Cladding-mode resonances in short- and long ... - OSA Publishing

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

Turan Erdogan

Cladding-mode resonances in short- and longperiod 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 grating when substantial coupling to cladding modes occurs is analyzed both experimentally and theoretically. A straightforward theory is presented that is based on the calculation of the modes of a three-layer step-index fiber geometry and on multimode coupled-mode theory that accurately models the measured transmission in gratings 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. © 1997 Optical Society of America [S0740-3232(97)02908-6]

1. INTRODUCTION Optical fiber phase gratings formed by ultraviolet irradiation1 have developed rapidly in recent years. Numerous applications have been demonstrated that utilize fiber gratings as mirrors, in which a forward-propagating mode guided by the fiber core couples to a backwardpropagating mode of the same type,1 and as mode converters, in which one type of guided core mode couples to a different type.2 Fiber gratings can also function as loss filters by enabling the guided core mode to couple to radiation modes of the fiber,3 which are effectively extinguished by leakage of light away from the fiber. When the cladding is surrounded by a medium with a refractive index lower than that of the glass, such as air, the core mode may couple to fiber cladding modes.4 These modes too are easily extinguished by scattering loss, by bending loss, and ultimately by leakage loss when the cladding mode reaches fiber that is coated with a material of index equal to or higher than that of the glass, at which point truly guided cladding modes no longer exist. Radiationmode coupling and cladding-mode coupling may occur as both counterpropagating and co-propagating interactions. Of the fiber grating applications reported to date that utilize radiation-mode and cladding-mode coupling, most are based on spectral filtering,5 although a few experiments have been described that utilize cladding-mode coupling gratings for sensors.6 The advantages of these devices as filters relative to competing technologies, such as bulk pig-tailed filters, include low insertion loss, low backreflection, and potentially low cost. In addition, the spectral characteristics of fiber-grating filters are quite flexible, since practically one can vary numerous parameters, including induced-index change, length of the grating, apodization, tilt of the grating fringes, chirp of the grating period, and whether the grating supports a counterpropagating or a copropagating interaction at the desired wavelength. Normalized filter bandwidths (Dl/l) of 0.1 to 1024 are achievable. These grating filters will likely be employed in optical communications applications, such as optical-amplifier gain-spectrum flattening, 0740-3232/97/0801760-14$10.00

spectral filtering in wavelength-division-multiplexed systems, and suppression of amplified spontaneous emission, and in other applications such as fiber laser components and sensor systems. In this paper we consider the details of the interaction between a guided core mode and the cladding modes of a typical, step-index fiber. To clarify the regime in which such an interaction occurs, consider the measured transmission spectra through an untilted fiber grating shown in Fig. 1. The grating is approximately 5 mm long, has a peak index change of approximately 2 3 1023 , and is designed to reflect the fundamental (core) mode of the fiber at ;1540 nm. Figure 1(a) shows the spectrum when the bare (uncoated) section of fiber that contains the grating is immersed in index-matching fluid to simulate an infinite cladding. There is a smooth transmission profile, demonstrating loss that is due to radiation-mode coupling for wavelengths shorter than 1540 nm. This behavior has been described in detail in a number of papers.3,7 In Fig. 1(b) the fiber is immersed in glycerin, which has a refractive index slightly greater than that of the cladding. The transmission spectrum now exhibits fringes that are caused by Fabry–Perot-like effects associated with the imperfect reflection of the radiation modes off the cladding–glycerin interface. We defer an analysis of this case to a later publication. In Fig. 1(c) the bare fiber is surrounded by air, such that clear resonance dips associated 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 over the length of the grating. Note that the spectrally integrated loss is the same for all three cases because material absorption is insignificant in these devices. The case shown in Fig. 1(c) is the subject of this paper. One can understand the core–cladding-mode interaction in a fiber grating by treating the coupling among the core mode and multiple cladding modes simultaneously at a particular wavelength, using coupled-mode theory. In some cases the individual resonances are sufficiently narrow and spectrally separated that coupling between the core mode and a single cladding mode well describes the © 1997 Optical Society of America

Turan Erdogan

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

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perimental measurements for short- and long-period gratings to verify the analysis. In the final section we discuss the results and consider limitations associated with approximations made in the analysis.

2. MODES AND FIELDS

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

transmission. In such cases simple two-mode coupledmode theory that involves constant-coefficient differential equations with slowly varying amplitudes can be employed. Once the coupling coefficient for the particular pair of modes is calculated, the analysis is identical to that described in numerous references.8,9 For a uniform (nonapodized) grating, an analytical solution for the reflectivity (counterpropagating) or transmission (copropagating) is available. However, in many cases of practical interest the core-mode–cladding-mode resonances overlap one another and even the core-mode–coremode reflection resonance. In these cases all modes that are nearly resonant at a particular wavelength must be included simultaneously in the theory. The analysis below demonstrates these calculations. The remainder of the paper is constructed as follows. Section 2 describes the method used to calculate the mode propagation constants and mode field profiles of both the fundamental core mode and the cladding modes in a typical step-index optical fiber. The cladding mode fields are obtained by using an exact, vector-field treatment, since the interface between the cladding and its surround might have a large index difference. Section 3 describes the calculation of the coupling coefficients associated with the interactions of interest. The coupled-mode theory formalism and its particular implementation for claddingmode coupling are described in Section 4. Both counterpropagating (short-period gratings) and co-propagating (long-period gratings) interactions are considered. In Section 5 some numerical results are compared with ex-

The interactions considered in this paper occur mainly between the fundamental core mode (LP01 or HE11) and the cladding modes of a step-index fiber. The coupled-mode theory and ensuing calculations (Sections 4 and 5) are general and apply to any fiber, assuming that all the necessary propagation constants and coupling coefficients have been calculated. To keep the analysis as clear as possible and focused on the mode interactions rather than calculations of the modes themselves, we assume here the simple three-layer, step-index fiber geometry shown in Fig. 2. With this assumption, we can readily calculate the fields of this structure and derive explicit expressions for the dispersion relations, the field profiles and intensity distributions, and the coupling coefficients that are used in Sections 4 and 5. Since we are interested mainly in low D fibers, where D 5 (n 1 2 n 2 )/n 1 is the normalized core–cladding index difference, the linearly polarized (LP) approximation10 should be sufficient to describe a mode guided by the fiber core. We use this description to find the mode propagation constant, since the LP-mode dispersion relation is simpler than the exact HE11 expression, but we write the field in terms of radial and azimuthal vector components because we ultimately seek overlap integrals between this field and the exact cladding-mode fields. In particular, the dispersion relation that we solve to obtain the LP01 mode effective index is V A1 2 b

J 1 ~ V A1 2 b ! J 0 ~ V A1 2 b !

5 V Ab

K 1 ~ V Ab ! K 0 ~ V Ab !

,

(1)

where J is a Bessel function of the first kind, K is a modified Bessel function of the second kind, V 5 (2 p /l)a 1 An 1 2 2 n 2 2 is the V number of the fiber at a wavelength l, b is the normalized effective index, given

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

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Turan Erdogan

The dispersion relation for a cladding mode with azimuthal dependence exp(6ilf), for which by definition n 3 , n eff , n2 , is given by

2 by b 5 (n eff 2 n22)/(n12 2 n22), and the rest of the parameters are as defined in Fig. 2. For the field expressions that describe the core mode we can approximate the exact radial and azimuthal vector components of the HE11 mode fields in the core region of the fiber (r < a 1 ) as11

z0 5

S

u 2 JK 1

1

s2

2u 2

S

u 32 n 22a 2

u2

S

n 32 n 22

2

n 2 a 1a 2

S

n 12a 1

u 32

JK 1

a2

D

u 21

J2

J2

D

K p l~ a 2 ! 1

n 3 2 u 21 2

n2 a1

n 12a 1a 2

D

D

u 32 n 12a 2

K p l~ a 2 ! 1

p l~ a 2 ! 2

n 32 n 12

q l~ a 2 ! 1 u 32 a2

~ r < a 1 ! , (2)

co E f co > 2E 01 J 0 ~ V A1 2 br/a 1 !

p n 2 A1 1 2bD

D

r l~ a 2 !

n 12a 1

u 21

r l~ a 2 !

a1

n 22 n 12u 2

.

(7)

s l~ a 2 !

(8)

s 2 [ iln eff Z 0 ,

(9)

u 32 [

S

(6)

s 1 [ iln eff /Z 0 ,

where the normalization constant E 01co, based on a total power of 1 W carried by the mode, is

>

,

Kq l ~ a 2 ! 1 Jr l ~ a 2 ! 2

u 21 [

1/2

s l~ a 2 !

u 21

q l~ a 2 ! 1

~ r < a 1 ! , (3)

3 exp~ i f ! exp@ i ~ b z 2 v t !#

Z 0b

u2

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

3 exp~ i f ! exp@ i ~ b z 2 v t !#

E 01co

1

p l ~ a 2 ! 2 Kq l ~ a 2 ! 1 Jr l ~ a 2 ! 2

s 1 s 2 u 21u 32

co E r co > iE 01 J 0 ~ V A1 2 br/a 1 !

(5)

where

s 1 s 2 u 21u 32

u2

z 08 5 s 1

z 0 5 z 08,

1 a 1 J 1 ~ V A1 2 b !

1 u2

2

1 w3

2

2 1

1 u 12

,

(10)

1 u 22

,

(11)

where (4)

and where Z 0 5 Am 0 / e 0 5 377 V is the electromagnetic impedance in vacuum. The z axis is along the axis of the fiber, and b is the propagation constant, given by b 5 (2 p /l)n eff . Here the notation 01 is used to denote the LP01 core mode. In Eqs. (2) and (3) we have assumed that the factors that multiply the Bessel functions in the exact expressions can be approximated by 1 1 n eff /n1 > 2 and 1 2 n eff /n1 > 0. The normalization constant in relation (4) is derived directly from that for the LP01 mode as described in Ref. 12. Note that both the propagation constant and the mode fields of the guided core mode are obtained in the infinite-cladding geometry (a 2 → `), implying the assumption that this mode does not sense the presence of the cladding–surround interface. Since Eqs. (2) and (3) are the only field components that we require for calculation of the coupling coefficients, and because the LP01 /HE11 fiber mode is so familiar, we do not list the transverse magnetic fields or the longitudinal field components here. The cladding modes are somewhat more complicated than the core modes for the geometry of Fig. 2, since we may not neglect one of the interfaces. The exact modes for such a three-layer fiber have been detailed in Ref. 11, for example. Our analysis follows that of Ref. 11, but we include an explicit listing of the dispersion relation and mode fields here in a clearer, ready-to-program form.

u j 2 [ ~ 2 p /l ! 2 ~ n j 2 2 n eff2 !

@ j P ~ 1, 2!# ,

(12)

w 3 2 [ ~ 2 p /l ! 2 ~ n eff2 2 n 3 2 ! ,

(13)

J[

J l8~ u 1a 1 ! , u 1J l~ u 1a 1 !

(14)

K[

K l8~ w 3a 2 ! , w 3K l~ w 3a 2 !

(15)

p l~ r ! [ J l~ u 2r ! N l~ u 2a 1 ! 2 J l~ u 2a 1 ! N l~ u 2r ! ,

(16)

q l~ r ! [ J l~ u 2r ! N l8~ u 2a 1 ! 2 J l8~ u 2a 1 ! N l~ u 2r ! ,

(17)

r l~ r ! [ J l8~ u 2r ! N l~ u 2a 1 ! 2 J l~ u 2a 1 ! N l8~ u 2r ! ,

(18)

s l~ r ! [ J l8~ u 2r ! N l8~ u 2a 1 ! 2 J l8~ u 2a 1 ! N l8~ u 2r ! .

(19)

In Eqs. (14)–(19) the prime notation indicates differentiation with respect to the total argument and N is a Bessel function of the second kind, or the Neumann function. The dispersion relation given by Eqs. (5)–(19) is straightforward to solve numerically. For a given azimuthal number l, there are typically several hundred cladding modes at near-infrared wavelengths in a 125-mmdiameter fiber. As is evident from the complexity of the dispersion relation for the cladding modes, the field expressions are also somewhat complex. Because we are interested

Turan Erdogan


mainly in ultraviolet-induced fiber gratings in which the index perturbation responsible for mode coupling coefficients generally exists only in the core of the fiber, here we list the cladding-mode fields only in the core. For completeness, the fields in the cladding and in the surround are listed in Appendix A. Furthermore, if we limit the analysis to untilted gratings or, more generally, to gratings that consist of a circularly symmetric index perturbation in any transverse plane of the fiber, the only nonzero coupling coefficients between the core mode and the cladding modes involve cladding modes of azimuthal order l 5 1 [Eq. (35)]. Therefore for the remainder of the analysis we are concerned only with l 5 1 cladding modes. The vector components of the electric field for the cladding modes in the fiber core (r < a 1 ) are given by

H

u1

cl E cl r 5 iE 1 n

2

J 2~ u 1r ! 1 J 0~ u 1r ! 2

J

s 2z 0 n 12

@ J 2~ u 1r !

2 J 0 ~ u 1 r !# exp~ i f ! exp@ i ~ b z 2 v t !# ~ r < a1!,

E fcl 5 E 1cln

u1 2

H

J 2~ u 1r ! 2 J 0~ u 1r ! 2

J

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Because s 1 and z 0 are both imaginary numbers, the radial component of the magnetic field is real and the azimuthal and longitudinal components are imaginary. As an example, the vector components of the electric field for the lowest order (l 5 1, v 5 1) cladding mode in a typical fiber considered here are plotted in Fig. 3 as a function of radial position. The fiber is described by the following parameters: D 5 0.0055 (n 1 5 1.458), n 2 5 1.45, n 3 5 1.0, a 1 5 2.625 m m, and a 2 5 62.5 m m. Figure 3(a) shows the radial and azimuthal components given by Eqs. (20) and (21) and by Eqs. (A1), (A2), (A9), and (A10) below; Fig. 3(b) shows the longitudinal component given by Eq. (22) and Eqs. (A3) and (A11) below. To demonstrate the continuity of the fields across the dielectric boundaries, we actually plot n j 2 E r cl in the jth layer instead of E cl r . Notice that the longitudinal component is nearly 2 orders of magnitude smaller than the transverse components. A further useful profile with which to visualize the transverse distribution of light in the fiber is the local intensity of light propagating along the z axis. For an l 5 1 cladding mode this quantity is a function of only the radial coordinate, and is given by

(20) I z~ r ! 5

s 2z 0

1 2

Re~ E 3 H* ! • zˆ 5

1 2

cl* cl* Re~ E cl E f cl! . r Hf 2 Hr

(26)

n 12

3 @ J 2 ~ u 1 r ! 1 J 0 ~ u 1 r !# exp~ i f ! exp@ i ~ b z 2 v t !# ~ r < a1!,

E z cl 5 E 1 n cl

u 12s 2z 0 n 12b

(21)

J 1 ~ u 1 r ! exp~ i f ! exp@ i ~ b z 2 v t !# ~ r < a1!,

(22)

where n is the cladding-mode number and E 1 n cl is the field normalization constant. Note that because s 2 z 0 is a real number according to Eqs. (5) and (6), the radial component of the electric field is imaginary, whereas the azimuthal and longitudinal components are real. The magnetic field components are cl H cl r 5 E 1n

u1 $ i s 1 @ J 2 ~ u 1 r ! 2 J 0 ~ u 1 r !# 2 i z 0 @ J 2 ~ u 1 r ! 2

1 J 0 ~ u 1 r !# % exp~ i f ! exp@ i ~ b z 2 v t !#

~ r < a1!,

(23) H fcl 5 2iE 1cln

u1 $ i s 1 @ J 2 ~ u 1 r ! 1 J 0 ~ u 1 r !# 1 i z 0 @ J 2 ~ u 1 r ! 2

2 J 0 ~ u 1 r !# % exp~ i f ! exp@ i ~ b z 2 v t !#

~ r < a1!,

(24) H z cl 5 2iE 1 n cl

u 12i s 1 J 1 ~ u 1 r ! exp~ i f ! exp@ i ~ b z 2 v t !# b ~ r < a1!.

(25)

Fig. 3. Plots of the vector components of the electric field for the lowest-order ( n 5 1) cladding mode of a fiber with the structural parameters listed in the text: (a) n 2 (r) (for the unperturbed fiber) times the radial component (solid curve) and the azimuthal component (dashed curve), (b) the longitudinal component.

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Turan Erdogan

normalization constant E 1cln , which is then readily calculated. The results of such a calculation are provided in Appendix B. The guided core mode as well as the cladding modes are thus completely characterized. In the remainder of this paper the mode fields are used to determine the strength of the coupling interaction, or the coupling coefficients, and the propagation constants (or effective indexes) are used to determine the spectral locations of the coupling resonances.

3. COUPLING COEFFICIENTS Fig. 4. Plots of the local intensity I z (r) as a function of radius for the four lowest-order l 5 1 cladding modes in a typical fiber. All modes are circularly symmetric and have been normalized to carry a power of 1 W.

Figure 4 shows the local intensity profiles of the first four l 5 1 cladding modes ( n 5 1, 2, 3, 4). Several important features are evident. First, the low-order even modes (e.g., n 5 2, 4) contain very little light in the fiber core, whereas the low-order odd modes (e.g., n 5 1, 3)

n ~ r, z ! 5

5

H

P 5 1/2 Re

E E 0

df

`

0

rdr ~ E r H f cl

cl*

2 Hr

cl*

F

n 1 ~ z ! 5 n 1 1 1 s ~ z ! 1 1 m cos

have a peak localized in the core. As a result, we expect the coupling between the low-order even modes and the LP01 core mode of the fiber to be very weak. Second, whereas one might naı¨vely assume that the slightly higher index of the core relative to the cladding results in only a minor perturbation to the structure and thus might be ignored in the cladding-mode calculation, Fig. 4 demonstrates that this is a poor assumption.13 In fact it is essential to treat the structure as a three-layer waveguide to reasonably approximate the fields and hence the coupling coefficients, which are the subject of Section 3. In analogy to the core-mode treatment, we can calculate an expression for the normalization constant E 1cln once we determine the mode effective index by specifying that each mode carry a power of 1 W. In particular, we set 2p

In this section we calculate the coupling coefficients that are used in the coupled-mode theory analysis in the remainder of the paper. Coefficients are calculated for coupling between the LP01 core mode and itself and between the LP01 core mode and the l 5 1 cladding modes. In the absence of an ultraviolet-induced phase grating the fiber structure is as pictured in Fig. 2. In our analysis we assume that, when a phase grating is induced in the fiber, it exists only in the fiber core, changing the core index to n 1 (z) but leaving the cladding and surround indexes unchanged, as follows:

E f ! 5 1 W. cl

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

S D GJ 2p z L

r < a1

n2

a1 , r < a2

n3

r . a2

.

(28)

Here n 1 is the unperturbed core index, L is the period of the grating, m is the induced-index fringe modulation, where 0 < m < 1, and s (z) is the slowly varying envelope of the grating. Thus the peak induced index change (at a grating tooth peak) at any z is s (z)n 1 (11 m), the minimum induced-index change (at a grating tooth valley) is s (z)n 1 (1 2 m), and the product s (z)n 1 describes the profile of the dc induced-index change, averaged over a grating period. In principle s (z)n 1 can be arbitrarily shaped, but we consider two practical special cases. First, a uniform grating is defined to be one with a constant index change s (z)n 1 [ s n 1 over a fixed length w. Another common grating profile is the Gaussian grating, in which s (z) takes the form

s ~ z ! [ s exp~ 24 ln 2z 2 /w 2 ! ,

(29)

where w is the full width at half-maximum (FWHM) of the grating profile. Figure 5 is an illustration of the changed core index n 1 (z) for a Gaussian grating, where the size of the grating period L relative to the length w has been exaggerated for clarity. The coupling coefficients that we need for the analysis in Section 4 are described in a number of publications. We follow most closely the notation of Kogelnik,8 the only difference being a factor of 4 that arises from our choice for the power normalization in Eq. (27) [cf. Kogelnik’s Eq. (2.2.51)]. The transverse coupling coefficient between two modes n and m is thus [cf. Kogelnik’s Eq. (2.6.21)]

Turan Erdogan


v e 0n 12s~ z ! 2

k ncl-co 201~ z ! 5

3

E

a1

0

E

2p

1765

df

0

rdr ~ E r clE r co* 1 E f clE f co* ! . (34)

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

E

2p

df exp@ i ~ l 2 1 ! f # 5 2 p d l1 ,

(35)

0

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

K nmt~ z ! 5

v 4

E E 2p

df

0

`

0

rdrD e ~ r, z ! En t ~ r, f !

• Em ~ r, f ! , t*

k 1cl-co n 201~ z !

F

K n m t ~ z ! 5 k n m ~ z ! 1 1 m cos

S DG 2p z L

,

(31)

named a constant by convention even though k n m can have a slowly varying z dependence, then we can write the coupling constant for core-mode–core-mode coupling specifically as

v e 0n 12s~ z ! 2

5 s~ z !

(30)

where the superscript t denotes transverse vector components only (radial and azimuthal) and the quantity D e (r, z) describes the ultraviolet-induced index perturbation, here assumed to be independent of f. We do not calculate the longitudinal coupling coefficients K n m z since we neglect the contribution from them in the coupledmode theory analysis, anyway. They can be neglected since the longitudinal field components are 1–2 orders of magnitude smaller than the transverse field components, as we saw in Section 2. Thus K n m z , which involves the product of two longitudinal fields, is generally 2–4 orders of magnitude smaller than K n m t . For a small index perturbation ( s ! 1), which is generally a good approximation for an ultraviolet-induced fiber grating, we can make the approximation D e 5 e 0 D(n 2 ) > 2 e 0 nDn. If we further define the coupling constant k n m through

co-co k 01201 ~z! 5

where d l1 is the Kronecker delta function, which is equal to 1 when l 5 1 and equal to 0 when l Þ 1. Therefore the only nonzero coupling constants are those between the LP01 core mode and the l 5 1 cladding modes. Inserting field components (2), (3), (20), and (21) into Eq. (34), using Eq. (35), and then performing the integral along the radial direction, we obtain

E E 2p

df

0

a1

3

2p l

Z 0 n 2 A1 1 2bD

D

1/2

n 12u 1 u 1 2 2 V 2 ~ 1 2 b ! /a 1 2

S

3 11

3

S

pb

s 2z 0 n 12

D F

E 1 n cl u 1 J 1 ~ u 1 a 1 !

J 0 ~ V A1 2 b ! J 1 ~ V A1 2 b !

2

V A1 2 b a1

G

J 0~ u 1a 1 ! . (36)

Note that the coupling constant is directly proportional to the normalized induced-index change s (z). The remaining factors in this expression are determined entirely by the dielectric structure of the fiber as shown in Fig. 2 and the resulting mode characteristics at a wavelength l. To obtain a sense of the variation in the strength of coupling between the LP01 core mode and the range of l 5 1 cladding modes, we calculate the coupling constants for the fiber geometry described above for Figs. 3 and 4. Figure 6 shows the coupling constant in Eq. (36) divided

rdr ~ u E r cou 2

0

1 u E f cou 2 ! .

(32)

Using the expressions for fields (2)–(4), and performing the integrals, we obtain co-co k 01201 ~ z ! 5 s~ z !

F

G

2p n 12b J 0 2 ~ V A1 2 b ! 11 . l n 2 A1 1 2bD J 1 2 ~ V A1 2 b !

(33) The coupling constant for core-mode–cladding-mode coupling is also straightforward to calculate. In particular, we need only simplify the expression

Fig. 6. Coupling constant k 1cl-co n 201 divided by s (z) for the 168 l 5 1 cladding modes in a typical fiber, showing odd and even modes separately.

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Turan Erdogan

by s (z) for all the l 5 1 cladding modes at a wavelength of 1550 nm. There are 168 modes below cutoff at this wavelength. Note that, as we anticipated from the local intensity profiles in Fig. 4, the coupling between the lowest-order even cladding modes and the core mode is very weak compared to that involving the lowest-order odd cladding modes. However, for cladding modes of order ;40 and higher, the even and the odd modes have coupling strengths comparable to one another. The slowly varying oscillation in the coupling strength versus cladding mode number is analogous to the oscillation with respect to wavelength observed in coupling from the core mode to the continuum of radiation modes of the fiber.7 Qualitatively, this oscillation arises because as the cladding-mode order increases (or the radiation-mode transverse wave vector increases) the mode fields exhibit more and more oscillations along the radial direction. As the number of oscillations increases within the fixed radial extent of the fiber, the cladding-mode fields begin to exhibit nulls in the fiber core. Each time a new null moves into the core, it is possible for the overlap integral in Eq. (34) to become zero, thus causing the zeros in the slowly varying envelope in Fig. 6.

transverse coefficients K n m t even for those coefficients that involve cladding-mode fields, as we showed in Section 3. A second approximation is to ignore the coupling among cladding modes, including cladding-mode selfscattering that results in a perturbation to the propagation constant (or phase evolution) of the mode. This approximation is reasonable since for the grating structures cl-co analyzed here we find k1cl-cl n21m ! k1n201; the index perturbation De exists only in the fiber core, which represents a much smaller fraction of the cladding-mode field extent than of the core-mode field extent [see Eqs. (32) and (34)]. co-co Note that it is also true that k1cl-co n201 ! k01201, but we do not ignore cladding-mode–core-mode coupling outright, since this is the very effect that we seek to understand, and for a given grating period this type of coupling generally occurs at a different wavelength from that of core-mode– core-mode coupling. With these approximations, the coupled-mode equations [(37) and (38)] that describe counterpropagating interactions in a short-period grating simplify to dA co m co-co co co-co co-co 5 i k 01201 A co 1 i k B exp~ 2i2 d 01201 z! dz 2 01201 1i

4. COUPLED-MODE THEORY The coupling interactions that we analyze here include the coupling of an LP01 core mode to itself (counterpropagating Bragg reflection) and the coupling of an LP01 core mode to both counterpropagating and copropagating l 5 1 cladding modes. The general coupled-mode equations that describe the changes in the forward- and backward-going amplitudes of a mode m that result from the presence of other modes n near a dielectric perturbation can be written as dA m 5i dz

(

A n ~ K n m t 1 K n m z ! exp@ i ~ b n 2 b m ! z #

n

1i

( n

B n ~ K n m t 2 K n m z ! exp@ 2i ~ b n 1 b m ! z # , (37)

dB m 5 2i dz

(

2i

n

( n

A n ~ K n m 2 K n m ! exp@ i ~ b n 2 b m ! z # t

z

B n ~ K n m t 1 K n m z ! exp@ 2i ~ b n 1 b m ! z # , (38)

where A m (z) is the amplitude for the transverse mode field traveling to the right (1z direction), B m (z) is the amplitude for the transverse mode field traveling to the left (2z direction), and K n m t and K n m z are the transverse and longitudinal coupling coefficients, respectively, between modes n and m. Equations (37) and (38) are identical to Eqs. (2.6.23) and (2.6.24) of Ref. 8, except we have replaced j with 2i because we assume that the fields have an exp(2ivt) harmonic time dependence. The first approximation that we make to Eqs. (37) and (38) is to neglect the longitudinal coupling coefficients K n m z , because they are substantially smaller than the

( n

m cl-co k B cl exp~ 2i2 d 1cl-co n 201z ! , 2 1 n 201 n

(39)

dB co m co-co co co-co co-co 5 2i k 01201 B co 2 i k A exp~ 1i2 d 01201 z !, dz 2 01201 (40)

( n

F

dB cl n dz

5 2i

G

m cl-co k A co exp~ 1i2 d 1cl-co n 201z ! , (41) 2 1 n 201

where A co and B co are the amplitudes for the core mode, cl A cl n and B n are the amplitudes for the n th cladding mode, and we have defined the small-detuning parameters:

S S

D

co-co d 01201 [

1 2p co 2 b 01 2 , 2 L

d 1cl-co n 201 [

1 2p co b 01 1 b 1cln 2 . 2 L

(42)

D

(43)

In writing Eqs. (39)–(41) we employed the usual synchronous approximation, in which we assume that only those interactions that we have kept are nearly phase matched and are thus capable of resonant coupling.8 In mathematical terms, we neglect all driving terms on the righthand sides that oscillate too rapidly to contribute significantly to the change of the mode amplitudes on the lefthand sides and keep only those terms that either do not oscillate at all or oscillate at a very small rate d. The remaining equations are thus appropriate in the wavelength range for which Eqs. (42) and (43) are nearly zero co-co for a given choice of L. The wavelength at which d 01201 5 0 is the resonant wavelength, or Bragg reflection wavelength, for core-mode–core-mode coupling and the wavelength at which d 1cl-co n 201 5 0 is the resonant wavelength for core-mode– n th-cladding-mode coupling. These phase-matching considerations for a given shortperiod grating are illustrated in Fig. 7(a). On the axes the filled circles represent core modes, for which n 2 , n eff , n1 ; the open circles represent cladding modes,

Turan Erdogan


for which n 3 , n eff , n2 ; the hatched regions represent the continuum of radiation modes for which n eff , n3 ; and the left halves of the axes (negative b) imply modes traveling in the 2z direction. The top axis is a propagation constant, or b, axis at the particular wavelength for which the lowest-order core mode is exactly phase matched by a grating of period L to a counterpropagating mode of the same type. The remaining three axes are drawn at the wavelengths for which the lowest-order core mode is exactly phase matched to counterpropagating, higher-order core, cladding, and radiation modes. Note that these interactions are phase matched at successively shorter wavelengths. The coupled-mode equations that describe copropagating interactions in a long-period grating simplify to dA co co-co 5 i k 01201 A co dz 1i

(F n

( n

m cl-co k A cl exp~ 2i2 d 1cl-co n 201z ! , 2 1 n 201 n

(44)

G

dA n cl m cl-co 5 1i k A co exp~ 1i2 d 1cl-co n 201z ! , dz 2 1 n 201

(45)

Fig. 7. Diagrams that illustrate the phase-matching conditions necessary for resonant coupling between two modes by a grating of period L. (a) For a short-period grating, counterpropagating coupling can occur between (top to bottom, longest to shortest wavelength) oppositely traveling similar core modes, two different core modes, a core mode and a cladding mode, and a core mode and radiation modes. (b) For a long-period grating, copropagating coupling can occur between (top to bottom, longest to shortest wavelength) a core mode and radiation modes, a core mode and a cladding mode, and two different core modes.

1767

where for the copropagating interactions the small detuning parameter is

d 1cl-co n 201 [

S

D

1 2p co b 01 2 b 1cln 2 . 2 L

(46)

As for Eqs. (39)–(41), we have simplified Eqs. (44) and (45) by using the synchronous approximation. The phase-matching considerations for copropagating interactions are illustrated in Fig. 7(b). Here the top axis is the b axis at the particular wavelength for which the lowestorder core mode is phase matched by a grating of period L to a copropagating radiation mode. The following two axes represent phase matching of the lowest-order core mode to a copropagating cladding mode and to a higherorder copropagating core mode. Note that these interactions are phased matched at successively shorter wavelengths. To summarize Fig. 7, if we were able to measure the transmission of the lowest-order core mode through a fiber grating of a given period over an extremely broad range of wavelengths, the measured loss in transmission would be caused by the following processes: At the shortest wavelength, loss is caused by copropagating coupling to higher-order core modes; at longer wavelengths, the loss results from coupling to cladding modes; at still longer wavelengths, radiation-mode coupling occurs, first copropagating, then normal to the fiber axis, then counterpropagating; at still longer wavelengths loss is caused by counterpropagating cladding-mode coupling; at even longer wavelengths counterpropagating coupling with higher-order core modes occurs; and, finally, Bragg reflection of the lowest-order core mode into a counterpropagating mode of the same type is phase matched at the longest wavelength. In principle the mathematical machinery to calculate the transmission of an LP01 core mode through both a short-period and a long-period fiber grating in a noninfinitely clad fiber is now in place. The boundary conditions for a short-period grating of length L (counterpropagating interaction) are A co(z 5 2L/2) 5 1, B co(z 5 L/2) 5 0, and B n cl(z 5 L/2) 5 0 for all n. For a long-period grating (copropagating interaction) the boundary conditions are A co(z 5 2L/2) 5 1, and A n cl(z 5 2L/2) 5 0. The transmission through the grating is simply T 5 A co(L/2)/A co(2L/2). Note that for a uniform grating we take L 5 w, whereas for a nonuniform grating we choose L to be a length that fully encloses the grating such that the grating perturbation is negligible outside the range 2L/2 < z < L/2. For example, for a Gaussian grating we choose L to be several times the grating width w. We then solve Eqs. (39)–(41) or (44) and (45) subject to these boundary conditions. In practice the difficulty of solving the coupled-mode equations depends on the strength and the spectral density of the resonances. Note that the sets of Eqs. (39)– (41) and (44) and (45) each describe a large number (typically several hundred) of coupled first-order differential equations. Since the solutions are computed at each wavelength, we can greatly simplify the task by recognizing that only one or several mode interactions are nearly resonant at the particular wavelength. Thus only those mode pairs with the smallest associated detuning param-

1768


Turan Erdogan

eters in Eqs. (42), (43), and (46) need to be kept. In our calculations we determine which resonances to include at each wavelength by estimating the spectral location and width of the nearby resonances; we keep only those that are centered a specified number (or fraction) of spectral bandwidths from the wavelength. The estimates are based on the closed-form solution of the coupled-mode equations9 for the single resonance in question (such that only two modes are included) and for a uniform grating that exactly or approximately represents that actual grating. For a Gaussian grating, a uniform grating of length equal to w and s equal to the peak s of the Gaussian is a good approximation. The approximate spectral location of the resonance also takes into account the effect of the ultraviolet-induced average index in the core by means of the core-mode self-scattering terms—the first terms on the right-hand sides of Eqs. (39) and (44). The estimates are based on the following results. The approximate center wavelength of the LP01 –LP01 Bragg reflection resonance in a short-period grating can be found by taking co-co co-co d 01201 1 k 01201 5 0,

(47)

co-co where the term k 01201 accounts for the wavelength shift that is due to the increased average index. For the resonance associated with the n th cladding mode and the LP01 mode in either a short-period or a long-period grating, the approximate spectral location is given by co-co d 1cl-co n 201 1 k 01201/2 5 0,

(48)

where d is defined by Eq. (43) or (46), depending on whether the grating is short period or long period, respectively, and where the factor of 1/2 arises because we are assuming that only the core mode is substantially affected by the increased average index in the core. The approximate (normalized) spectral bandwidth of a counterpropagating resonance obtained from the closed-form solution of the two-mode coupled-mode equations9 is cl-co 1 n 201

F S DG

p Dl lk > 11 l p n avg kL

length located less than two spectral bandwidths Dl from the wavelength of interest be included in the calculation. Calculation of the transmission spectrum thus proceeds as follows: At each wavelength we locate the closest resonance; if the grating is sufficiently weak that the spectral bandwidths Dl of neighboring resonances are much smaller than the separations of those resonances from the wavelength of interest, then we keep only the closest resonance and thus have only two modes. For a uniform grating, a closed-form solution is available.9 For a nonuniform grating, we integrate the pair of coupled first-order differential equations, using a fourth-order, adaptive-step-size Runge–Kutta algorithm subject to the boundary conditions discussed above. If the grating is not sufficiently weak, we numerically integrate the set of coupled differential equations including as many modes as required by our estimate. Note that in this case a numerical solution is necessary even for a uniform grating, as we no longer obtain constant-coefficient differential equations as in the two-mode case.

5. RESULTS In this section we compare experimentally measured transmission spectra with calculations based on the theory described above to examine the accuracy of the theory and to demonstrate typical grating transmission spectra when cladding-mode coupling is present. Figure 8(a) shows the measured transmission through a grating written in Corning Flexcore fiber that was loaded with deuterium.14 The fiber parameters are approximately those noted above in the discussion of Figs. 3

2 1/2

.

(49)

For LP01 –LP01 Bragg reflection, l is determined by Eq. co-co , and n avg is n eff for the LP01 mode; for (47), k is k 01201 cladding-mode LP01 scattering, l is determined by Eq. co cl (48), k is k 1cl-co n 201 , and n avg is (n eff 1 n eff)/2, or the average of the effective indexes for the core and cladding modes. The approximate (normalized) spectral bandwidth of a copropagating resonance9 is

S

Dl l 4kL > 11 l DnL p

D

1/2

,

(50)

where for cladding-mode LP01 scattering in a long-period grating l is determined by Eq. (48); k is k 1cl-co n 201 , and Dn co cl 2 n eff . Using Eqs. (47) and (48) and relations 5 n eff (49) and (50), we can estimate straightforwardly how many resonances should be included at a particular wavelength. For example, for a conservative calculation we might specify that all resonances with a center wave-

Fig. 8. (a) Experimentally measured and (b) theoretically calculated transmission spectra through a relatively weak, Gaussian short-period grating, demonstrating both core-mode–core-mode and core-mode–cladding-mode coupling.

Turan Erdogan

Fig. 9. (a) Experimentally measured and (b) theoretically calculated transmission spectra through a strong, Gaussian shortperiod grating.

and 4 (here the best theoretical fits are obtained with D 5 0.0050 and a 1 5 2.5 m m). The grating is a Gaussian grating with a width w 5 4 mm and a peak inducedindex change of s n 1 5 7.2 3 1024 . It was written by interfering ultraviolet beams from an excimer-laserpumped, frequency-doubled dye laser producing 15-ns pulses at a 30-Hz repetition rate. The exposure utilized 10 mW of average power for ;60 s, with the beams focused on the fiber to an approximate spot size of 4 mm 3 50 mm. The strong resonance at 1540 nm is due to LP01 –LP01 Bragg reflection; the peak reflectivity is ;98%. The resonances at shorter wavelengths are caused by coupling to the n 5 1, 3, 5, ... cladding modes. Evidence of coupling to the even cladding modes is not visible on this trace. Note that the transmission spectrum for this grating approximates a comb of flat-topped transmission windows, suggesting unique potential applications. Figure 8(b) shows a theoretical calculation of the transmission through such a grating. For this relatively weak grating the cladding-mode resonances are far apart 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 the two-mode, closed-form solutions. For this case there is excellent agreement in terms of both the locations and the strengths (relative and absolute) of the resonances. Figure 9(a) shows the measured transmission through a similar grating but with a larger peak induced-index change of s n 1 5 2.8 3 1023 . To write this grating we increased the average power and exposure time to 20 mW and 100 s, respectively. This grating is nearly 100% reflecting, with a reflection bandwidth of ;2 nm. Figure 9(b) shows the corresponding theoretical calculation of the transmission. Note that the resonances overlap sub-


1769

stantially over most of the wavelength range shown in the plot. As a result, an accurate calculation could be obtained only by including multiple cladding-mode resonances simultaneously at each wavelength. Here all resonances that were estimated to be within two spectral bandwidths of the calculation wavelength were included. Again there is excellent agreement between theory and experiment. In particular, note that the theory accurately reproduces the fine structure on the shortwavelength side of each resonance. This structure is mainly a result of the Fabry–Perot-like property of a nonuniform, tapered grating in which interference occurs when light at the short-wavelength edge of the resonance sees only the wings of the grating and not the center.3 The fine structure is also due to the emergence of coupling to the n 5 even modes at the shortest wavelengths. For this grating the limited measurement resolution (0.1 nm) causes a slightly larger apparent discrepancy with theory than in Fig. 8. Transmission spectra (in decibels) calculated for typical long-period gratings are shown in Fig. 10. Figure 10(a) shows the transmission for a relatively weak grating with a length of 25 mm, a peak induced-index change of 1 3 1024 , and a uniform or a Gaussian profile. The five main dips seen in these spectra correspond to coupling to the n 5 1, 3, 5, 7, 9 cladding modes. Figure 10(b) shows the transmission for a stronger, uniform grating with a peak induced-index change of 3.6 3 1024 . In each case the grating period is adjusted to yield coupling at 1550 nm between the LP01 core mode and the n 5 7 cladding mode. That is, the gratings in Figs. 10(a) and 10(b) have periods of L 5 600 m m and L 5 570 m m, respectively. Notice that the dip in Fig. 10(b) associated with coupling

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

1770


Fig. 11. (a) Experimentally measured and (b) theoretically calculated transmission spectra through a relatively weak, uniform long-period grating. The resonance is associated with coupling to the lowest-order (l 5 1, n 5 1) cladding mode.

to the n 5 9 cladding mode at 1800 nm is slightly weaker than the dip associated with coupling to the n 5 7 cladding mode at 1550 nm, despite the stronger coupling to the n 5 9 mode shown in Fig. 5. The reason for this behavior is that, over the length of the grating, incident core-mode light at 1800 nm is fully converted to the cladding mode and then begins to convert back to the core mode, resulting in a smaller loss in transmission. The measured transmission (in decibels) through a relatively weak long-period grating (L 5 396 m m) is shown in Fig. 11(a). This grating was written in an AT&T dispersion-shifted communications fiber loaded with hydrogen by direct exposure to a 248-nm KrF excimer laser beam through a chrome Ronchi-ruled mask. The profile of the grating is nearly uniform, with a length of w > 50 mm. The peak induced-index change is s n 1 5 1.9 3 1024 , as determined by our calculation. The single resonance that is visible within the wavelength range plotted is associated with coupling to the n 5 1 cladding mode; this is the shortest-wavelength resonance. Figure 11(b) shows the calculated transmission spectrum through this grating, where agreement of the transmission minimum was obtained by adjusting the normalized index change s. Again there is excellent overall agreement.

6. CONCLUSION A straightforward theory has been presented that accurately models the measured transmission through a number of practical fiber gratings that exhibit substantial

Turan Erdogan

cladding-mode coupling. There are two parts to the theory. The first is the specific method used to calculate core and cladding modes, the associated propagation constants and fields, and the coupling coefficients. For this part a simple three-layer, step-index fiber geometry was chosen so that we could write down the needed results. Even for more-complicated fiber geometries, one can often reasonably approximate the modes with those of a threelayer structure. The second part of the theory is the multimode coupled-mode theory. This part is more general and requires only the propagation constants and coupling coefficients associated with the fiber modes, which can be obtained with a more sophisticated method if desired. A number of approximations were made in the theory; the accuracy with which the measured spectra are modeled for the most part justifies these. Some approximations, such as the number of cladding-mode resonances to include in the calculation at each wavelength, are easily verified by simply including more and more resonances until the result converges to a stable solution. Others require direct testing of the theory with and without the neglected component for certainty of the validity in a variety of cases. For the examples considered here, the neglect of both longitudinal coupling coefficients and claddingmode to cladding-mode scattering is reasonable: These would not alter the calculated transmission visibly. However, these approximations should be reconsidered for very strong gratings and for gratings that couple the core mode to fairly high-order cladding modes. An issue that was not addressed specifically is how carefully dispersion should be taken into account in the calculations. In particular, ideally all of the mode properties should be recomputed at each wavelength, but if one is calculating transmission at several hundreds or even thousands of wavelengths, this approach might be intractable. Instead, if one is concerned with only one long-period-grating resonance or several short-periodgrating resonances, it is a reasonable approximation to compute the mode properties once at a central wavelength and then simply integrate the coupled-mode equations at each wavelength (even when the mode resonances are strongly overlapping). We have also used an intermediate approach in which we compute the mode properties at a number of wavelengths that are less densely spaced than the transmission calculation grid, and then we simply interpolate the quantities that are specific to the modes at each new wavelength from the sparse grid. Finally it should be pointed out that our experimental and theoretical discussion is limited to gratings with a circularly symmetric index perturbation, thus excluding gratings with tilted fringes and other nonuniformities across the core. Although our reason for doing so is partly simplicity, it is also true that it is more difficult to utilize individual cladding-mode resonances for applications in circularly asymmetric gratings, as so many more resonances show up. One quickly finds such a large overlap of resonances that the spectrum mimics a smooth strictly radiation-mode coupling spectrum but with a spiky, almost random modulation impressed on it. For most applications this result is less desirable than the simpler radiation-mode coupling spectrum that can be

Turan Erdogan


flexibly altered by using grating tilt, for example.7 However, despite the apparent experimental and theoretical difficulties in investigating asymmetric cladding-mode couplers, there might very well be enticing reasons for pursuing them.

cl H cl z 5 2iE 1 n

2

1771

F

p a 1u 12u 22i s 1J 1~ u 1a 1 ! F 2 p 1~ r ! 2b

G

1 q ~ r ! exp@ i f 1 i ~ b z 2 v t !# , u2 1

(A6)

where the quantities F 2 and G 2 are defined to be

APPENDIX A

F2 [ J 2

In this appendix expressions are listed for the claddingmode fields in the cladding and in the surround for the three-layer fiber geometry shown in Fig. 2. The vector components of the electric field for the l 5 1 cladding modes in the fiber cladding region (a 1 < r < a 2 ) are

H

p a 1u 1 J 1~ u 1a 1 ! 2

cl E cl r 5 iE 1 n

s2

2

n 22

F

2

F2

2

p 1~ r ! 1

r

n 22z 0

u 2G 2r 1~ r ! 2

s 1~ r !

n 12

GJ

1 u 2r

q 1~ r !

3 exp@ i f 1 i ~ b z 2 v t !# ,

H F

p a 1u 1 J 1~ u 1a 1 ! 2

E fcl 5 E 1cln 2

2 n 22z 0

(A1)

G

s2

G2

n 22

r

n 12u 2r

1

n 12u 2

2n 2 2 b

G

F

E fcl 5 E 1cln

1i

2 n 22z 0 n 12u 2r

H

G2

2i

5

iE 1cln

(A3)

r

2

2 iu 2 G 2 r 1 ~ r ! 1 i

H F is1

n 22z 0 n 12

r

s 1~ r !

3 exp@ i f 1 i ~ b z 2 v t !# ,

J

@ K 2 ~ w 3 r ! 1 K 0 ~ w 3 r !#

p 1~ r ! 2

J

1 u 2r

s 2G 3 n 32

H

cl E cl z 5 E 1n

2F 3 @ K 2 ~ w 3 r !

@ K 2 ~ w 3 r ! 2 K 0 ~ w 3 r !#

p a 1u 12u 22s 2J 1~ u 1a 1 ! 2n 3 2 b K 1 ~ w 3 a 2 !

J (A9)

3 exp@ i f 1 i ~ b z 2 v t !# ,

J

(A10)

G 3K 1~ w 3r !

3 exp@ i f 1 i ~ b z 2 v t !# .

cl H cl r 5 E 1n

(A4)

F2

2F 3 @ K 2 ~ w 3 r !

(A11)

Finally, the vector components of the magnetic field for the l 5 1 cladding modes in the surround region (r > a 2 ) are

p 1~ r !

q 1 ~ r ! 1 i s 1 @ u 2 F 2 r 1 ~ r ! 2 s 1 ~ r !#

p a 1u 12J 1~ u 1a 1 !

n 32

4w 3 K 1 ~ w 3 a 2 !

1 K 0 ~ w 3 r !# 1

3 exp@ i f 1 i ~ b z 2 v t !# ,

H fcl

s 2G 3

p a 1u 12u 22J 1~ u 1a 1 !

G 2 p 1~ r !

q 1 ~ r ! exp@ i f 1 i ~ b z 2 v t !#

p a 1u 12J 1~ u 1a 1 !

H

4w 3 K 1 ~ w 3 a 2 !

3 exp@ i f 1 i ~ b z 2 v t !# ,

and the vector components of the magnetic field for the l 5 1 cladding modes in the fiber cladding region (a 1 < r < a 2 ) are

cl H cl r 5 E 1n

(A8)

(A2)

p a 1u 12u 22s 2J 1~ u 1a 1 !

n 22z 0

u 21s 1 . a1

p a 1u 12u 22J 1~ u 1a 1 !

2 K 0 ~ w 3 r !# 1

3 exp@ i f 1 i ~ b z 2 v t !# ,

cl E cl z 5 2E 1 n

(A7)

Note that F 2 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 the azimuthal and longitudinal components of the magnetic field are imaginary and the other three components are real. The vector components of the electric field for the l 5 1 cladding modes in the surround region (r > a 2 ) are

p 1~ r !

J

,

n 12a 1

G 2 [ z 0J 1

cl E cl r 5 iE 1 n

q 1~ r ! 1 u 2F 2r 1~ r ! 2 s 1~ r !

u 21s 2 z 0

q 1~ r !

G

p a 1u 12u 22J 1~ u 1a 1 ! $ 2i s 1 @ K 2 ~ w 3 r ! 4w 3 K 1 ~ w 3 a 2 !

1 K 0 ~ w 3 r !# 2 iG 3 @ K 2 ~ w 3 r ! 2 K 0 ~ w 3 r !# % 3 exp@ i f 1 i ~ b z 2 v t !# , H fcl 5 iE 1cln

(A12)

p a 1u 12u 22J 1~ u 1a 1 ! $ i s 1@ K 2~ w 3r ! 4w 3 K 1 ~ w 3 a 2 !

2 K 0 ~ w 3 r !# 2 G 3 @ K 2 ~ w 3 r ! 1 K 0 ~ w 3 r !# % (A5)

3 exp@ i f 1 i ~ b z 2 v t !# ,

(A13)

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cl H cl z 5 iE 1 n

Turan Erdogan

p a 1u 12u 22i s 1J 1~ u 1a 1 ! F 3K 1~ w 3r ! 2 b K 1~ w 3a 2 !

3 exp@ i f 1 i ~ b z 2 v t !# .

(A14) 3

In Eqs. (A9)–(A14) the quantities F 3 and G 3 are defined to be

F 3 [ 2F 2 p 1 ~ a 2 ! 1

1 q ~ a !, u2 1 2

3

(A15)

F

p 3a 12u 14u 22J 12~ u 1a 1 !

P 2 5 ~ E 1cln ! 2

G

HS S S S S

16

n eff Z0

n eff Z0

1 11

n 32 n 22z 0 G 3 [ 2 2 G 2 p 1~ a 2 ! 1 q 1~ a 2 ! . n2 n 12u 2

1 11 (A16)

2 11

As anticipated, the radial component of the electric field and the azimuthal and longitudinal components of the magnetic field are imaginary, and the other three components are real.

F 22 2

2

n eff Z 0 n 22

n eff Z 0 n 2 2 z 0 2

n eff2 n 22 n eff2 n 22 n eff2 n2

2

D D DF

˜! 1 3 ~S 2 S

n 14

D

˜!1 G 22 ~ Q 1 Q

D

1 u 22

˜! ~R 1 R

˜! F 2 Im~ G 2 !~ Q 2 Q n 22 n 12u 22

˜! Im~ z 0 !~ R 2 R

n 2 2 Im~ z 0 ! 2

n1 u2

S

2n eff Z 0 z 0 u2

n1

2

F2 1 G2 2

1 u2 1 Z0

Im~ G 2 !

D

G

J

˜! . F2 ~S 1 S (B3)

The new quantities defined in Eq. (B3) are

APPENDIX B In this appendix it is shown how the normalization constant E 1cln for the cladding modes can be calculated for a particular normalization choice (i.e., P 5 1 W). As explained in the text, we evaluate the integral in Eq. (27) by using the mode field expressions given in Eqs. (20)–(25), (A1)–(A6), and (A9)–(A14), and then we set the result equal to the power normalization choice. Solving the resulting equation for E 1cln provides the desired result. The total power carried by the cladding mode is the sum of the powers carried in the core, the cladding, and the surround region, or P 5 P1 1 P2 1 P3 ,

(B1)

where P j , the power carried in the jth region of the fiber, can be calculated by means of the integral in Eq. (27) but with the limits along the radial direction replaced by those appropriate for the region of interest. Consider first the core, or region 1 (r < a 1 ). We do not provide the details of the calculation, but the result is

P1 5

~ E 1cln ! 2

S

p a 12u 12

1 11

4 n eff2 n1

2

D

Z0

2

G

F

n eff Z0

2

2 2 u JN J 1 ~ u 2 a 1 ! n 1 ~ u 2 a 1 ! ,

2 2˜u JN J 1 ~ u 2 a 1 ! N 1 ~ u 2 a 1 ! , R[

n eff Z 0 z 0 n 12

1 4

u J @ N 2 ~ u 2 a 1 ! 2 N 0 ~ u 2 a 1 !# 2 1

2 J 0 ~ u 2 a 1 !# 2 2

1 2

1 4

˜ [ R

1 4

u JN @ N 2 ~ u 2 a 1 ! 2 N 0 ~ u 2 a 1 !#

˜u @ N ~ u a ! 2 N ~ u a !# 2 1 J 2 2 1 0 2 1

2 J 0 ~ u 2 a 1 !# 2 2

1 2

(B6) 1 4

˜u @ J ~ u a ! N 2 2 1

˜u @ N ~ u a ! 2 N ~ u a !# JN 2 2 1 0 2 1

3 @ J 2 ~ u 2 a 1 ! 2 J 0 ~ u 2 a 1 !# , S[

1 2

1

1 2

u N J 1 ~ u 2 a 1 !@ J 0 ~ u 2 a 1 ! 2 J 2 ~ u 2 a 1 !#

2

1 2

u JN $ N 1 ~ u 2 a 1 !@ J 0 ~ u 2 a 1 ! 2 J 2 ~ u 2 a 1 !#

S J

2 11

2

n eff

n 12

3 Im~ z 0 ! @ J 0 2 ~ u 1 a 1 ! 1 J 1 2 ~ u 1 a 1 !# . Next consider the cladding region (a 1 < r < a 2 ). power is given by

D

1 2

The

(B8)

˜u N ~ u a !@ N ~ u a ! 2 N ~ u a !# J 1 2 1 0 2 1 2 2 1

1

1 2

˜u J ~ u a !@ J ~ u a ! 2 J ~ u a !# N 1 2 1 0 2 1 2 2 1

2

1 2

˜u $ N ~ u a !@ J ~ u a ! 2 J ~ u a !# JN 1 2 1 0 2 1 2 2 1

1 J 1 ~ u 2 a 1 !@ N 0 ~ u 2 a 1 ! 2 N 2 ~ u 2 a 1 !# % , (B2)

(B7)

u J N 1 ~ u 2 a 1 !@ N 0 ~ u 2 a 1 ! 2 N 2 ~ u 2 a 1 !#

1 J 1 ~ u 2 a 1 !@ N 0 ~ u 2 a 1 ! 2 N 2 ~ u 2 a 1 !# % , ˜ [ S

(B5)

u N@ J 2~ u 2a 1 !

3 @ J 2 ~ u 2 a 1 ! 2 J 0 ~ u 2 a 1 !# ,

n 12

2

(B4)

˜ [ ˜u N 2 ~ u a ! 1 ˜u J 2 ~ u a ! Q J 1 2 1 N 1 2 1

n eff Z 0 z 0 2

Im~ z 0 ! @ J 2 2 ~ u 1 a 1 ! 2 J 1 ~ u 1 a 1 !

3 J 3 ~ u 1 a 1 !# 1

G

HF

n eff

Q [ u JN 12~ u 2a 1 ! 1 u N J 12~ u 2a 1 !

(B9)

where these expressions utilize the definitions

u J [ a 2 2 @ J 2 2 ~ u 2 a 2 ! 2 J 1 ~ u 2 a 2 ! J 3 ~ u 2 a 2 !# 2 a 1 2 @ J 2 2 ~ u 2 a 1 ! 2 J 1 ~ u 2 a 1 ! J 3 ~ u 2 a 1 !# ,

(B10)

Turan Erdogan


u N [ a 2 2 @ N 2 2 ~ u 2 a 2 ! 2 N 1 ~ u 2 a 2 ! N 3 ~ u 2 a 2 !# 2 a 1 2 @ N 2 2 ~ u 2 a 1 ! 2 N 1 ~ u 2 a 1 ! N 3 ~ u 2 a 1 !# , 1 2

u JN [ a 2 2 $ J 2 ~ u 2 a 2 ! N 2 ~ u 2 a 2 ! 2

ACKNOWLEDGMENTS (B11)

@ J 1~ u 2a 2 ! N 3~ u 2a 2 !

1 J 3 ~ u 2 a 2 ! N 1 ~ u 2 a 2 !# % 2 a 1 2 $ J 2 ~ u 2 a 1 ! N 2 ~ u 2 a 1 ! 2

1 2

@ J 1 ~ u 2 a 1 ! N 3 ~ u 2 a 1 ! 1 J 3 ~ u 2 a 1 ! N 1 ~ u 2 a 1 !# % ,

(B12)

The author is grateful to John Sipe for many insightful discussions concerning the calculation of interactions in fiber gratings, to Suresh Pereira for critical analysis of the theory, and to Justin Judkins and Lucent Technologies Bell Laboratories for providing the experimental data for the long-period grating and for partially supporting this research. This work is funded in part by the National Science Foundation (award ECS-9502670) and by the Alfred P. Sloan Foundation. The author can be reached at tel: 716-275-7227; fax: 716-244-4936; e-mail: [email protected].

˜u [ a 2 @ J 2 ~ u a ! 1 J 2 ~ u a !# 2 a 2 @ J 2 ~ u a ! J 2 2 2 2 1 2 2 1 2 2 1 1 J 1 2 ~ u 2 a 1 !# ,

(B13)

˜u [ a 2 @ N 2 ~ u a ! 1 N 2 ~ u a !# 2 a 2 @ N 2 ~ u a ! N 2 2 2 2 1 2 2 1 2 2 1 1 N 1 ~ u 2 a 1 !# , 2

(B14)

˜u [ a 2 @ J ~ u a ! N ~ u a ! 1 J ~ u a ! N ~ u a !# JN 2 0 2 2 0 2 2 1 2 2 1 2 2

REFERENCES 1. 2.

2 a 1 @ J 0 ~ u 2 a 1 ! N 0 ~ u 2 a 1 ! 1 J 1 ~ u 2 a 1 ! N 1 ~ u 2 a 1 !# . 2

(B15)

3.

Finally, the power carried in the surround region (r > a 2 ) is given by

4.

P3 5

~ E 1cln ! 2

3

HF S

p 3a 12a 22u 14u 24J 12~ u 1a 1 ! 16w 3 2 K 1 2 ~ w 3 a 2 !

n eff Z 0 n3

2 11

2

G 32 2

n eff2 n 32

D

n eff Z0

5.

F 32

F 3 Im~ G 3 !

6.

G

7. 8.

3 @ K 2 2 ~ w 3 a 2 ! 2 K 1 ~ w 3 a 2 ! K 3 ~ w 3 a 2 !# 1

F S

n eff Z 0 n 32

1 11

1773

G 32 2

n eff2 n3

2

D

n eff Z0

9.

F 32

F 3 Im~ G 3 !

G

10. 11.

J

3 @ K 0 2 ~ w 3 a 2 ! 2 K 1 2 ~ w 3 a 2 !# .

12.

(B16)

Using Equations (B2), (B3), and (B16), we can obtain a value for the normalization constant E 1cln by solving for this parameter as the only unknown in the equation P 1 1 P 2 1 P 3 5 1 W.

13. 14.

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