Abstract:
Disclosed is an article that comprises an optical waveguide mode converter for converting light of wavelength λ in a few-moded optical waveguide from a given guided mode (e.g., LP 01,f ) to another guided mode (e.g., LP 02,b ). The converter comprises a tilted refractive index grating in the core of the waveguide. Appropriate choice of the refractive index profile n(r), photosensitivity p(r) and tilt angle θ makes possible substantial nulling of the coupling between some guided modes (e.g., LP 01,f  to LP 01,b  and LP 01,f  to LP 11,b ), and substantial maximization of the coupling between other guided modes (e.g., LP 01,f  to LP 02,b ). Mode converters according to the invention can be advantageously used in optical fiber communication systems in add/drop multiplexers.

Description:
RELATED APPLICATIONS 
     This application is related to concurrently filed U.S. patent application Ser. No. 09/584,071 by T. A. Strasser et al., titled “Article Comprising a Tilted Grating in a Single Mode Waveguide”, filed May 31, 2000, incorporated herein by reference. 
    
    
     FIELD OF THE INVENTION 
     This application pertains to few-moded optical waveguides with a refractive index (Bragg) grating, and to optical communication systems that comprise such waveguides. 
     BACKGROUND 
     Bragg gratings (also referred to as refractive index gratings) in optical waveguides are known. Conventionally such gratings couple a forward-propagating core-guided mode in single mode fiber to the backreflected core mode. 
     Mode conversion gratings are also known. See, for instance, U.S. Pat. Nos. 5,717,798 and 5,740,292. The latter discloses reflective gratings that inter alia couple light in the fundamental mode (LP 01 ) to the LP 11  mode. 
     Mode coupling gratings can find a variety of uses in optical waveguide systems. For instance, they can serve as wavelength routing filters in WDM networks. 
     However, it has been found that reflective grating mode converters that efficiently convert LP 01  radiation to LP 11  radiation frequently are difficult to manufacture, due to the spatial degeneracy of the LP 11  mode. Whereas one of the LP 11 spatial modes typically can be converted to LP 01  by a LP 01  to LP 11  mode converter, the other LP 11  spatial mode typically can not be so converted, due to the interference of two nearby degenerate spatial modes. This spatial degree of freedom is difficult to control. Thus, it would be desirable to have available a mode converter which does not involve coupling to or from a spatially degenerate mode. 
     In particular, there is a need for a mode converter which couples the LP 01  mode to the LP 02  mode, without coupling the LP 01  mode to any other guided mode, e.g., LP 11  and reflected LP 01 . Such an LP 01 -LP 02  mode converter could be used, for instance, in an add/drop multiplexer. This would avoid the need for an expensive and lossy circulator. Circulators are essential in implementing prior art Bragg grating-based add/drop filtering applications. 
     The LP 02  mode is not spatially degenerate and thus can be efficiently converted to LP 01 . However, a LP 01  to LP 02  mode converting reflective grating typically must be designed such that both the LP 01  to LP 11  mode conversion and the LP 01  to LP 01  mode conversion are substantially nulled. The LP 11  mode must exist in the optical fiber since, in order for the LP 01  to LP 02  mode conversion to be strong, the optical fiber must guide the LP 02  mode, and therefore must also guide the LP 11  mode. To the best of our knowledge, the prior art does not provide a technique for making such a reflective LP 01  to LP 02  mode converter. This application inter alia discloses such a mode converter. 
     C. X. Shi, IEEE Journal of Quantum Electronics, Vol. 32(8), August, 1996, page 1360, provides a theoretical treatment of a fiber-optic Fabry-Perot resonator with two mode conversion (LP 01  to LP 02 ) “mirrors”. See also C. X. Shi et al., Optics Letters, Vol. 17(23), page 1655, December 1992; and F. Bilodeau et al., Electronics Letters, Vol. 27(8), page 682, April 1991. 
     M. J. Holmes et al., ECOC &#39;99, Sep. 26-30, 1999, Nice, France, pages I-216-217 disclose a fiber for sidetap filters. The fiber had a non-photosensitive core dopant for normalized radius less than 0.4, a combination of a non-photosensitive core dopant and germania for normalized radius 0.4-1, and a photosensitive cladding doped with germania out to a normalized radius of 3.5, to which boron was added to reduce the cladding index to match the deposition tube. The germania concentrations for the regions 0.4-1.0 and 1.0-3.5 were in the ratio 0.6:1.0 in order to obtain the required relative photosensitivity. The Holmes et al. paper thus discloses fiber in which the core had two different photosensitivity levels, with the cladding also being photosensitive. The photosensitivity profile was chosen to optimize the wavelength dependence of the cladding mode loss spectrum for applications, and not to obtain a mode converter of the herein relevant type. 
     All cited references are incorporated herein by reference. 
     GLOSSARY AND DEFINITIONS 
     For ease of exposition the discussion herein will generally refer to optical fibers. It will be appreciated, however, that similar results are obtainable in other optical waveguides, e.g., in planar waveguides. 
     The “coupling strength” between two guided core modes in a few-moded optical fiber is conventionally expressed in terms of an overlap integral, as shown in equation 2) below. The coupling strength typically depends on the refractive index profile n(r), the photosensitivity profile p(r), and the tilt angle θ. 
     “Minimizing” the coupling between two guided core modes in a given waveguide means adjusting the tilt angle of a tilted grating such that the coupling strength between the two modes is less than −30 dB. 
     “Maximizing” the coupling between two guided core modes in a given waveguide means adjusting the tilt angle of a tilted grating such that the coupling strength between the two modes is at least about −10 dB. 
     By a “regular null” we mean herein a tilt angle region in a tilted (“blazed”) fiber Bragg grating that has a core mode coupling strength for light of a predetermined wavelength that is less than −30 dB over only a small (typically less than 0.1°) angular range of the tilt angle. Regular nulls occur for many tilt angles. 
     By a “super null” we mean two (or possibly more) regular nulls that occur at closely spaced tilt angles, thereby making the core mode coupling at the predetermined wavelength very low (typically less than −30 dB) over a relatively large (more than 0.1°, desirably more than 0.2°, or even 0.5° or more) range of tilt angles between the regular nulls. 
     Modes of the guided light are designated LP mn  in conventional fashion, with m and n being integers. Por instance, LP 01  is the fundamental mode. LP 01,f  refers to the forward propagating fundamental mode, and LP 01,b  refers to the backward propagating fundamental mode. 
     “Photosensitivity” refers to the refractive index change in the waveguide that results if an appropriately doped waveguide is exposed to actinic radiation, typically UV radiation. 
     A “few-moded” optical waveguide supports the fundamental mode and one or more higher order modes, typically no more than about 10 guided modes total. 
     The description of the invention herein is generally in terms of conversion between the fundamental mode and a higher order mode such as LP 02 . This is for the sake of concreteness only, and the invention at least in principle can be embodied in an article for mode conversion between two appropriate higher order modes. 
     SUMMARY OF THE INVENTION 
     In an exemplary mode converter according to the invention, it is necessary that LP 01,f  light be strongly coupled into the LP 02,b  mode. However, in such a mode converter the coupling between LP 01,f  and all other guided reflected modes (exemplarily LP 11,b  and LP 01,b ) has to be small, exemplarily at least 20 dB less than LP 01,f  to LP 02,b  coupling. This simultaneous “nulling” of the coupling between LP 01,f  and the other guided modes (i.e., other than the desired coupling) can not be achieved with optical fiber that has uniform photosensitivity throughout the core, necessitating use of a more complex fiber design, as is described below. 
     The coupling strengths between the various guided modes in an optical fiber depend on the refractive index profile of the fiber and the electric fields of the various modes. Both of these parameters generally are fixed at the time of grating formation, and thus can not be varied to achieve a desired coupling. The only grating parameter which can be changed to significantly alter the relative coupling strengths is the tilt of the grating with respect to the core axis. However, with uniform photosensitivity in the fiber core, the control over the various coupling strengths that is achievable by introduction of a tilt in the grating is limited. In particular, with a uniform radial photosensitivity profile, it is impossible to “null” simultaneously an even-even reflection (e.g., LP 01,f  to LP 01,b ) and an even-odd reflection (eg., LP 01,f  to LP 11,b ). Thus, we have determined that an additional degree of freedom has to be provided. This degree of freedom is the photosensitivity profile of the optical fiber. This profile has at least two distinct levels of photosensitivity in the core (of which one or more can be zero), and may, but need not, have substantially no photosensitivity in the cladding. 
     Thus, by way of example, forming a Bragg grating in an optical fiber wherein photosensitivity is removed (or substantially reduced) in certain regions of the fiber core makes it possible to achieve a much broader range of relative coupling strength as a function of the tilt angle of the fiber than is possible with fiber having uniform photosensitivity in the core. In particular, it is possible to simultaneously null both an even-even (e.g., LP 01,f  to LP 01,b ) and an even-odd (e.g., LP 01,f  to LP 11,b ) reflection, with strong LP 01,f  to LP 02,b  coupling, something that is not possible with a tilted refractive index grating that has uniform photosensitivity in the core. 
     More generally, the invention is embodied in an article that comprises an optical waveguide mode converter for converting light of wavelength λ (exemplarily about 1.5 μm) from a forward-propagating given guided mode to another predetermined guided mode. The mode converter comprises a tilted refractive index grating in the waveguide, the grating having a tilt angle θ with respect to the waveguide axis and extending longitudinally over at least a portion of the waveguide. The waveguide is a few-moded waveguide for light of wavelength λ and has a core and a cladding that contactingly surrounds the core. The fiber has a dopant distribution selected to provide the fiber with a refractive index profile n(r) and a photosensitivity profile p(r), with both profiles being functions of the radial coordinate r of the waveguide. 
     The mode converter has two or more non-zero coupling strengths among core guided modes, and p(r) has at least two different levels of photosensitivity in the core. Furthermore, n(r), p(r) and θ are selected such that more than one of said non-zero coupling strengths are simultaneously nulled. For instance, n(r), p(r) and θ are selected such that a given guided mode (e.g., LP 01,f ) is nulled with at least one other guided mode (e.g., LP 01,b  and LP 11,b ), and is strongly coupled to at least one other guided mode (e.g., LP 02,b ). In a preferred embodiment of the invention, the fundamental mode LP 01,f  is nulled simultaneously with an even and an odd backward-propagating guided mode (LP 01,b , and LP 11,b , respectively), and LP 01,f  is simultaneously strongly coupled to a higher order even mode (e.g., LP 02,b ). 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIGS. 1 a  and  1   b  schematically show a refractive index profile and a photosensitivity profile for an exemplary fiber according to the invention; 
     FIGS. 2 a - 2   c  schematically show the index profile, photosensitivity profile and mode electric fields of a fiber according to the invention; 
     FIG. 3 shows various coupling constants as a function of tilt angle; 
     FIGS. 4 a - 4   b  schematically depict an exemplary photosensitivity profile and refractive index profile, respectively; and 
     FIG. 5 schematically shows an optical fiber communication system comprising a mode converter according to the invention. 
    
    
     DETAILED DESCRIPTION 
     Below we provide mathematical expressions that can be used to determine a photosensitivity profile that at least approximately provides the desired coupling strengths. If desired, optimized results can then be obtained by, typically minor, variation of the tilt angle, or possibly of the photosensitivity profile. Trimming of the photosensitivity profile by UV exposure can also be used for optimization. 
     The coupling between a first and a second guided mode (designated LP mn  and LP pq ) in an optical waveguide depends on the coupling strength κ, which is proportional to the following θ-dependent integral 
     
       
         κ mn-pq  (θ)=∫E mn E pq  H(r)rdr,  1) 
       
     
     where H(r) depends on the mode indices and on the grating tilt angle (see, for instance, T. Erdogan et al., “Tilted Fiber Phase Gratings”, J. Optical Soc. America, A. Vol. 13(2), pages 296-313, 1996), incorporated herein by reference. 
     For instance, if an optical fiber supports the LP 01  mode as well as the LP 11  mode then a tilted grating will couple the LP 01  mode to the LP 11  mode with a strength that depends on the LP 01  to LP 11  overlap integral. That is to say: 
     
       
         κ 01-11  (θ)=∫E 01 E 11 J 1  (K grating r sin θ) W(r)rdr,  2) 
       
     
     where E 01  and E 11  are the radially dependent electric field amplitude (normalized to unity, i.e., ∫ 0   ∞ E 2   01  rdr=1) of the LP 01  and LP 11  modes, K grating  is the wave vector of the grating (K gating =2π/Δ grating ) θ is the tilt angle of the grating with respect to the fiber axis, and W(r) is a radially dependent weighting function which expresses the radial variations of p(r), the photosensitivity profile of the grating. The Bessel function J 1  arises from the azimuthal integration and is zero when θ=0, since the LP 11  mode is odd and the LP 01  mode is even. 
     The weighting function W(r) can be defined via the full index modulation of the tilted grating, namely 
     
       
         δn(r,Φ,z)=δn W(r) exp[(iK grating )(sin θ r cos Φ+cos θz)].  3) 
       
     
     In equation 3, Φ is the azimuthal angle in cylindrical coordinates and δn is the amplitude of the index modulation. In a uniformly photosensitive fiber, W(r) is the same as the index profile n(r) and is unity up to the core radius. However, herein we consider fibers in which W(r) is not uniform and may or may not have the same radial dependence as the index profile n(r). 
     The above expressions can be used to determine the tilt angle θ that yields the desired coupling between two given guided modes, for a selected photosensitivity profile. If the mathematically determined value of θ does not directly yield the desired coupling strength then a minor amount of routine experimentation will typically suffice to determine a corrected tilt angle that yields the desired coupling, e.g., that nulls the coupling between the modes. After determination of the tilt angle that provides the desired coupling strengths, a grating having the tilt angle and a desired length and strength is manufactured in conventional manner. 
     In order to achieve efficient mode conversion between two predetermined guided modes in a few-moded optical fiber it is typically necessary to substantially null all couplings except the mode conversion coupling, and substantially maximize the mode conversion coupling. By way of example, if the fiber supports LP 01 , LP 11  and LP 02 , and does not support any other higher order modes (e.g., LP 21 ), and if the desired mode conversion is the LP 01,f  to LP 02,b  mode conversion, then the LP 01,f  to LP 01,b  coupling strength and the LP 01,f  to LP 11,b  coupling strength desirably are nulled, and the LP 01,b  to LP 02,b  coupling strength desirably is maximized. 
     For the sake of clarity the description below is for a LP 01  to LP 02  mode converter in a three-moded optical fiber. The approach can be extended to gratings in higher-moded optical fibers, and to coupling between any two spatial modes. 
     If a fiber supports an LP 01  and LP 11  mode then a tilted grating in the fiber will couple the LP 01,f  mode to the LP 11,b  mode with a strength that depends on the LP 01 -LP 11  overlap integral. See equation 2 above. Analogous statements can be made about LP 01,f  to LP 01,b  coupling and LP 01,f  to LP 02,b  coupling. By appropriate choice of the photosensitivity profile p(r) of the fiber it is possible to simultaneously null the LP 01,f  to LP 01,b  coupling and the LP 01,f  to LP 11,b  coupling, and to obtain large LP 01,f  to LP 02,b  coupling. 
     In order to null both the LP 01  to LP 11  and the LP 01  to LP 01  couplings at the same value of tilt angle θ, the photosensitivity profile p(r) must be appropriately selected. Simultaneous nulling is achieved when the photosensitivity is removed (or substantially lowered) over a radial range such that in the two coupling strength integrals the integrands are both positive in one of the regions and both negative in the other region, and both cancel each other in the total integral. Alternately, the simultaneous LP 01  to LP 01  and LP 01  to LP 11  nulling can be understood as the result of formation of a “supernull” for the LP 01 -LP 01  coupling in which two regular nulls come close together for some value of tilt angle. This large angular range can then be made to overlap the LP 01 -LP 11  angular null. 
     FIG. 1 a  schematically shows the refractive index profile of an exemplary three-moded fiber according to the invention, and FIG. 1 b  schematically shows the photosensitivity profile of the fiber. In FIG. 1 a , no refers to the refractive index of silica. The innermost core region  11  is doped with Ge and Al, making the region partly photosensitive. The intermediate core region  12  is doped with Al, making it non-photosensitive, and the outermost core region  13  is doped with Ge, making it strongly photosensitive. See FIG. 1 b , wherein the three core regions are designated  14 - 16 , respectively. 
     FIGS. 2 a-b  schematically show the refractive index profile and photosensitivity profile, and FIG. 2 c  shows the electric field strengths of LP 01  (ref. numeral  21 ), LP 02  (ref. numeral  22 ) and LP 11  (ref. numeral  23 ), respectively. 
     FIG. 3 shows the computed values of various coupling strengths as a function of tilt angle, for the fiber of FIGS. 2 a-c . As can readily be seen from FIG. 3, at θ˜6.5° the LP 01,f  to LP 01,b  coupling strength  31  has a “super null”, and the LP 01,f  to LP 11,b  coupling strength  32  has a regular null which overlaps the LP 01  to LP 01  supernull. At the same tilt angle, the LP 01,f  to LP 02,b  coupling strength  33  has very nearly a maximum, thereby facilitating efficient mode conversion. 
     It will be appreciated that practice of the instant invention is not limited to the photosensitivity profile specifically disclosed and is also not limited to LP 01,f  to LP 02,b  mode converters. Few-moded optical fibers are known and do not require further discussion. 
     Independent manipulation of the refractive index profile and photosensitivity profile of a few-moded optical waveguide is not limited to the above-described particular embodiment but can be applied in a more general design procedure to simultaneously null several higher order mode couplings. 
     As an example of this general technique, the photosensitivity may be set at different levels p(r) in different annular regions  0  to r 1 , r 1  to r 2 , etc., as exemplified by FIG. 4 a . The refractive index n(r) may likewise be set at different values in a separate set of annular regions  0  to r 4 , r 4 -r 5 , etc. The desired set of mode overlap integrals may then be calculated through a known mathematical optimization procedure. The procedure involves minimizing the unwanted couplings (and maximizing the desired coupling strengths) as a function of the several variables that define the fiber, namely the radii defining the photosensitivity and refractive index profiles, the photosensitivity levels, the refractive index levels, and the tilt angle. 
     FIGS. 4 a  and  4   b  schematically depict an exemplary photosensitivity and refractive index profile of a few-mode fiber. The variables are p 1 , p 2  and p 3 ; r 1 , r 2  . . . r 6 ; n 1 , n 2  and n 3 ; and tilt angle θ. The optimization procedure involves evaluation of overlap integrals, substantially as shown above. The optimization procedure is directed towards minimization (nulling) of predetermined coupling strengths (e.g., κ 01-01 , κ 01-11 , κ 01-02 , κ 11-11  and κ 02-02 ), and maximizing another predetermined coupling strength, e.g., κ 01-02 . As another example, in a three moded fiber, LP 01  is nulled with LP 02  and LP 11 . By way of further example, in a few-moded fiber (more than 3 guided modes) LP 01  is nulled with all guided modes except one, or LP 01  is nulled with all guided modes. 
     Mode converters as described above can find a variety of uses in an optical fiber communication system. FIG. 5 schematically depicts an exemplary fiber optic communication system  50  wherein numeral  51  refers to a WDM transmitter,  52  refers to optical transmission fiber,  53  and  54  refer to demultiplexers, and  55  to  57  light of wavelengths λ 1 , λ 2  . . . , refer to receivers. Fiber  52  guides only the fundamental mode LP 01 , to the first de-multiplexer  53 , which comprises a mode converter according to the invention. A channel (e.g., λ 1 ) is converted into LP 02 , dropped from the signal stream and received by receiver  55 . De-multiplexer  54  similarly drops channel λ 2  which is detected by receiver  56 . Other channels are dropped in similar manner, until only one channel (e.g., λ n ) remains and is detected by receiver  57 .