- Email: [email protected]
Power coupling between fiber and multimode slab composite waveguide
1 April 2002
Optics Communications 204 (2002) 171–178 www.elsevier.com/locate/optcom
Power coupling between fiber and multimode slab composite waveguide Valeri V. Kapoustine a,*, Sergey O. Yarovikov b a
KA&V Company, 1591 Riverside Dr., Suite 204, Ottawa, Ont., Canada K1G 4A7 b Ulyanovsk State University, 42 L. Tolstoi St., Ulyanovsk 432700, Russia Received 26 September 2001
Abstract We discuss power coupling in devices composed of a fiber with partly or completely removed cladding and multimode slab overlay waveguide of any complexity. A fiber mode excites with a high efficiency only one mode of the composite structure in the wide range of system parameters. But only when the effective refractive index of this mode almost equals to the one of the initial fiber mode the modal value of electric field is high enough to produce effective interactions with active and passive elements in the region of the overlay waveguide. In the narrow range of system parameters last two modes are excited with a nearly equal efficiency providing a regular directional coupler. The powercoupling coefficient essentially depends on the thickness of remaining fiber cladding and an optimal value of this cladding is about 1.5–2 lm. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 42.82.Et Keywords: Optical coupler; Fiber; Composite planar waveguide; Power-coupling coefficient
Fiber-to-planar waveguide couplers find their applications in different kinds of optoelectronic devices such as polarizers [1], switchers [2], channel dropping filters [3], and others, and the interest in these structures keeps increasing due to their simplicity, low insertion loss, and high mechanical stability. The basic theoretical approach to investigate parameters of these devices is a coupling mode theory [4,5] when only one mode of the
*
Corresponding author. Tel.: +1-613-738-5177; fax: +1-613738-5177. E-mail address: [email protected] (V.V. Kapoustine).
overlay planar waveguide is considered to match the fiber mode. But these structures are usually multimode [6,7], and even if we have only twomode composite system we need to assume that even and odd modes are excited with an equal efficiency [8] that is not obvious. In this work we propose a simple method to calculate the coefficients of an excitation of all modes of the multimode composite structure by fiber mode on the boundary: fiber–fiber with overlay waveguide. The accuracy of this method is the same as for an exact treatment of directional couplers by computing the compound modes [9] and it can be very useful to calculate the complex optoelectronic device
0030-4018/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 ( 0 2 ) 0 1 2 4 7 - 6
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parameters. In many simple cases it can compete with 2D beam propagation method. We will consider our method for two examples shown in Fig. 1, a fiber core with remaining cladding and overlay planar waveguide with a varying thickness and refractive index (Fig. 1(a)), and two fiber cores, one with remaining claddings (lower), sandwiched with an interlayer with varying parameters (Fig. 1(b)). For the sake of simplicity fibers have been substituted by equivalent planar waveguides. Parameters of the equivalent waveguides can be chosen using a variational method assuming that the propagation constant and a fiber mode field distribution are matched to those of the appropriate planar waveguides [10]. A region of a light transmission from the fiber into the multilayer composite waveguide can be considered as a sharp boundary between them to calculate power-coupling coefficients. If field distributions of all compound modes in the composite slab waveguide and equivalent fiber are normalized
Fig. 1. Structures for an investigation: (a) an equivalent fiber with an overlay waveguide; (b) two fibers or a fiber and a waveguide with an intermediate layer, the lower fiber has remaining cladding; df ¼ df1 ¼ 6:982 lm, df2 ¼ 5 or 9 lm; nc ¼ 1:44645, nf ¼ nf1 ¼ 1:44989, nf2 ¼ 1:45, n0 ¼ 1:5 or 1.7.
on the unit power per unit length from the equality of transverse electric fields on the boundary between fiber and composite waveguide for TE-modes X Ef ¼ a n En n
with neglect of a reflection on this boundary one can obtain the expression for power-coupling coefficients (power per unit length in nth mode of the composite structure) in the form: Z 2 bf bn bf 1 2 Pn ¼ jan j ¼ Ef En dx ; ð1Þ 2 bn ð2cl0 Þ 1 where Ef and En are normalized electric fields of the equivalent fiber mode and nth waveguide compound mode, respectively, bf , bn are normalized effective refractive indices of fiber and nth waveguide mode, c is the speed of light in free space, l0 is the free space permeability, and an are coefficients. Since we take into account only bounded waveguide modes, the system of orthonormal mode functions is usually not complete and the computed total power in all bounded modes of the composite waveguide is usually less than unit. In the case of TM-modes a similar approach gives the next expression for the power-coupling coefficients: 2 Z b b 1 Hf Hn Pn ¼ n f 2 dx : ð10 Þ eðxÞ ð2ce0 Þ 1 Here Hf and Hn are normalized on unit power per unit length transverse magnetic fields of the fiber mode and nth waveguide mode, respectively, e0 is the free space permittivity, and eðxÞ is a transverse dependence of dielectric permittivity of the composite waveguide. The transverse matrix method was used for the calculation of propagation constants for all possible compound modes in the described waveguides. Only TE-modes were taken into account since in the absence of an anisotropy the results for TM-modes would be mostly the same. The dispersion equation for TE-modes can be written in the following way: M21 ðM22 þ M11 Þðn2c b2 Þ1=2 þ M12 ðn2c b2 Þ ¼ 0; ð2Þ
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where b is a normalized propagation constant of an appropriate compound mode, nc is a refractive index of cladding and substrate, and Mij are the elements of the total transfer matrix of the composite multilayer planar structure. The elements of the transmission matrix for the kth layer of thickness lk and refractive index nk are represented by formulae: m11 ¼ m22 ¼ cosðgk lk Þ; ik0 m12 ¼ sinðgk lk Þ; gk gk m21 ¼ sinðgk lk Þ: ik0
ð3Þ
Here a transverse propagation constant equals 1=2 gk ¼ k0 ðn2k b2 Þ , k0 ¼ 2p=k, and k is the free space wavelength. Eq. (2) with (3) was solved numerically for different system parameters. A modal electric field distribution was calculated for each compound mode as a sum of hyperbolic sine and cosine coordinate functions with coefficients getting from the join of solutions in the adjacent layers on the boundaries and normalized on unit power per unit length [11]. Numerical checking of the modes orthogonality for the composite waveguide has shown that overlap integrals between all couples of modes have not exceeded 2 104 in all cases of interest. The field
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distribution was as well calculated and normalized for equivalent fiber to obtain the mode excitation coefficient (1) on the boundary fiber–composite slab waveguide. Thickness and refractive indices of the slab waveguide equivalent to the lower fiber in Fig. 1 were taken from [7] getting a thickness of 6:982 lm and refractive indices 1.44989 and 1.44645 for the waveguide core and cladding, respectively. These parameters correspond to a real fiber with a core diameter of 8 lm and refractive indices of the core and cladding, 1.451 and 1.447, respectively, at k ¼ 1:3 lm and will be used in all subsequent calculations. Fig. 2 shows dependences of the power-coupling coefficient on the thickness of the overlay waveguide d0 (Fig. 1(a)) in the case when the refractive index of that waveguide equals n0 ¼ 1:7 (a) and 1.5 (b). Wavelength here and thereinafter is 1:3 lm. Digits in figures denote modal numbers, and digits with and without strokes correspond to the cases when a thickness of the remaining fiber cladding equals 2 lm or 0, respectively. Fig. 3(a) represents dependencies of modal effective refractive indices on the overlay waveguide thickness when n0 ¼ 1:5. One can see from Figs. 2(b) and 3(a) that powercoupling coefficient essentially depends on the thicknesses of the overlay waveguide and remaining
Fig. 2. Power-coupling coefficient for different compound modes versus overlay waveguide thickness; n0 ¼ 1:7 (a) and n0 ¼ 1:5 (b). Digits denote modal numbers. Digits with a stroke are for the thickness of the remaining fiber cladding 2 lm and ones without stroke are for the fiber without cladding.
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Fig. 3. Dependencies of the modal effective refractive index (a) and parameters of 2nd mode (b) on the overlay waveguide thickness; n0 ¼ 1:5, dc ¼ 2 lm. Digits in (a) correspond to numbers of modes. In (b) 1 – reduced in 10 times values of electric field E, normalized on unity, on the upper boundary in Fig. 1(a), 2 – reflection coefficient R of a grating placed on the upper boundary, 3 – power-coupling coefficient P, 4 – value of a product of the reflection coefficient by power-coupling coefficient. Grating parameters are given in the text.
cladding and approaches a maximum value for a given compound mode when it has almost the same effective refractive index as the fiber mode. In Fig. 3(b) some parameters of second mode are shown versus overlay waveguide thickness to understand mechanisms of wave propagation through the composite waveguide. Since such devices intend for a light interaction with active and passive elements placed in the region of the overlay waveguide curve 1 shows a behavior of modal electric field, normalized on unity, on the upper boundary in Fig. 1(a), where for example a grating can be arranged. Electric field values do not correspond to the dependence of the power-coupling coefficient for a given mode (curve 3). Where the electric field has maximums the power-coupling coefficient has minimums and vice versa. Curve 2 models a behavior of the reflection coefficient of a grating placed on the upper boundary in Fig. 1(a). A grating length was chosen 5 mm with a groove 0:3 lm and a period met a resonance condition for a given mode and given overlay waveguide thickness. Electric field distributions used for reflection coefficient calculations were normalized on the unit power per unit length [10]. Therefore a real value of the reflected power can be defined as a product of the reflection coefficient by power-coupling coefficient (curve 4). When n0 ¼ 1:5 as one can see from
Fig. 3(b) the maximum reflection can be realized at d0 ¼ 3:42 and 2:02 lm. Note that the maximum at the larger overlay waveguide thickness is higher but does not exceed 90%. Larger values of the reflection can be achieved with larger values of n0 (Fig. 4).
Fig. 4. Dependence of the value (reduced in five times) of electric field E normalized on the power-coupling coefficient on the upper boundary in Fig. 1(a) (1), and value of a product of the power-coupling coefficient P by reflection coefficient R of the grating placed on the upper boundary (2,3,4) on the remaining fiber cladding thickness for 2nd mode. 1,2 – d0 ¼ 3:3 lm and n0 ¼ 1:5; 3 – d0 ¼ 3:4 lm, n0 ¼ 1:5; 4 – d0 ¼ 1:44 lm, n0 ¼ 1:7. Grating parameters are given in the text.
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When the thickness of remaining cladding increases the power-coupling coefficient P rises tending to the unity, but in this case the electric field value E on the upper boundary of the system in Fig. 1(a) obviously decreases. So, the optimal value of the cladding thickness depends on an application, and in Fig. 4 the dependences of the product of the power-coupling coefficient by reflection coefficient of the grating placed on the upper boundary in Fig. 1(a) are shown as well as the electric field values in the grating region versus the cladding thickness for the second mode. As one can see from Fig. 4 the optimal cladding thickness is about 1.5–2 lm in our case and we can obtain more than 97% of the initial fiber power in the reflection if n0 ¼ 1:7. All other modes practically are not excited by fiber mode and therefore do not participate in the reflection process. The full-width at half maximum (FWHM) of the reflected signal dependence on the wavelength is about 0.5 nm. Note that in the case of a coincidence of effective refractive indices of the fiber mode and composite waveguide mode (at d0 ¼ 3:3 lm, n0 ¼ 1:5) the maximum value of the reflection coefficient is less in comparison with one obtained at d0 ¼ 3:4 lm, and therefore this coincidence does not play a significant role in the reflection though we can say that the maximum reflection always occurs at the sys-
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tem parameters almost but not exactly corresponding to the coincidence of effective refractive indices of the fiber mode and composite waveguide mode. When the thickness of the overlay waveguide varies the modal field distributions vary significantly as well (Fig. 5). In the resonance condition, when the effective refractive index of the compound mode is equal to the one of the fiber, only one mode is excited with high efficiency (Fig. 5(a)). As a rule the only one mode with highest index has the maximum efficiency of an excitation with the exception of a small range of overlay waveguide thicknesses when two modes are excited simultaneously (Fig. 2). At some value of the overlay waveguide thickness we can obtain almost 50% excitation of each of two modes (Fig. 5(b), curves 1 and 2). At n0 ¼ 1:5, d0 ¼ 3:567 lm. In that case we have a regular directional coupler and values of effective refractive indices of these two modes can be used for a definition of the power exchange length [8]. Sum and difference of these two modal distributions show a power rearrangement between two waveguides on the power exchange length (Fig. 5(b), curves 3 and 4). So, in the case when the fiber mode propagates through a boundary between the fiber and composite waveguide comprising the fiber with some
Fig. 5. Modal electric field distribution of the composite multilayer waveguide in Fig. 1(a). Thickness of the remaining fiber cladding 2 lm, refractive index of the overlay waveguide n0 ¼ 1:5, thickness d0 ¼ 3:3 lm (a) and 3:567 lm (b). In (b) 1,2 – even and odd modes of a directional coupler, 3,4 – sum of these modes and difference between them, respectively. The initial fiber mode is shown for the sake of convenience in (a).
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remaining cladding and overlay waveguide we should expect that there are definite conditions depending on the overlay waveguide parameters and thickness of cladding at which practically the only one mode with maximum index will be excited, and that mode will have a spatial distribution similar to one of the fiber mode in the fiber region. However there are some very narrow conditions (parameters of the overlay waveguide) when the value of the effective refractive index of the composite waveguide is almost equal to the one of the fiber and in that case we can expect the electric field value in the overlay waveguide region is large enough to interact with active or passive elements placed there. For another very narrow conditions we can obtain a regular directional coupler but an equal power distribution between two modes can be achieved only for one set of the system parameters. Now let us consider directional couplers with two fibers sandwiched with an interlayer structure (Fig. 1(b)). The initial mode of the lower fiber will excite modes of the composite waveguide. Fig. 6 represents the calculation results of the powercoupling coefficients for a symmetrical directional coupler with two equivalent fibers separated by intermediate layer made of a cladding material.
Fig. 6. Power-coupling coefficient on the boundary: fiber – multilayer structure in the symmetrical directional coupler: 1 – an even mode, 2 – odd mode. Thickness of the fiber cores is 6:982 lm; refractive index – 1.44989; refractive index of the intermediate layer and substrate is 1.44645.
Parameters of both fibers are the same as in the previous case. Note that the excitation coefficients for even and odd modes get equal only for a large enough distance between waveguides. It means that if the distance between waveguides in the symmetrical directional coupler is small enough, we cannot use normalized field distributions to represent a power exchange between waveguides. In the case of non-symmetrical coupled waveguides a situation looks like the one with the overlay waveguide. In Fig. 1(b) the lower waveguide has the same parameters as former and can have or not cladding. The upper waveguide does not have cladding, and its refractive index is 1.45. Fig. 7 shows dependences of the power-coupling coefficient on the thickness of the intermediate layer with a refractive index of n0 ¼ 1:7 when a thickness of the upper waveguide equals 5 lm (a) and 9 lm (b). Notations are the same as in Fig. 2. Dependencies are very similar to the case of the fiber with the overlay waveguide. However there are essential distinctions. There are two independent modes tied with different waveguides that cannot interact with each other and only one is excited by fiber mode. It is the last mode when df2 ¼ 9 lm (Fig. 8(b)) and last but one mode when df2 ¼ 5 lm (Fig. 8(a)). It is interesting to see that in Fig. 7(b) the maximums are shorter than in Fig. 7(a). It is caused by occurrence of little peaks in the region of the next effective refractive index resonance in Fig. 7(b). Therefore in this case using fibers with smaller core diameters is preferable to obtain better device parameters. When two last modes are excited with equal or almost equal efficiency we have a regular directional coupler and in the narrow range of intermediate layer parameters the power-coupling coefficient can reach almost 50% for each such a mode. Thus as well as in the case of the fiber with overlay waveguide two fibers sandwiched with the intermediate layer behave in such a way that in the wide range of intermediate layer parameters practically the only one mode is excited by fiber mode with high efficiency. But fortunately simultaneously there is one more mode (previous or next in index) tied with another fiber, which is not excited by fiber mode almost at all and can be coupled with the first fiber mode by grating. In the
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Fig. 7. Power-coupling coefficient versus intermediate layer thickness d0 for the non-symmetrical structure; n0 ¼ 1:7, nf2 ¼ 1:45, df2 ¼ 5 lm (a) and df2 ¼ 9 lm (b). Digits are modal numbers. Digits with strokes are for the remaining fiber cladding thickness of 2 lm and ones without strokes for the fiber without cladding.
Fig. 8. Modal electric field distribution in the non-symmetrical structure with an intermediate layer in Fig. 1(b); n0 ¼ 1:7, d0 ¼ 1:4, dc ¼ 2 lm, nf2 ¼ 1:45, df2 ¼ 5 lm (a) and df2 ¼ 9 lm (b). The initial fiber mode is shown for the sake of convenience.
narrow range of parameters the system can work as a regular directional coupler and only in the region near a resonance the system has a value of electric field large enough to provide an effective interaction with active and passive elements placed in the regions of the intermediate layer or second fiber. Really the intermediate layer can be arbitrary complex; it does not change a system behavior.
In the conclusion we have proposed a simple method for the calculation of power-coupling coefficients in the devices comprising the fiber with a partly removed cladding or without it and slab overlay waveguide of any complexity. In the case of directional couplers our method allows to get the initial power distribution between even and odd modes for an exact treatment. Independently of the system has a second fiber or not the first
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fiber mode excites with high efficiency only one mode of the composite waveguide in the wide range of system parameters. But in the case of two fibers with intermediate layers simultaneously there is one more mode (previous or next in index depending on the system parameters) tied with another fiber that is not excited by initial fiber mode and can be coupled with the first fiber mode by grating. In the narrow range of parameters two last modes can be excited with nearly equal 50% efficiency. In this case the system behaves as a regular directional coupler. In the case when only one mode is effectively excited there is one more limitation. Only when the effective refractive index of this mode almost equals to one of the initial fiber modes the value of modal electric field is high enough to produce effective interactions with active and passive elements in the region of overlay waveguide. Therefore such devices have to be very sensitive to the overlay waveguide parameters. The power-coupling coefficient also significantly depends on the thickness of remaining fiber cladding. Our calculations have showed that the opti-
mal value of remaining fiber cladding is about 1.5–2 lm.
References [1] K.-T. Kim, H.-W. Kwon, J.-W. Song, S. Lee, W.-G. Jung, S.-W. Kang, Opt. Commun. 180 (2000) 37. [2] K. McCallion, W. Johnstone, G. Thursby, Electron. Lett. 28 (1992) 410. [3] C.A. Millar, M.C. Brierly, S.R. Mallinson, Opt. Lett. 12 (1987) 284. [4] D. Marcuse, Theory of Dielectric Optical Waveguides, Academic Press, New York, 1974. [5] A.W. Snyder, J.D. Love, Optical Waveguide Theory, Chapman & Hall, London, 1983. [6] J. Deng, Y. Huang, J. Lightwave Technol. 16 (1998) 1062. [7] V. Kutsaenko, E. Lavretskii, W. Johnstone, J. Lightwave Technol. 14 (1996) 2294. [8] D. Marcuse, J. Lightwave Technol. 5 (1987) 113. [9] E.A.J. Marcatili, Bell. Syst. Tech. J. 48 (2071) 1969. [10] A. Sharma, J. Kompella, P.K. Mishra, J. Lightwave Technol. 8 (1990) 143. [11] A. Yariv, Quantum Electronics, second ed., Wiley, New York, 1975.
Optics Communications 204 (2002) 171–178 www.elsevier.com/locate/optcom
Power coupling between fiber and multimode slab composite waveguide Valeri V. Kapoustine a,*, Sergey O. Yarovikov b a
KA&V Company, 1591 Riverside Dr., Suite 204, Ottawa, Ont., Canada K1G 4A7 b Ulyanovsk State University, 42 L. Tolstoi St., Ulyanovsk 432700, Russia Received 26 September 2001
Abstract We discuss power coupling in devices composed of a fiber with partly or completely removed cladding and multimode slab overlay waveguide of any complexity. A fiber mode excites with a high efficiency only one mode of the composite structure in the wide range of system parameters. But only when the effective refractive index of this mode almost equals to the one of the initial fiber mode the modal value of electric field is high enough to produce effective interactions with active and passive elements in the region of the overlay waveguide. In the narrow range of system parameters last two modes are excited with a nearly equal efficiency providing a regular directional coupler. The powercoupling coefficient essentially depends on the thickness of remaining fiber cladding and an optimal value of this cladding is about 1.5–2 lm. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 42.82.Et Keywords: Optical coupler; Fiber; Composite planar waveguide; Power-coupling coefficient
Fiber-to-planar waveguide couplers find their applications in different kinds of optoelectronic devices such as polarizers [1], switchers [2], channel dropping filters [3], and others, and the interest in these structures keeps increasing due to their simplicity, low insertion loss, and high mechanical stability. The basic theoretical approach to investigate parameters of these devices is a coupling mode theory [4,5] when only one mode of the
*
Corresponding author. Tel.: +1-613-738-5177; fax: +1-613738-5177. E-mail address: [email protected] (V.V. Kapoustine).
overlay planar waveguide is considered to match the fiber mode. But these structures are usually multimode [6,7], and even if we have only twomode composite system we need to assume that even and odd modes are excited with an equal efficiency [8] that is not obvious. In this work we propose a simple method to calculate the coefficients of an excitation of all modes of the multimode composite structure by fiber mode on the boundary: fiber–fiber with overlay waveguide. The accuracy of this method is the same as for an exact treatment of directional couplers by computing the compound modes [9] and it can be very useful to calculate the complex optoelectronic device
0030-4018/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 ( 0 2 ) 0 1 2 4 7 - 6
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parameters. In many simple cases it can compete with 2D beam propagation method. We will consider our method for two examples shown in Fig. 1, a fiber core with remaining cladding and overlay planar waveguide with a varying thickness and refractive index (Fig. 1(a)), and two fiber cores, one with remaining claddings (lower), sandwiched with an interlayer with varying parameters (Fig. 1(b)). For the sake of simplicity fibers have been substituted by equivalent planar waveguides. Parameters of the equivalent waveguides can be chosen using a variational method assuming that the propagation constant and a fiber mode field distribution are matched to those of the appropriate planar waveguides [10]. A region of a light transmission from the fiber into the multilayer composite waveguide can be considered as a sharp boundary between them to calculate power-coupling coefficients. If field distributions of all compound modes in the composite slab waveguide and equivalent fiber are normalized
Fig. 1. Structures for an investigation: (a) an equivalent fiber with an overlay waveguide; (b) two fibers or a fiber and a waveguide with an intermediate layer, the lower fiber has remaining cladding; df ¼ df1 ¼ 6:982 lm, df2 ¼ 5 or 9 lm; nc ¼ 1:44645, nf ¼ nf1 ¼ 1:44989, nf2 ¼ 1:45, n0 ¼ 1:5 or 1.7.
on the unit power per unit length from the equality of transverse electric fields on the boundary between fiber and composite waveguide for TE-modes X Ef ¼ a n En n
with neglect of a reflection on this boundary one can obtain the expression for power-coupling coefficients (power per unit length in nth mode of the composite structure) in the form: Z 2 bf bn bf 1 2 Pn ¼ jan j ¼ Ef En dx ; ð1Þ 2 bn ð2cl0 Þ 1 where Ef and En are normalized electric fields of the equivalent fiber mode and nth waveguide compound mode, respectively, bf , bn are normalized effective refractive indices of fiber and nth waveguide mode, c is the speed of light in free space, l0 is the free space permeability, and an are coefficients. Since we take into account only bounded waveguide modes, the system of orthonormal mode functions is usually not complete and the computed total power in all bounded modes of the composite waveguide is usually less than unit. In the case of TM-modes a similar approach gives the next expression for the power-coupling coefficients: 2 Z b b 1 Hf Hn Pn ¼ n f 2 dx : ð10 Þ eðxÞ ð2ce0 Þ 1 Here Hf and Hn are normalized on unit power per unit length transverse magnetic fields of the fiber mode and nth waveguide mode, respectively, e0 is the free space permittivity, and eðxÞ is a transverse dependence of dielectric permittivity of the composite waveguide. The transverse matrix method was used for the calculation of propagation constants for all possible compound modes in the described waveguides. Only TE-modes were taken into account since in the absence of an anisotropy the results for TM-modes would be mostly the same. The dispersion equation for TE-modes can be written in the following way: M21 ðM22 þ M11 Þðn2c b2 Þ1=2 þ M12 ðn2c b2 Þ ¼ 0; ð2Þ
V.V. Kapoustine, S.O. Yarovikov / Optics Communications 204 (2002) 171–178
where b is a normalized propagation constant of an appropriate compound mode, nc is a refractive index of cladding and substrate, and Mij are the elements of the total transfer matrix of the composite multilayer planar structure. The elements of the transmission matrix for the kth layer of thickness lk and refractive index nk are represented by formulae: m11 ¼ m22 ¼ cosðgk lk Þ; ik0 m12 ¼ sinðgk lk Þ; gk gk m21 ¼ sinðgk lk Þ: ik0
ð3Þ
Here a transverse propagation constant equals 1=2 gk ¼ k0 ðn2k b2 Þ , k0 ¼ 2p=k, and k is the free space wavelength. Eq. (2) with (3) was solved numerically for different system parameters. A modal electric field distribution was calculated for each compound mode as a sum of hyperbolic sine and cosine coordinate functions with coefficients getting from the join of solutions in the adjacent layers on the boundaries and normalized on unit power per unit length [11]. Numerical checking of the modes orthogonality for the composite waveguide has shown that overlap integrals between all couples of modes have not exceeded 2 104 in all cases of interest. The field
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distribution was as well calculated and normalized for equivalent fiber to obtain the mode excitation coefficient (1) on the boundary fiber–composite slab waveguide. Thickness and refractive indices of the slab waveguide equivalent to the lower fiber in Fig. 1 were taken from [7] getting a thickness of 6:982 lm and refractive indices 1.44989 and 1.44645 for the waveguide core and cladding, respectively. These parameters correspond to a real fiber with a core diameter of 8 lm and refractive indices of the core and cladding, 1.451 and 1.447, respectively, at k ¼ 1:3 lm and will be used in all subsequent calculations. Fig. 2 shows dependences of the power-coupling coefficient on the thickness of the overlay waveguide d0 (Fig. 1(a)) in the case when the refractive index of that waveguide equals n0 ¼ 1:7 (a) and 1.5 (b). Wavelength here and thereinafter is 1:3 lm. Digits in figures denote modal numbers, and digits with and without strokes correspond to the cases when a thickness of the remaining fiber cladding equals 2 lm or 0, respectively. Fig. 3(a) represents dependencies of modal effective refractive indices on the overlay waveguide thickness when n0 ¼ 1:5. One can see from Figs. 2(b) and 3(a) that powercoupling coefficient essentially depends on the thicknesses of the overlay waveguide and remaining
Fig. 2. Power-coupling coefficient for different compound modes versus overlay waveguide thickness; n0 ¼ 1:7 (a) and n0 ¼ 1:5 (b). Digits denote modal numbers. Digits with a stroke are for the thickness of the remaining fiber cladding 2 lm and ones without stroke are for the fiber without cladding.
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Fig. 3. Dependencies of the modal effective refractive index (a) and parameters of 2nd mode (b) on the overlay waveguide thickness; n0 ¼ 1:5, dc ¼ 2 lm. Digits in (a) correspond to numbers of modes. In (b) 1 – reduced in 10 times values of electric field E, normalized on unity, on the upper boundary in Fig. 1(a), 2 – reflection coefficient R of a grating placed on the upper boundary, 3 – power-coupling coefficient P, 4 – value of a product of the reflection coefficient by power-coupling coefficient. Grating parameters are given in the text.
cladding and approaches a maximum value for a given compound mode when it has almost the same effective refractive index as the fiber mode. In Fig. 3(b) some parameters of second mode are shown versus overlay waveguide thickness to understand mechanisms of wave propagation through the composite waveguide. Since such devices intend for a light interaction with active and passive elements placed in the region of the overlay waveguide curve 1 shows a behavior of modal electric field, normalized on unity, on the upper boundary in Fig. 1(a), where for example a grating can be arranged. Electric field values do not correspond to the dependence of the power-coupling coefficient for a given mode (curve 3). Where the electric field has maximums the power-coupling coefficient has minimums and vice versa. Curve 2 models a behavior of the reflection coefficient of a grating placed on the upper boundary in Fig. 1(a). A grating length was chosen 5 mm with a groove 0:3 lm and a period met a resonance condition for a given mode and given overlay waveguide thickness. Electric field distributions used for reflection coefficient calculations were normalized on the unit power per unit length [10]. Therefore a real value of the reflected power can be defined as a product of the reflection coefficient by power-coupling coefficient (curve 4). When n0 ¼ 1:5 as one can see from
Fig. 3(b) the maximum reflection can be realized at d0 ¼ 3:42 and 2:02 lm. Note that the maximum at the larger overlay waveguide thickness is higher but does not exceed 90%. Larger values of the reflection can be achieved with larger values of n0 (Fig. 4).
Fig. 4. Dependence of the value (reduced in five times) of electric field E normalized on the power-coupling coefficient on the upper boundary in Fig. 1(a) (1), and value of a product of the power-coupling coefficient P by reflection coefficient R of the grating placed on the upper boundary (2,3,4) on the remaining fiber cladding thickness for 2nd mode. 1,2 – d0 ¼ 3:3 lm and n0 ¼ 1:5; 3 – d0 ¼ 3:4 lm, n0 ¼ 1:5; 4 – d0 ¼ 1:44 lm, n0 ¼ 1:7. Grating parameters are given in the text.
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When the thickness of remaining cladding increases the power-coupling coefficient P rises tending to the unity, but in this case the electric field value E on the upper boundary of the system in Fig. 1(a) obviously decreases. So, the optimal value of the cladding thickness depends on an application, and in Fig. 4 the dependences of the product of the power-coupling coefficient by reflection coefficient of the grating placed on the upper boundary in Fig. 1(a) are shown as well as the electric field values in the grating region versus the cladding thickness for the second mode. As one can see from Fig. 4 the optimal cladding thickness is about 1.5–2 lm in our case and we can obtain more than 97% of the initial fiber power in the reflection if n0 ¼ 1:7. All other modes practically are not excited by fiber mode and therefore do not participate in the reflection process. The full-width at half maximum (FWHM) of the reflected signal dependence on the wavelength is about 0.5 nm. Note that in the case of a coincidence of effective refractive indices of the fiber mode and composite waveguide mode (at d0 ¼ 3:3 lm, n0 ¼ 1:5) the maximum value of the reflection coefficient is less in comparison with one obtained at d0 ¼ 3:4 lm, and therefore this coincidence does not play a significant role in the reflection though we can say that the maximum reflection always occurs at the sys-
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tem parameters almost but not exactly corresponding to the coincidence of effective refractive indices of the fiber mode and composite waveguide mode. When the thickness of the overlay waveguide varies the modal field distributions vary significantly as well (Fig. 5). In the resonance condition, when the effective refractive index of the compound mode is equal to the one of the fiber, only one mode is excited with high efficiency (Fig. 5(a)). As a rule the only one mode with highest index has the maximum efficiency of an excitation with the exception of a small range of overlay waveguide thicknesses when two modes are excited simultaneously (Fig. 2). At some value of the overlay waveguide thickness we can obtain almost 50% excitation of each of two modes (Fig. 5(b), curves 1 and 2). At n0 ¼ 1:5, d0 ¼ 3:567 lm. In that case we have a regular directional coupler and values of effective refractive indices of these two modes can be used for a definition of the power exchange length [8]. Sum and difference of these two modal distributions show a power rearrangement between two waveguides on the power exchange length (Fig. 5(b), curves 3 and 4). So, in the case when the fiber mode propagates through a boundary between the fiber and composite waveguide comprising the fiber with some
Fig. 5. Modal electric field distribution of the composite multilayer waveguide in Fig. 1(a). Thickness of the remaining fiber cladding 2 lm, refractive index of the overlay waveguide n0 ¼ 1:5, thickness d0 ¼ 3:3 lm (a) and 3:567 lm (b). In (b) 1,2 – even and odd modes of a directional coupler, 3,4 – sum of these modes and difference between them, respectively. The initial fiber mode is shown for the sake of convenience in (a).
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remaining cladding and overlay waveguide we should expect that there are definite conditions depending on the overlay waveguide parameters and thickness of cladding at which practically the only one mode with maximum index will be excited, and that mode will have a spatial distribution similar to one of the fiber mode in the fiber region. However there are some very narrow conditions (parameters of the overlay waveguide) when the value of the effective refractive index of the composite waveguide is almost equal to the one of the fiber and in that case we can expect the electric field value in the overlay waveguide region is large enough to interact with active or passive elements placed there. For another very narrow conditions we can obtain a regular directional coupler but an equal power distribution between two modes can be achieved only for one set of the system parameters. Now let us consider directional couplers with two fibers sandwiched with an interlayer structure (Fig. 1(b)). The initial mode of the lower fiber will excite modes of the composite waveguide. Fig. 6 represents the calculation results of the powercoupling coefficients for a symmetrical directional coupler with two equivalent fibers separated by intermediate layer made of a cladding material.
Fig. 6. Power-coupling coefficient on the boundary: fiber – multilayer structure in the symmetrical directional coupler: 1 – an even mode, 2 – odd mode. Thickness of the fiber cores is 6:982 lm; refractive index – 1.44989; refractive index of the intermediate layer and substrate is 1.44645.
Parameters of both fibers are the same as in the previous case. Note that the excitation coefficients for even and odd modes get equal only for a large enough distance between waveguides. It means that if the distance between waveguides in the symmetrical directional coupler is small enough, we cannot use normalized field distributions to represent a power exchange between waveguides. In the case of non-symmetrical coupled waveguides a situation looks like the one with the overlay waveguide. In Fig. 1(b) the lower waveguide has the same parameters as former and can have or not cladding. The upper waveguide does not have cladding, and its refractive index is 1.45. Fig. 7 shows dependences of the power-coupling coefficient on the thickness of the intermediate layer with a refractive index of n0 ¼ 1:7 when a thickness of the upper waveguide equals 5 lm (a) and 9 lm (b). Notations are the same as in Fig. 2. Dependencies are very similar to the case of the fiber with the overlay waveguide. However there are essential distinctions. There are two independent modes tied with different waveguides that cannot interact with each other and only one is excited by fiber mode. It is the last mode when df2 ¼ 9 lm (Fig. 8(b)) and last but one mode when df2 ¼ 5 lm (Fig. 8(a)). It is interesting to see that in Fig. 7(b) the maximums are shorter than in Fig. 7(a). It is caused by occurrence of little peaks in the region of the next effective refractive index resonance in Fig. 7(b). Therefore in this case using fibers with smaller core diameters is preferable to obtain better device parameters. When two last modes are excited with equal or almost equal efficiency we have a regular directional coupler and in the narrow range of intermediate layer parameters the power-coupling coefficient can reach almost 50% for each such a mode. Thus as well as in the case of the fiber with overlay waveguide two fibers sandwiched with the intermediate layer behave in such a way that in the wide range of intermediate layer parameters practically the only one mode is excited by fiber mode with high efficiency. But fortunately simultaneously there is one more mode (previous or next in index) tied with another fiber, which is not excited by fiber mode almost at all and can be coupled with the first fiber mode by grating. In the
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Fig. 7. Power-coupling coefficient versus intermediate layer thickness d0 for the non-symmetrical structure; n0 ¼ 1:7, nf2 ¼ 1:45, df2 ¼ 5 lm (a) and df2 ¼ 9 lm (b). Digits are modal numbers. Digits with strokes are for the remaining fiber cladding thickness of 2 lm and ones without strokes for the fiber without cladding.
Fig. 8. Modal electric field distribution in the non-symmetrical structure with an intermediate layer in Fig. 1(b); n0 ¼ 1:7, d0 ¼ 1:4, dc ¼ 2 lm, nf2 ¼ 1:45, df2 ¼ 5 lm (a) and df2 ¼ 9 lm (b). The initial fiber mode is shown for the sake of convenience.
narrow range of parameters the system can work as a regular directional coupler and only in the region near a resonance the system has a value of electric field large enough to provide an effective interaction with active and passive elements placed in the regions of the intermediate layer or second fiber. Really the intermediate layer can be arbitrary complex; it does not change a system behavior.
In the conclusion we have proposed a simple method for the calculation of power-coupling coefficients in the devices comprising the fiber with a partly removed cladding or without it and slab overlay waveguide of any complexity. In the case of directional couplers our method allows to get the initial power distribution between even and odd modes for an exact treatment. Independently of the system has a second fiber or not the first
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fiber mode excites with high efficiency only one mode of the composite waveguide in the wide range of system parameters. But in the case of two fibers with intermediate layers simultaneously there is one more mode (previous or next in index depending on the system parameters) tied with another fiber that is not excited by initial fiber mode and can be coupled with the first fiber mode by grating. In the narrow range of parameters two last modes can be excited with nearly equal 50% efficiency. In this case the system behaves as a regular directional coupler. In the case when only one mode is effectively excited there is one more limitation. Only when the effective refractive index of this mode almost equals to one of the initial fiber modes the value of modal electric field is high enough to produce effective interactions with active and passive elements in the region of overlay waveguide. Therefore such devices have to be very sensitive to the overlay waveguide parameters. The power-coupling coefficient also significantly depends on the thickness of remaining fiber cladding. Our calculations have showed that the opti-
mal value of remaining fiber cladding is about 1.5–2 lm.
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