 12

J2 u 1r

J0 u 1r exp i exp i z t r a 1 , (20)

Ecl

E1cl

u 1

2 J2 u 1r J0 u1 r 20

n 12

J2 u 1r J0 u1 r exp i exp i z t r a 1 , (21)

Ezcl

E1cl

u 1220

n12

J1 u 1r exp i exp i z t

r a 1 , (22)

where is the cladding-mode number and E1cl is the field

normalization constant. Note that because 20 is a realnumber according to Eqs. (5) and (6), the radial compo-nent of the electric field is imaginary, whereas the azi-muthal and longitudinal components are real. The mag-netic field components are

Hrcl

E1cl

u 1

2 i1 J2 u 1r J0 u 1r i0 J2 u 1r

J0 u 1r exp i exp i z t r a 1 ,

(23)

Hcl

iE 1cl

u 1

2 i1 J2 u1 r J0 u 1 r i0 J2 u 1r

J0 u 1r exp i exp i z t r a 1 ,

(24)

Hzcl

iE 1cl

u 12i1

J1 u 1r exp i exp i z t

r a 1 . (25)

Because 1 and 0 are both imaginary numbers, the ra-dial component of the magnetic field is real and the azi-muthal and longitudinal components are imaginary.

As an example, the vector components of the electricfield for the lowest order (l 1, v 1) cladding mode ina typical fiber considered here are plotted in Fig. 3 as afunction of radial position. The fiber is described by thefollowing parameters: 0.0055 (n 1 1.458), n 2 1.45, n 3 1.0, a1 2.625 m, and a2 62.5 m.

Figure 3(a) shows the radial and azimuthal componentsgiven by Eqs. (20) and (21) and by Eqs. (A1), (A2), (A9),and (A10) below; Fig. 3(b) shows the longitudinal compo-nent given by Eq. (22) and Eqs. (A3) and (A11) below. Todemonstrate the continuity of the fields across the dielec-tric boundaries, we actually plot nj

2Ercl in the jth layer

instead ofErcl . Notice that the longitudinal component is

nearly 2 orders of magnitude smaller than the transversecomponents.

A further useful profile with which to visualize thetransverse distribution of light in the fiber is the local in-tensity of light propagating along the z axis. For anl 1 cladding mode this quantity is a function of onlythe radial coordinate, and is given by

Iz r 12 Re E H* z

12 Re Er

clHcl*

Hrcl*E

cl .

(26)

Fig. 3. Plots of the vector components of the electric field for thelowest-order ( 1) cladding mode of a fiber with the structuralparameters listed in the text: (a) n 2(r) (for the unperturbed fi-ber) times the radial component (solid curve) and the azimuthalcomponent (dashed curve), (b) the longitudinal component.



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Figure 4 shows the local intensity profiles of the first fourl 1 cladding modes ( 1, 2, 3, 4). Several impor-

tant features are evident. First, the low-order evenmodes (e.g., 2, 4) contain very little light in the fibercore, whereas the low-order odd modes (e.g., 1, 3)

have a peak localized in the core. As a result, we expectthe coupling between the low-order even modes and the

LP01 core mode of the fiber to be very weak. Second,whereas one might navely assume that the slightlyhigher index of the core relative to the cladding results inonly a minor perturbation to the structure and thus mightbe ignored in the cladding-mode calculation, Fig. 4 dem-onstrates that this is a poor assumption.13 In fact it isessential to treat the structure as a three-layer wave-guide to reasonably approximate the fields and hence thecoupling coefficients, which are the subject of Section 3.

In analogy to the core-mode treatment, we can calcu-late an expression for the normalization constant E1

cl

once we determine the mode effective index by specifyingthat each mode carry a power of 1 W. In particular, weset

P 1/2 Re0

2

d0

rdr ErclH

cl* Hr

cl*Ecl 1 W.

(27)

The integral along the radial direction can be broken intothree pieces, with the field expressions (20)(25), (A1)(A6), and (A9)(A14) used for the integrands in eachpiece, respectively. The solution to the integral can bewritten in a closed form, but the expression is quite long.When the integral is set equal to 1 W, as in Eq. (27), theonly unknown in the resulting equation is the desired

normalization constant E1cl , which is then readily calcu-

lated. The results of such a calculation are provided inAppendix B. The guided core mode as well as the clad-ding modes are thus completely characterized. In the re-mainder of this paper the mode fields are used to deter-mine the strength of the coupling interaction, or thecoupling coefficients, and the propagation constants (oreffective indexes) are used to determine the spectral loca-tions of the coupling resonances.

3. COUPLING COEFFICIENTS

In this section we calculate the coupling coefficients thatare used in the coupled-mode theory analysis in the re-mainder of the paper. Coefficients are calculated for cou-pling between the LP01 core mode and itself and betweenthe LP01 core mode and the l 1 cladding modes.

In the absence of an ultraviolet-induced phase gratingthe fiber structure is as pictured in Fig. 2. In our analy-sis we assume that, when a phase grating is induced in

the fiber, it exists only in the fiber core, changing the coreindex to n 1(z) but leaving the cladding and surround in-dexes unchanged, as follows:

Here n 1 is the unperturbed core index, is the period ofthe grating, m is the induced-index fringe modulation,

where 0 m 1, and (z) is the slowly varying enve-lope of the grating. Thus the peak induced index change(at a grating tooth peak) at any z is (z)n1(1 m), theminimum induced-index change (at a grating tooth val-ley) is (z)n 1(1 m), and the product (z)n 1 describesthe profile of the dc induced-index change, averaged overa grating period. In principle (z)n 1 can be arbitrarilyshaped, but we consider two practical special cases.First, a uniform grating is defined to be one with a con-stant index change (z)n 1 n1 over a fixed length w.

Another common grating profile is the Gaussian grating,in which (z) takes the form

z exp 4 ln 2z2/w 2 , (29)

where w is the full width at half-maximum (FWHM) ofthe grating profile. Figure 5 is an illustration of thechanged core index n1(z) for a Gaussian grating, wherethe size of the grating period relative to the length whas been exaggerated for clarity.

The coupling coefficients that we need for the analysisin Section 4 are described in a number of publications.We follow most closely the notation of Kogelnik,8 the onlydifference being a factor of 4 that arises from our choicefor the power normalization in Eq. (27) [cf. Kogelniks Eq.(2.2.51)]. The transverse coupling coefficient betweentwo modes and is thus [cf. Kogelniks Eq. (2.6.21)]

Fig. 4. Plots of the local intensity Iz(r) as a function of radiusfor the four lowest-order l 1 cladding modes in a typical fiber.All modes are circularly symmetric and have been normalized tocarry a power of 1 W.

n r , z n 1 z n 1 1 z 1 m cos 2 z r a 1

n 2 a1 r a2

n 3 r a2

. (28)



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Kt z

4 0

2

d0

rdr r , z Et r,

Et* r , , (30)

where the superscript t denotes transverse vector compo-nents only (radial and azimuthal) and the quantity(r , z) describes the ultraviolet-induced index perturba-tion, here assumed to be independent of . We do notcalculate the longitudinal coupling coefficients K

z sincewe neglect the contribution from them in the coupled-mode theory analysis, anyway. They can be neglectedsince the longitudinal field components are 12 orders ofmagnitude smaller than the transverse field components,as we saw in Section 2. Thus K

z, which involves theproduct of two longitudinal fields, is generally 24 ordersof magnitude smaller than K

t. For a small index per-turbation ( 1), which is generally a good approxima-

tion for an ultraviolet-induced fiber grating, we can makethe approximation 0(n

2) 20nn . If we fur-ther define the coupling constant through

Kt z z 1 m cos 2 z , (31)

named a constant by convention even though canhave a slowly varying z dependence, then we can writethe coupling constant for core-modecore-mode couplingspecifically as

0101co-co

z 0n 1

2 z

2 0

2

d0

a1

rdr Erco 2

Eco 2 . (32)

Using the expressions for fields (2)(4), and performingthe integrals, we obtain

0101co-co

z z2

n12b

n 21 2b 1 J02 V1 b

J12 V1 b

.(33)

The coupling constant for core-modecladding-modecoupling is also straightforward to calculate. In particu-lar, we need only simplify the expression

01cl-co

z 0n1

2 z

2 0

2

d

0

a1

rdr ErclEr

co* E

clEco* . (34)

Note that if we include all cladding modes (any l) in Eq.(34), then the azimuthal integral becomes

0

2

d exp i l 1 2l1 , (35)

where l1 is the Kronecker delta function, which is equalto 1 when l 1 and equal to 0 when l 1. Thereforethe only nonzero coupling constants are those betweenthe LP01 core mode and the l 1 cladding modes. In-serting field components (2), (3), (20), and (21) into Eq.(34), using Eq. (35), and then performing the integralalong the radial direction, we obtain

101cl-co

z z2

b

Z 0n 21 2b

1/2

n 12u 1

u12

V2 1 b /a12

1 20n1

2 E1cl u 1J1 u 1a 1

J0 V1 b

J1 V1 b

V1 b

a1J0 u 1a 1 .

(36)

Note that the coupling constant is directly proportional tothe normalized induced-index change (z). The remain-

ing factors in this expression are determined entirely bythe dielectric structure of the fiber as shown in Fig. 2 andthe resulting mode characteristics at a wavelength .

To obtain a sense of the variation in the strength ofcoupling between the LP01 core mode and the range of l 1 cladding modes, we calculate the coupling constantsfor the fiber geometry described above for Figs. 3 and 4.Figure 6 shows the coupling constant in Eq. (36) divided

Fig. 5. Diagram of the ultraviolet-induced refractive-indexchange in the core for a grating with a Gaussian profile along thefiber (z) axis. The size of the grating period () relative to thegrating width ( w) has been exaggerated for clarity.

Fig. 6. Coupling constant 101cl-co divided by (z) for the 168

l 1 cladding modes in a typical fiber, showing odd and evenmodes separately.



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by (z) for all the l 1 cladding modes at a wavelengthof 1550 nm. There are 168 modes below cutoff at thiswavelength. Note that, as we anticipated from the localintensity profiles in Fig. 4, the coupling between thelowest-order even cladding modes and the core mode is

very weak compared to that involving the lowest-orderodd cladding modes. However, for cladding modes of or-der 40 and higher, the even and the odd modes havecoupling strengths comparable to one another. The

slowly varying oscillation in the coupling strength versuscladding mode number is analogous to the oscillation withrespect to wavelength observed in coupling from the coremode to the continuum of radiation modes of the fiber.7

Qualitatively, this oscillation arises because as thecladding-mode order increases (or the radiation-modetransverse wave vector increases) the mode fields exhibitmore and more oscillations along the radial direction. Asthe number of oscillations increases within the fixed ra-dial extent of the fiber, the cladding-mode fields begin toexhibit nulls in the fiber core. Each time a new nullmoves into the core, it is possible for the overlap integralin Eq. (34) to become zero, thus causing the zeros in theslowly varying envelope in Fig. 6.

4. COUPLED-MODE THEORY

The coupling interactions that we analyze here includethe coupling of an LP01 core mode to itself (counterpropa-gating Bragg reflection) and the coupling of an LP01 coremode to both counterpropagating and copropagating l 1 cladding modes. The general coupled-mode equa-tions that describe the changes in the forward- andbackward-going amplitudes of a mode that result fromthe presence of other modes near a dielectric perturba-tion can be written as

dAdz i A

Kt

Kz exp i z

i

B Kt

Kz exp i z ,

(37)

dBdz

i

A Kt

Kz exp i z

i

B Kt

Kz exp i z ,

(38)

where A(z) is the amplitude for the transverse modefield traveling to the right (z direction), B(z) is the am-plitude for the transverse mode field traveling to the left(z direction), and K

t and Kz are the transverse and

longitudinal coupling coefficients, respectively, betweenmodes and . Equations (37) and (38) are identical toEqs. (2.6.23) and (2.6.24) of Ref. 8, except we have re-placed j with i because we assume that the fields havean exp(it) harmonic time dependence.

The first approximation that we make to Eqs. (37) and(38) is to neglect the longitudinal coupling coefficients

Kz, because they are substantially smaller than the

transverse coefficients Kt even for those coefficients

that involve cladding-mode fields, as we showed in Sec-tion 3. A second approximation is to ignore the couplingamong cladding modes, including cladding-mode self-scattering that results in a perturbation to the propaga-tion constant (or phase evolution) of the mode. This ap-proximation is reasonable since for the grating structuresanalyzed here we find 11

cl-cl101

cl-co ; the index perturba-tion exists only in the fiber core, which represents amuch smaller fraction of the cladding-mode field extentthan of the core-mode field extent [see Eqs. (32) and (34)].Note that it is also true that 101

cl-co 0101

co-co , but we do notignore cladding-modecore-mode coupling outright, sincethis is the very effect that we seek to understand, and fora given grating period this type of coupling generally oc-curs at a different wavelength from that of core-modecore-mode coupling.

With these approximations, the coupled-mode equa-tions [(37) and (38)] that describe counterpropagating in-teractions in a short-period grating simplify to

dAco

dz i0101

co-co Aco im

20101

co-co Bco exp i20101co-co z

i

m

2101

cl-co Bcl exp i2101

cl-co z , (39)

dBco

dz i0101

co-co Bco im

20101

co-co Aco exp i20101co-co z ,

(40)

dBcldz

im

2101

cl-co Aco exp i2101cl-co z , (41)

where Aco and Bco are the amplitudes for the core mode,A

cl and Bcl are the amplitudes for the th cladding mode,

and we have defined the small-detuning parameters:

0101co-co

12 201co 2

, (42)

101cl-co

1

2 01co 1cl 2

. (43)

In writing Eqs. (39)(41) we employed the usual synchro-nous approximation, in which we assume that only thoseinteractions that we have kept are nearly phase matchedand are thus capable of resonant coupling.8 In math-ematical terms, we neglect all driving terms on the right-hand sides that oscillate too rapidly to contribute signifi-cantly to the change of the mode amplitudes on the left-hand sides and keep only those terms that either do not

oscillate at all or oscillate at a very small rate . The re-maining equations are thus appropriate in the wave-length range for which Eqs. (42) and (43) are nearly zerofor a given choice of . The wavelength at which 0101

co-co

0 is the resonant wavelength, or Bragg reflectionwavelength, for core-modecore-mode coupling and thewavelength at which 101

cl-co 0 is the resonant wave-

length for core-modeth-cladding-mode coupling.These phase-matching considerations for a given short-

period grating are illustrated in Fig. 7(a). On the axesthe filled circles represent core modes, for which n 2 n eff n1 ; the open circles represent cladding modes,



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for which n 3 n eff n2 ; the hatched regions representthe continuum of radiation modes for which neff n3 ; andthe left halves of the axes (negative ) imply modes trav-eling in the z direction. The top axis is a propagationconstant, or , axis at the particular wavelength for whichthe lowest-order core mode is exactly phase matched by agrating of period to a counterpropagating mode of thesame type. The remaining three axes are drawn at thewavelengths for which the lowest-order core mode is ex-actly phase matched to counterpropagating, higher-ordercore, cladding, and radiation modes. Note that these in-teractions are phase matched at successively shorterwavelengths.

The coupled-mode equations that describe copropagat-ing interactions in a long-period grating simplify to

dAco

dz i0101

co-co Aco

i

m

2101

cl-co Acl exp i2101

cl-co z , (44)

dA

cl

dz i

m

2 101cl-co

Aco exp i2101cl-co

z

, (45)

where for the copropagating interactions the small detun-ing parameter is

101cl-co

1

2 01co 1cl 2

. (46)

As for Eqs. (39)(41), we have simplified Eqs. (44) and(45) by using the synchronous approximation. Thephase-matching considerations for copropagating interac-

tions are illustrated in Fig. 7(b). Here the top axis is theaxis at the particular wavelength for which the lowest-order core mode is phase matched by a grating of period to a copropagating radiation mode. The following twoaxes represent phase matching of the lowest-order coremode to a copropagating cladding mode and to a higher-order copropagating core mode. Note that these interac-tions are phased matched at successively shorter wave-lengths.

To summarize Fig. 7, if we were able to measure thetransmission of the lowest-order core mode through a fi-ber grating of a given period over an extremely broadrange of wavelengths, the measured loss in transmissionwould be caused by the following processes: At the short-

est wavelength, loss is caused by copropagating couplingto higher-order core modes; at longer wavelengths, theloss results from coupling to cladding modes; at stilllonger wavelengths, radiation-mode coupling occurs, firstcopropagating, then normal to the fiber axis, then coun-terpropagating; at still longer wavelengths loss is causedby counterpropagating cladding-mode coupling; at evenlonger wavelengths counterpropagating coupling withhigher-order core modes occurs; and, finally, Bragg reflec-tion of the lowest-order core mode into a counterpropagat-ing mode of the same type is phase matched at the longestwavelength.

In principle the mathematical machinery to calculatethe transmission of an LP01 core mode through both a

short-period and a long-period fiber grating in a noninfi-nitely clad fiber is now in place. The boundary condi-tions for a short-period grating of length L (counterpropa-gating interaction) are Aco(z L/2) 1, Bco(z L/2) 0, and B

cl(z L/2) 0 for all . For a long-periodgrating (copropagating interaction) the boundary condi-tions are Aco(z L/2) 1, and A

cl(z L/2) 0.The transmission through the grating is simply T Aco(L/2)/Aco(L/2). Note that for a uniform gratingwe take L w, whereas for a nonuniform grating wechoose L to be a length that fully encloses the gratingsuch that the grating perturbation is negligible outsidethe range L/2 z L/2. For example, for a Gaussiangrating we choose L to be several times the grating width

w. We then solve Eqs. (39)

(41) or (44) and (45) subjectto these boundary conditions.

In practice the difficulty of solving the coupled-modeequations depends on the strength and the spectral den-sity of the resonances. Note that the sets of Eqs. (39)(41) and (44) and (45) each describe a large number (typi-cally several hundred) of coupled first-order differentialequations. Since the solutions are computed at eachwavelength, we can greatly simplify the task by recogniz-ing that only one or several mode interactions are nearlyresonant at the particular wavelength. Thus only thosemode pairs with the smallest associated detuning param-

Fig. 7. Diagrams that illustrate the phase-matching conditionsnecessary for resonant coupling between two modes by a gratingof period . (a) For a short-period grating, counterpropagatingcoupling can occur between (top to bottom, longest to shortestwavelength) oppositely traveling similar core modes, two differ-ent core modes, a core mode and a cladding mode, and a coremode and radiation modes. (b) For a long-period grating, co-propagating coupling can occur between (top to bottom, longestto shortest wavelength) a core mode and radiation modes, a coremode and a cladding mode, and two different core modes.



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eters in Eqs. (42), (43), and (46) need to be kept. In ourcalculations we determine which resonances to include ateach wavelength by estimating the spectral location andwidth of the nearby resonances; we keep only those thatare centered a specified number (or fraction) of spectralbandwidths from the wavelength. The estimates arebased on the closed-form solution of the coupled-modeequations9 for the single resonance in question (such thatonly two modes are included) and for a uniform grating

that exactly or approximately represents that actual grat-ing. For a Gaussian grating, a uniform grating of lengthequal to w and equal to the peak of the Gaussian is agood approximation. The approximate spectral locationof the resonance also takes into account the effect of theultraviolet-induced average index in the core by means ofthe core-mode self-scattering termsthe first terms onthe right-hand sides of Eqs. (39) and (44).

The estimates are based on the following results. Theapproximate center wavelength of the LP01LP01 Braggreflection resonance in a short-period grating can befound by taking

0101

co-co

0101co-co

0, (47)where the term 0101

co-co accounts for the wavelength shiftthat is due to the increased average index. For the reso-nance associated with the th cladding mode and theLP01 mode in either a short-period or a long-period grat-ing, the approximate spectral location is given by

101cl-co

0101co-co /2 0, (48)

where 101cl-co is defined by Eq. (43) or (46), depending on

whether the grating is short period or long period, respec-tively, and where the factor of 1/2 arises because we areassuming that only the core mode is substantially af-fected by the increased average index in the core. Theapproximate (normalized) spectral bandwidth of a coun-terpropagating resonance obtained from the closed-formsolution of the two-mode coupled-mode equations9 is

n avg1

L

2 1/2. (49)For LP01LP01 Bragg reflection, is determined by Eq.(47), is 0101

co-co , and navg is n eff for the LP01 mode; forcladding-mode LP01 scattering, is determined by Eq.(48), is 101

cl-co , and n avg is (n effco

n effcl )/2, or the average

of the effective indexes for the core and cladding modes.The approximate (normalized) spectral bandwidth of a co-

propagating resonance9

is

nL 1 4L

1/2

, (50)

where for cladding-mode LP01 scattering in a long-periodgrating is determined by Eq. (48); is 101

cl-co , and n n eff

co neff

cl . Using Eqs. (47) and (48) and relations(49) and (50), we can estimate straightforwardly howmany resonances should be included at a particular wave-length. For example, for a conservative calculation wemight specify that all resonances with a center wave-

length located less than two spectral bandwidths fromthe wavelength of interest be included in the calculation.

Calculation of the transmission spectrum thus pro-ceeds as follows: At each wavelength we locate the clos-est resonance; if the grating is sufficiently weak that thespectral bandwidths of neighboring resonances aremuch smaller than the separations of those resonancesfrom the wavelength of interest, then we keep only theclosest resonance and thus have only two modes. For a

uniform grating, a closed-form solution is available.9 Fora nonuniform grating, we integrate the pair of coupledfirst-order differential equations, using a fourth-order,adaptive-step-size RungeKutta algorithm subject to theboundary conditions discussed above. If the grating isnot sufficiently weak, we numerically integrate the set ofcoupled differential equations including as many modesas required by our estimate. Note that in this case a nu-merical solution is necessary even for a uniform grating,as we no longer obtain constant-coefficient differentialequations as in the two-mode case.

5. RESULTSIn this section we compare experimentally measuredtransmission spectra with calculations based on thetheory described above to examine the accuracy of thetheory and to demonstrate typical grating transmissionspectra when cladding-mode coupling is present.

Figure 8(a) shows the measured transmission througha grating written in Corning Flexcore fiber that wasloaded with deuterium.14 The fiber parameters are ap-proximately those noted above in the discussion of Figs. 3

Fig. 8. (a) Experimentally measured and (b) theoretically calcu-lated transmission spectra through a relatively weak, Gaussianshort-period grating, demonstrating both core-modecore-modeand core-modecladding-mode coupling.



10/14

and 4 (here the best theoretical fits are obtained with 0.0050 and a1 2.5 m). The grating is a Gaussiangrating with a width w 4 mm and a peak induced-index change ofn 1 7.2 10

4. It was written by in-terfering ultraviolet beams from an excimer-laser-pumped, frequency-doubled dye laser producing 15-nspulses at a 30-Hz repetition rate. The exposure utilized10 mW of average power for 60 s, with the beams fo-

cused on the fiber to an approximate spot size of4 mm 50 m. The strong resonance at 1540 nm is dueto LP01LP01 Bragg reflection; the peak reflectivity is98%. The resonances at shorter wavelengths arecaused by coupling to the 1, 3, 5, ... cladding modes.Evidence of coupling to the even cladding modes is not

visible on this trace. Note that the transmission spec-trum for this grating approximates a comb of flat-toppedtransmission windows, suggesting unique potential appli-cations. Figure 8(b) shows a theoretical calculation ofthe transmission through such a grating. For this rela-tively weak grating the cladding-mode resonances are farapart relative to the width of the resonances, and thus,with the exception of the few lowest-order cladding

modes, the transmission could be calculated by using thetwo-mode, closed-form solutions. For this case there isexcellent agreement in terms of both the locations and thestrengths (relative and absolute) of the resonances.

Figure 9(a) shows the measured transmission througha similar grating but with a larger peak induced-indexchange of n 1 2.8 10

3. To write this grating weincreased the average power and exposure time to 20 mWand 100 s, respectively. This grating is nearly 100% re-flecting, with a reflection bandwidth of 2 nm. Figure9(b) shows the corresponding theoretical calculation ofthe transmission. Note that the resonances overlap sub-

stantially over most of the wavelength range shown in theplot. As a result, an accurate calculation could be ob-tained only by including multiple cladding-mode reso-nances simultaneously at each wavelength. Here allresonances that were estimated to be within two spectralbandwidths of the calculation wavelength were included.

Again there is excellent agreement between theory andexperiment. In particular, note that the theory accu-rately reproduces the fine structure on the short-

wavelength side of each resonance. This structure ismainly a result of the FabryPerot-like property of a non-uniform, tapered grating in which interference occurswhen light at the short-wavelength edge of the resonancesees only the wings of the grating and not the center.3

The fine structure is also due to the emergence of couplingto the even modes at the shortest wavelengths. Forthis grating the limited measurement resolution (0.1 nm)causes a slightly larger apparent discrepancy with theorythan in Fig. 8.

Transmission spectra (in decibels) calculated for typicallong-period gratings are shown in Fig. 10. Figure 10(a)shows the transmission for a relatively weak grating witha length of 25 mm, a peak induced-index change of 1 104, and a uniform or a Gaussian profile. The fivemain dips seen in these spectra correspond to coupling tothe 1, 3, 5, 7, 9 cladding modes. Figure 10(b) showsthe transmission for a stronger, uniform grating with apeak induced-index change of 3.6 104. In each casethe grating period is adjusted to yield coupling at 1550nm between the LP01 core mode and the 7 claddingmode. That is, the gratings in Figs. 10(a) and 10(b) haveperiods of 600 m and 570 m, respectively.Notice that the dip in Fig. 10(b) associated with coupling

Fig. 9. (a) Experimentally measured and (b) theoretically calcu-lated transmission spectra through a strong, Gaussian short-period grating.

Fig. 10. Theoretically calculated transmission spectra through(a) a relatively weak and (b) a relatively strong long-period grat-ing, each designed to couple the LP01 core mode to the 5cladding mode at 1550 nm.



11/14

to the 9 cladding mode at 1800 nm is slightly weakerthan the dip associated with coupling to the 7 clad-ding mode at 1550 nm, despite the stronger coupling tothe 9 mode shown in Fig. 5. The reason for this be-havior is that, over the length of the grating, incidentcore-mode light at 1800 nm is fully converted to the clad-ding mode and then begins to convert back to the coremode, resulting in a smaller loss in transmission.

The measured transmission (in decibels) through arelatively weak long-period grating ( 396 m) isshown in Fig. 11(a). This grating was written in an

AT&T dispersion-shifted communications fiber loadedwith hydrogen by direct exposure to a 248-nm KrF exci-mer laser beam through a chrome Ronchi-ruled mask.The profile of the grating is nearly uniform, with a lengthof w 50 mm. The peak induced-index change is n 1 1.9 104, as determined by our calculation. Thesingle resonance that is visible within the wavelengthrange plotted is associated with coupling to the 1cladding mode; this is the shortest-wavelength resonance.Figure 11(b) shows the calculated transmission spectrumthrough this grating, where agreement of the transmis-sion minimum was obtained by adjusting the normalizedindex change . Again there is excellent overall agree-ment.

6. CONCLUSION

A straightforward theory has been presented that accu-rately models the measured transmission through a num-ber of practical fiber gratings that exhibit substantial

cladding-mode coupling. There are two parts to thetheory. The first is the specific method used to calculatecore and cladding modes, the associated propagation con-stants and fields, and the coupling coefficients. For thispart a simple three-layer, step-index fiber geometry waschosen so that we could write down the needed results.Even for more-complicated fiber geometries, one can oftenreasonably approximate the modes with those of a three-layer structure. The second part of the theory is the mul-timode coupled-mode theory. This part is more generaland requires only the propagation constants and couplingcoefficients associated with the fiber modes, which can beobtained with a more sophisticated method if desired.

A number of approximations were made in the theory;the accuracy with which the measured spectra are mod-eled for the most part justifies these. Some approxima-tions, such as the number of cladding-mode resonances toinclude in the calculation at each wavelength, are easily

verified by simply including more and more resonancesuntil the result converges to a stable solution. Others re-quire direct testing of the theory with and without the ne-glected component for certainty of the validity in a varietyof cases. For the examples considered here, the neglectof both longitudinal coupling coefficients and cladding-mode to cladding-mode scattering is reasonable: Thesewould not alter the calculated transmission visibly.However, these approximations should be reconsideredfor very strong gratings and for gratings that couple thecore mode to fairly high-order cladding modes.

An issue that was not addressed specifically is howcarefully dispersion should be taken into account in thecalculations. In particular, ideally all of the mode prop-erties should be recomputed at each wavelength, but ifone is calculating transmission at several hundreds oreven thousands of wavelengths, this approach might beintractable. Instead, if one is concerned with only one

long-period-grating resonance or several short-period-grating resonances, it is a reasonable approximation tocompute the mode properties once at a central wave-length and then simply integrate the coupled-mode equa-tions at each wavelength (even when the mode reso-nances are strongly overlapping). We have also used anintermediate approach in which we compute the modeproperties at a number of wavelengths that are lessdensely spaced than the transmission calculation grid,and then we simply interpolate the quantities that arespecific to the modes at each new wavelength from thesparse grid.

Finally it should be pointed out that our experimentaland theoretical discussion is limited to gratings with a

circularly symmetric index perturbation, thus excludinggratings with tilted fringes and other nonuniformitiesacross the core. Although our reason for doing so ispartly simplicity, it is also true that it is more difficult toutilize individual cladding-mode resonances for applica-tions in circularly asymmetric gratings, as so many moreresonances show up. One quickly finds such a large over-lap of resonances that the spectrum mimics a smoothstrictly radiation-mode coupling spectrum but with aspiky, almost random modulation impressed on it. Formost applications this result is less desirable than thesimpler radiation-mode coupling spectrum that can be

Fig. 11. (a) Experimentally measured and (b) theoretically cal-culated transmission spectra through a relatively weak, uniformlong-period grating. The resonance is associated with couplingto the lowest-order (l 1, 1) cladding mode.



12/14

flexibly altered by using grating tilt, for example.7 How-ever, despite the apparent experimental and theoreticaldifficulties in investigating asymmetric cladding-modecouplers, there might very well be enticing reasons forpursuing them.

APPENDIX A

In this appendix expressions are listed for the cladding-mode fields in the cladding and in the surround for thethree-layer fiber geometry shown in Fig. 2. The vectorcomponents of the electric field for the l 1 claddingmodes in the fiber cladding region ( a1 r a2) are

Ercl

iE 1cl

a 1u 12J1 u 1a 1

2 F2

rp 1 r

1

u 2rq 1 r

2

n 22 u 2G 2r 1 r n 220n 12 s 1 r

exp i i z t , (A1)

Ecl

E1cl

a1u 12J1 u 1a 1

2 2

n 22 G 2r p 1 r

n 220

n 12u 2r

q 1 r u 2F2r 1 r s 1 r exp i i z t , (A2)

Ezcl

E1cl

a1u12u2

22J1 u 1a 1

2n22

G 2p 1 r

n220

n 12u 2

q 1 r exp i i z t (A3)and the vector components of the magnetic field for thel 1 cladding modes in the fiber cladding region(a1 r a2) are

Hrcl

E1cl

a 1u 12J1 u 1a 1

2 i G 2

rp 1 r

in 2

20

n 12u2rq

1r

i1

u2F

2r

1r

s1

r

exp i i z t , (A4)

Hcl

iE 1cl

a 1u 12J1 u 1a 1

2 i1 F2

rp 1 r

1

u 2rq 1 r

iu2G 2r 1 r in2

20

n 12

s 1 r exp i i z t , (A5)

Hzcl

iE1cl

a1u 12u 2

2i1J1 u 1a 1

2 F2p 1 r

1

u 2q 1 r exp i i z t , (A6)

where the quantities F2 and G 2 are defined to be

F2 J

u 2120

n12a1

, (A7)

G 2 0J u211

a 1. (A8)

Note that F2 is purely real and G 2 is purely imaginary.As a result, just as for the fields in the core [in Eqs. (20)(25)], the radial component of the electric field and theazimuthal and longitudinal components of the magneticfield are imaginary and the other three components arereal.

The vector components of the electric field for thel 1 cladding modes in the surround region ( r a2) are

Ercl

iE1cl

a1u 12u 2

2J1 u 1a 1

4w3K1 w 3a 2 F3 K2 w 3r

K0 w 3r 2G3

n 32

K2 w 3r K0 w 3r exp i i z t , (A9)

Ecl

E1cl

a1u 12u2

2J1 u 1a 1

4w 3K1 w3a 2 F3 K2 w3 r

K0 w 3r

2G 3

n32 K2 w 3r K0 w 3r

exp i i z t , (A10)

Ezcl

E1cl

a1u 12u2

22J1 u 1a 1

2n 32K1 w 3a 2

G 3K1 w 3r

exp i i z t . (A11)

Finally, the vector components of the magnetic field forthe l 1 cladding modes in the surround region(r a2) are

Hrcl E1cl a1u 12

u 22

J1 u 1a 14w 3K1 w 3a 2

i1 K2 w 3r

K0 w3 r iG 3 K2 w3 r K0 w3 r

exp i i z t , (A12)

Hcl

iE 1cl

a1u 12u2

2J1 u 1a 1

4w3K1 w3a 2 i1 K2 w 3r

K0 w 3r G 3 K2 w 3r K0 w 3r

exp i i z t , (A13)



13/14

Hzcl

iE1cl

a1u 12u 2

2i1J1 u 1a 1

2K1 w 3a 2F3K1 w 3r

exp i i z t . (A14)

In Eqs. (A9)(A14) the quantities F3 and G 3 are definedto be

F3 F2p 1 a 2 1

u 2q 1 a 2 , (A15)

G 3 n3

2

n22 G2p 1 a 2 n 220n 12u 2 q 1 a 2 .

(A16)

As anticipated, the radial component of the electric fieldand the azimuthal and longitudinal components of themagnetic field are imaginary, and the other three compo-nents are real.

APPENDIX B

In this appendix it is shown how the normalization con-stant E1

cl for the cladding modes can be calculated for aparticular normalization choice (i.e., P 1 W). As ex-plained in the text, we evaluate the integral in Eq. (27) byusing the mode field expressions given in Eqs. (20)(25),(A1)(A6), and (A9)(A14), and then we set the resultequal to the power normalization choice. Solving the re-sulting equation for E1

cl provides the desired result.The total power carried by the cladding mode is the

sum of the powers carried in the core, the cladding, andthe surround region, or

P P 1 P 2 P3 , (B1)

where Pj , the power carried in the jth region of the fiber,can be calculated by means of the integral in Eq. (27) butwith the limits along the radial direction replaced bythose appropriate for the region of interest. Considerfirst the core, or region 1 ( r a1). We do not provide thedetails of the calculation, but the result is

P 1 E1cl 2

a12u1

2

4 n eff

Z0

n effZ 002

n12

1 n eff2n 12 Im 0 J22 u 1a 1 J1 u 1a 1 J3 u 1a 1 neff

Z0

n effZ 002

n12

1 n eff2

n 12

Im 0 J02 u 1a 1 J12 u1a 1 . (B2)Next consider the cladding region (a 1 r a2) . Thepower is given by

P 2 E 1cl 2

3a 12u 1

4u 22J1

2 u 1a 1

16

n effZ0

F22

n effZ0

n22

G 22 Q Q 1

u 22

n effZ0

neffZ0n220

2

n14 R R

1 n eff2

n22

F2 Im G 2 Q Q

1 n eff2

n22

n22

n 12u2

2Im 0 R R

1 n eff2

n22 n 22 Im 0n 12u2 F2

1

u 2Im G 2

S S 2n eff

u 2 Z00

n 12

G 2 1

Z 0F2 S S

.(B3)

The new quantities defined in Eq. (B3) are

Q JN12 u2a 1 NJ1

2 u2a 1

2JNJ1 u 2a 1 n 1 u 2a 1 , (B4)

Q JN12 u 2a 1

NJ1

2 u2a 1

2JNJ1 u 2a 1 N1 u 2a 1 , (B5)

R 14 J N2 u2a 1 N0 u 2a 1

2

14 N J2 u 2a 1

J0 u 2a 1 2

12 JN N2 u2a 1 N0 u 2a 1

J2 u 2a 1 J0 u 2a 1 , (B6)

R 14

J N2 u 2a 1 N0 u 2a 1

2

14

N J2 u 2a 1

J0 u 2a 1 2

12

JN N2 u 2a 1 N0 u2a 1

J2 u2a 1 J0 u 2a 1 , (B7)

S 12 JN1 u 2a 1 N0 u 2a 1 N2 u 2a 1

12 NJ1 u 2a 1 J0 u 2a 1 J2 u2a 1

12 JN N1 u2a 1 J0 u 2a 1 J2 u 2a 1

J1 u 2a 1 N0 u 2a 1 N2 u 2a 1 , (B8)

S

12 JN1 u 2a 1 N0 u 2a 1 N2 u 2a 1

12

NJ1 u 2a 1 J0 u 2a 1 J2 u2a 1

12

JN N1 u2a 1 J0 u 2a 1 J2 u 2a 1

J1 u 2a 1 N0 u 2a 1 N2 u 2a 1 , (B9)

where these expressions utilize the definitions

J a 22 J2

2 u 2a 2 J1 u 2a 2 J3 u 2a 2

a 12 J2

2 u 2a 1 J1 u 2a 1 J3 u 2a 1 , (B10)



14/14

N a 22 N2

2 u2a 2 N1 u 2a 2 N3 u2a 2

a12 N2

2 u2a 1 N1 u 2a 1 N3 u 2a 1 , (B11)

JN a 22 J2 u 2a 2 N2 u 2a 2

12 J1 u2a 2 N3 u 2a 2

J3 u 2a 2 N1 u 2a 2 a12 J2 u 2a 1 N2 u 2a 1

12 J1 u 2a 1 N3 u 2a 1 J3 u 2a 1 N1 u 2a 1 ,

(B12)

J a 22 J2

2 u 2a 2 J12 u 2a 2 a1

2 J22 u 2a 1

J12 u 2a 1 , (B13)

N a22 N2

2 u2a 2 N12 u 2a 2 a1

2 N22 u2a 1

N12 u2a 1 , (B14)

JN a22 J0 u 2a 2 N0 u 2a 2 J1 u2a 2 N1 u 2a 2

a12 J0 u 2a 1 N0 u 2a 1 J1 u2a 1 N1 u 2a 1 .

(B15)Finally, the power carried in the surround region(r a2) is given by

P3 E1cl 2

3a12a2

2u 14u2

4J12 u 1a 1

16w32K1

2 w 3a 2

n eff Z0n 3

2G 3

2

n eff

Z 0F3

2

1 n eff2

n 32

F3 Im G 3

K

2

2

w

3a

2 K

1w

3a

2K

3w

3a

2

n effZ 0n 3

2G3

2

n eff

Z 0F3

2

1 n eff2

n 32

F3 Im G 3 K0

2 w 3a 2 K12 w 3a 2 . (B16)

Using Equations (B2), (B3), and (B16), we can obtain avalue for the normalization constant E1

cl by solvingfor this parameter as the only unknown in the equation

P1 P2 P 3 1 W.

ACKNOWLEDGMENTS

The author is grateful to John Sipe for many insightfuldiscussions concerning the calculation of interactions infiber gratings, to Suresh Pereira for critical analysis ofthe theory, and to Justin Judkins and Lucent Technolo-gies Bell Laboratories for providing the experimentaldata for the long-period grating and for partially support-ing this research. This work is funded in part by the Na-

tional Science Foundation (award ECS-9502670) and bythe Alfred P. Sloan Foundation.

The author can be reached at tel: 716-275-7227; fax:716-244-4936; e-mail: [email protected].

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cladding-mode resonances in short- and longperiod

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