Optical filter with harmonic elements

Optical interference filters are designed with harmonic elements that can be related to Fourier series approximations of desired filter responses. The harmonic elements can be fashioned as individual waveguides of an array arranged in various formats including planar or concentric geometries. The expansion coefficients of the Fourier series correspond to normalized power distributions among the waveguides, and the harmonic components of the series correspond to incremental optical path length differences between the waveguides.

FIELD OF THE INVENTION
 The invention relates generally to the field of optical communications, and
 more particularly to devices and methods employing optical interference
 phenomena for differentially attenuating selected wavelengths of light to
 provide filtering for, e.g., gain flattening, as well as dispersion
 compensation and for other wavelength management purposes.
 BACKGROUND
 Interference filters rely on constructive and destructive interference to
 provide and shape filter responses. The interference is created by
 overlapping different phase-shifted portions of the same beam. The beam
 divisions are generally overlapping portions separated only by time (i.e.,
 phase delay). However, the beams can also be divided spatially into
 transverse sections subject to different phase delays, which must be
 recombined to produce the required overlap.
 Examples of interference filters with temporally separated beam portions
 include dielectric filters, Fabry-Perot etalons, Bragg gratings, long
 period gratings, and micro-optic devices. Most of these filters,
 particularly Fabry-Perot etalons and long period gratings, have limited
 response profiles and must be concatenated to produce more complex
 response profiles. For example, simple response profiles, such as Gaussian
 profiles, can be combined by conventional curve fitting techniques to
 approximate the desired response profiles, however, the multiple filter
 components can be cumbersome to assemble and are subject to both
 fabrication and assembly errors.
 Co-assigned U.S. Pat. No. 5,841,583 to Bhagavatula entitled "Multi-path
 Interference Filter" discloses examples of interference filters with
 spatially separated beam portions. Complex response profiles can be
 supported by the multi-path filters, but fitting the filters' performances
 to desired response profiles is more difficult.
 SUMMARY OF THE INVENTION
 The present invention is directed to optical interference filters and
 filtering methods. It promotes a simplified design of optical interference
 filters for approximating a wide range of spectral responses. The filters
 include a series of harmonically related elements that can be combined to
 produce more complex periodic spectral responses. The harmonically related
 elements correspond to beam divisions distinguished by phase angles and
 optical power.
 Many desired spectral response functions can be approximated by a series of
 harmonic functions (such as a Fourier series), and such a representation
 can be directly related to the physical design of an interference filter.
 According to an embodiment of the invention, the filter divides a beam of
 light into beam portions that traverse different optical paths lengths and
 that subsequently interfere with each other. Optical energy of the beam is
 apportioned between the beam portions in accordance with relative
 magnitudes of the series of harmonic functions, and the different optical
 path lengths are equated to periods of the harmonic functions.
 In a preferred aspect of the invention, the filter has an array format
 composed of a series of individual waveguides, but other formats for
 dividing beams either spatially or temporally can also be directly related
 to such harmonic series. A Fourier series analysis of the desired response
 function, for example, can be converted directly into the physical
 characteristics of the filter array required to achieve the Fourier series
 approximation. The coefficients of the Fourier series convert directly
 into divisions of optical power among the waveguides, which can be
 accomplished by adjusting the position or the relative size of the
 waveguides in the beam field. The phase angle terms of the Fourier series
 representing integer multiples of a fundamental frequency convert directly
 into relative differences in the optical path lengths of the waveguides,
 which can be accomplished by adjusting the physical path lengths or the
 propagation constants of the waveguides.
 In another embodiment, an interference filtering system provides a desired
 spectral response through beam division and apportionment and subsequent
 selective interference.
 A further embodiment describes a phased-array interference filter for
 modifying the spectral characteristics of a multi-wavelength light beam,
 and includes a waveguide array having a plurality of optical waveguides
 with different optical path lengths, at least one optical coupler
 connecting the waveguide array to input and output waveguides wherein the
 waveguides of the array are sized and positioned in relation to each other
 for conveying unequal portions of the total optical power of the beam, and
 further wherein the waveguides of the array have optical path lengths that
 differ from one another by a multiple of a given optical path length
 difference.
 Another embodiment of the invention describes a method of designing an
 optical interference filter to approximate a desired spectral response. An
 aspect of this embodiment involves representing the desired spectral
 response by a series of harmonic functions, converting magnitudes of the
 harmonic functions into apportionments of power among optical pathways
 through the filter, and converting periods of the harmonic functions into
 optical path length differences among the optical pathways.

DETAILED DESCRIPTION OF A PREFERRED EMBODIMENT
 FIG. 1 shows a desired spectral response curve 1. This curve is
 approximated by a plot 12 of a Fourier series, which can take the
 following form:
 ##EQU1##
 where "F(.lambda.)" is a function of beam power attenuation over a domain
 of wavelengths ".lambda.", "a.sub.0 and a.sub.k " are the Fourier
 expansion coefficients, "j" is an imaginary number "-1", "k" is an integer
 that distinguishes the series terms, and ".omega." is equal to an
 expression 2.pi./T with "T" as the overall period of the function. The
 curve 12 covers one entire period.
 FIG. 2 shows a series of curves 14a-14f which depict the harmonic content
 of the Fourier series approximation 12 of FIG. 1. Each of the series terms
 contributes a simple harmonic function plotted as individual curves
 14a-14f, which combine to produce the series approximation 12. Those of
 skill in this art are well aware of the curve-fitting possibilities of
 such Fourier series and can readily fit them where appropriate to many
 other spectral response curves (e.g., those satisfying Dirichlet
 conditions).
 I have found a remarkable correspondence between such Fourier series
 representations of spectral response curves and the physical attributes of
 interference filters. The correspondence can be readily demonstrated with
 reference to FIG. 3 by a new planar filter array 20. The filter array 20
 is symmetric with respect to a vertical line (not shown) centered between
 components 26 and 28, and includes an input waveguide 22, an output
 waveguide 24, and two end-by-end optical couplers 26 and 28 connecting the
 input and output waveguides 22 and 24 to opposite ends of array waveguides
 30. Phase adjusting regions 32 and 36 of the array waveguides 30
 compensate for any inadvertent phase shifting between the array waveguides
 30 caused by the couplings 26 and 28, and a phase shifting region 34 of
 the array waveguides 30 provides intentional phase shifting between the
 waveguides 30 in accordance with a predetermined multiple of an optical
 path length difference. Such a filter 20 can be manufactured similar to
 conventional phased arrays used for purposes of multiplexing and
 demultiplexing.
 A field "F.sub.out " exiting the filter array 20 can be described by a
 product of four matrices "M1, M2, M3, M4". The matrix "M1" describes an
 input field, the matrix "M2" describes a field exiting the coupler 26 and
 the first phase adjusting region 32, the matrix "M3" describes a field
 exiting a phase shifting region 34 and a second phase adjusting region 36,
 and matrix "M4" describes a field exiting the coupler 28. The matrices are
 expressed as follows:
 ##EQU2##
 where ".PHI..sub.m " is equal to the expression ".phi..sub.0 +m j .beta.
 .DELTA.z"; and
EQU M4=M2
EQU F.sub.out =M4.times.M3.times.M2.times.M1
 Only the first element of this matrix multiplication is non-zero. The
 expanded expression is given by:
EQU F.sub.out =-.alpha..sub.0 exp[j.phi..sub.0 ]-.alpha..sub.1 exp[j.phi..sub.0
 +j.beta..DELTA.Z]-.beta..sub.2 exp[j.phi..sub.0 +2j.beta..DELTA.Z] . . .
 -.beta..sub.n exp[j.phi..sub.0 +nj.beta..DELTA.Z]
 Setting the term ".phi..sub.0 " equal to zero, which corresponds to an
 arbitrary phase shift across all of the waveguides of the array 30, the
 expression for "F.sub.out " can be rewritten as follows:
 ##EQU3##
 The simplified expression for "F.sub.out " clearly matches the form of a
 Fourier series expression such as written above for the spectral response
 function "F(.lambda.)". The Fourier expansion coefficients ".alpha..sub.0
 and .alpha..sub.m ", which replace the coefficients "a.sub.0 and a.sub.k
 ", correspond to normalized amounts of optical power conveyed by each of
 the array waveguides 30, which are numbered from "m=0" to "m=n". The size
 of the matrices is set equal to the integer "n", which also corresponds to
 the total number of array waveguides 30. The integer "m", which replaces
 the integer "k", distinguishes the series terms as well as the array
 waveguides 30. Although only a limited number "n" of array waveguides 30
 is depicted in FIG. 3, a symmetric distribution of array waveguides 30 is
 preferably provided on an opposite side of a longitudinal axis 39, and
 more or less waveguides "n" can be used where appropriate to approximate
 the desired filter function.
 The expression ".beta. .DELTA.z" replaces the expression ".omega..lambda.".
 ".beta." corresponds to the propagation constant of the array waveguides
 30, and ".DELTA.z" corresponds to an incremental difference in physical
 path length between adjacent waveguides of the array 30.Together, the
 expression ".beta. .DELTA.z" dictates the optical path distance
 modification between the waveguides.
 The normalized power of each waveguide given by the expansion coefficients
 ".alpha..sub.0 -.alpha..sub.n " can be controlled by adjusting an opening
 size "S.sub.m " of each waveguide corresponding to a percentage of the
 transmitted beam field or by adjusting the position of each waveguide
 within the beam field, assuming the field has a variable intensity
 profile. The optical path distance modification ".beta. .DELTA.z" is
 preferably carried out by adjusting the physical path length difference
 ".DELTA.z" between the array waveguides. However, the propagation constant
 ".beta." could also be varied to produce similar results, such as by
 incrementally varying the waveguide widths "W.sub.m " or their refractive
 index.
 Instead of arranging the harmonic components of my new filter in a planar
 array as shown in FIG. 3, a similarly performing filter can be formed with
 corresponding harmonic components arranged in a co-axial filter array 40
 as shown in FIG. 4. Tapered input and output couplers 42 and 44 connect an
 array of co-axial waveguide rings 46 to the ends of single mode fibers 48
 and 50. The number of waveguide rings 46, the normalized power in each of
 the rings ".alpha..sub.0 -.alpha..sub.n ", and the optical path length
 modification between rings ".beta. .DELTA.z" are all dictated by the
 Fourier series "F.sub.out ", whose terms are equated to the Fourier series
 "F(.lambda.)" approximating a desired spectral response for the coaxial
 filter array 40.
 Many other interference filter designs can benefit from this invention
 regardless of whether the filtered signal is separated spatially or
 temporally along optical pathways distinguished by incremental optical
 path length differences. However, the filters must permit control over the
 division of power among the different pathways to satisfy the Fourier
 expansion coefficients. Several more examples of such filters are
 disclosed in U.S. Pat. No. 5,841,583 to Bhagavatula, the disclosure of
 which is hereby incorporated by reference. These include examples in which
 different portions of a collimated beam are interrupted by optical path
 length modifiers (such as axially offset reflectors or spacers with
 different refractive indices). The relative size and positioning of the
 modifiers can be adjusted to control the division of power among the
 different beam portions. Another example includes a stack of partially
 reflective surfaces. Layer widths and refractive index differences can be
 adjusted to control both the optical path length variation between layers
 and the relative amounts of power reflected by each layer.
 The phase delay characteristics of these filters also support their use for
 performing dispersion compensation. A desired spectral response (i.e.,
 transfer function) can be specified for purposes of dispersion
 compensation. An example is given in a paper by Takeshi Ozeki entitled
 "Optical equalizers", published in Optics Letters, Mar. 1, 1992, Vol. 17,
 No. 5, which is hereby incorporated by reference. A Fourier series
 approximation can be made of the suggested transfer function and related
 as explained above to the physical characteristics of the filter.
 Although the invention has been described in detail for the purpose of
 illustration, it is understood that such detail is solely for that
 purpose, and variations can be made therein by those skilled in the art
 without departing from the spirit and scope of the invention that is
 defined by the following claims.