Patent Publication Number: US-6671094-B2

Title: Composite birefringent crystal and filter

Description:
This application is a Divisional application of Jay N, Damask-6, Ser. No. 09/664,579, filed on Sep. 18, 2000. 
     CROSS-REFERENCE TO RELATED APPLICATIONS 
     Related subject matter is disclosed in the applications filed on March 21, 2000 and entitled “POLARIZATION DIVERSITY FOR BIREFRINGENT FILTERS” and “DOUBLE-PASS POLARIZATION DIVERSIFIED BIREFRINGENT FILTER,” Ser. Nos. 09/532,143 and 09/532,150, respectively, both filed by J. N. Damask and C. R. Doerr and both assigned to the same Assignee. 
    
    
     FIELD OF THE INVENTION 
     The invention relates generally to birefringent crystals and filters and, more particularly, to a composite birefringent crystal and a folded birefringent filter using such a composite birefringent crystal. 
     BACKGROUND OF THE INVENTION 
     The birefringent filter of FIG. 1 ( a,b ) is a construct well-known to astronomers since the 1930s. Lyot [1] and Evans [2] used such filters in combination with their telescopes to image the sun within narrow frequency bands, for example isolating the helium line to provide for the imaging of helium gas dynamics on the sun&#39;s surface. (Note in this specification, a reference to another document is designated by a number in brackets to identify its location in a list of references found in the Appendix) The birefringent filter appears to be introduced to optical telecommunications in the late 1980s by C. Burher, who demonstrated a simple periodic filter [3, 4, 5]. In the late 1990s the birefringent filter has become a lead contender for the attractive operation of interleaving and de-interleaving DWDM wavelengths. 
     A birefringent filter proper,  110  or  111 , rotates the state of polarization (SOP),  106 , at the output,  105 , with respect to the input,  103 , as a function of the optical frequency. The frequency response of the induced polarization rotation is tailored by the relative orientation of the ordinary birefringent axes,  109 , from one birefringent element to the next. The relative orientations can be calculated using filter-synthesis procedures as described in Harris [6]. Advantageous filter designs can be realized by use of, all birefringent elements having the same length, e.g.  110 , or all birefringent elements having integral-Multiple lengths of a unit length, e.g.  111 . 
     In order to generate an amplitude response from the polarization rotation produced by a birefringent filter proper, Lyot and Evans both added input and output polarizers,  120  and  121 . In this manner, input light  101  with input SOP  102  is first polarized to a linear state,  104 . Birefringent-filter proper  110  or  111  receives linearly polarized input beam  103  and produces in general an elliptically polarized beam  105  with output SOP  106 . Output polarizer  121  then analyzes the SOP  106  and produces linearly polarized beam  107  with SOP  108 . As the optical frequency of input beam  101  changes, the output SOP  106  changes, producing an amplitude change of beam  107 . 
     The use of polarizers for an optical telecommunications application is generally disadvantageous because of the polarization-dependent loss that results. Buhrer proposed and demonstrated the substitution of input and output polarizers with a polarization-diversity scheme. FIG. 2 illustratively shows the use of input polarization diversity  201  and output polarization diversity  202  elements in place of polarizers. Buhler&#39;s U.S. Pat. No. 4,987,567 [4] provides one such architecture. The above-referenced Damask and Doerr patents, Ser. Nos. 09/532,143 and 09/532,150, cover alternative schemes. 
     FIG. 2 illustrates input beam  101  with SOP  102  being split into two parallel yet offset beams  210  and  211  by input polarization diversity element  201 . Beams  210  and  211  have polarizations  220  and  221  that may be either orthogonal or parallel, depending on the method of polarization diversity implemented. The clear aperture of the birefringent filter proper,  110 , is designed large enough to accept both beams  210  and  211 . Beams  212  and  213  are output from the filter with polarizations  222  and  223 , which may be orthogonal or parallel, depending on the method of polarization diversity implemented. Output polarization diversity element  202  then combines the orthogonal polarization elements of each SOP,  222  and  223 , creating beams  214  and  215  with SOPs  224  and  225 . In this scheme, no optical power is in principle lost. The intensities of beams  214  and  215  alternate as a function of input optical frequency such that the sum of their optical powers remains constant. 
     Whether input and output polarizers or polarization diversity elements are used to implement the transformation from SOP rotation to amplitude response, it is the core birefringent filter  110  which dictates the shape and periodicity of the frequency response of the filter. The relative orientations of the birefringent ordinary axes dictate the filter magnitude and phase response; the thickness of the birefringent plates dictates the periodicity of the response. The number of birefringent elements required to realize a specific filter shape dependents on the filter specifics. Typically, though, three or more stages are used. 
     As a practical matter, the unit crystal length of any one birefringent element, Lo, is on the order of 17.65 mm to achieve a frequency periodicity, called the free-spectral range (FSR), of 100 GHz using calcite as the birefringent material. A three-stage filter is then 52.95 mm long. Many important filters require more stages. Accordingly, as a practical matter, the length and material cost of the birefringent filter proper can be large. Moreover, the length tolerance from one birefringent element to the next is stringent. To shift the response of any one birefringent element by one FSR, the crystal length need change 100 GHz/193 THz, or about 0.05%, assuming an approximate and illustrative 1545 nm optical wavelength. Therefore the length control for calcite elements must be a small fraction of 0.05% of 17.65 mm, or less then about 9 microns. Typically crystal length can be controlled to about +/−3 microns. The impact of small variations of crystal length along a birefringent filter is to distort the desired spectral response. 
     What is needed is a birefringent filter design that overcomes the above limitations of existing designs. 
     SUMMARY OF THE INVENTION 
     In accordance with the present invention, the problems of prior birefringent filter designs are overcome using a folded birefringent filter incorporating a multi-pass architecture. More generally, I have invented a novel composite birefringent crystal which includes a stack of two or more birefringent crystals having complimentary properties arranged to have zero net beam walk-off at off-normal beam incidence and a finite free-spectral range. In another embodiment, the birefringent crystal materials are selected to provide reduced temperature dependence. The result is an optically and mechanically stable composite birefringent crystal. In one application, a folded birefringent filter is implemented so that an input beam has multiple transits of the composite birefringent crystal. The folded birefringent filter uses the composite birefringent crystal together with one or more highly reflective devices and waveplates arrays to form a variety of single- or multiple-order folded birefringent filters. 
     More particularly, my composite birefringent crystal comprises: 
     a. a stack of two or more uniaxial birefringent crystals, each having front and back substantively parallel surfaces in which plane the crystalline extraordinary axis substantively lies and each located one behind the next with surfaces parallel, which receives an input optical beam that is non-normal to the front surface of the first crystal and produces from the back surface of the last crystal first and second orthogonally polarized optical beams; 
     b. where at least one crystal in the stack exhibits positive uniaxial birefringence; 
     c. where at least one crystal in the stack exhibits negative uniaxial birefringence; 
     d. where the extraordinary axis of at least one of the positive uniaxial crystals and the extraordinary axis of at least one of the negative uniaxial crystals have non-parallel alignment; 
     e. where with respect to the length of the first crystal, the ratio of the length of each subsequent crystal to the length of the first crystal is selected to produce at the bottom surface of the last crystal 
     i. zero net spatial displacement between first and second orthogonally polarized optical beams, and 
     ii. temporal delay between first and second orthogonally polarized optical beams. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     In the drawings 
     FIG. 1 a  shows an illustration of a prior art single-order cascaded birefringent filter with input and output polarizers; 
     FIG. 1 b  shows an illustration of a prior art multiple-order cascaded birefringent filter with input and output polarizers; 
     FIG. 2 shows an illustration of a prior art single-order cascaded birefringent filter with input and output polarization diversity; 
     FIG. 3 a  shows an illustration of a single-order cascaded birefringent filter; 
     FIG. 3 b  shows a single-order unfolded birefringent filter equivalent to the filter of FIG. 3 a;    
     FIG. 4 a  shows a birefringent filter proper with multiple principle waveplate thicknesses; 
     FIG. 4 b  shows a first unfolded construction of the filter of FIG. 4 a  using half-wave waveplate (HWP) pairs; 
     FIG. 4 c  shows a more advantageous unfolded construction of the filter of FIG. 4 a  using half-wave waveplate (HWP) pairs; 
     FIG. 5 a  illustratively shows a folded birefringent filter equivalent to FIG. 3 b;    
     FIG. 5 b  shows the face of an illustrative quarter-wave plate (QWP) array; 
     FIG. 6 a  shows a folded birefringent filter equivalent to FIG. 4 c;    
     FIG. 6 b  illustratively shows the e-axis orientations and reflection patterns located at five planes within the cavity of the filter of FIG. 6 a;    
     FIG. 7 illustrates the transit of input optical beam through the birefringent plates and the HWP; 
     FIGS. 8 a  and  8   b  illustratively show the refraction of an input beam through two stacked birefringent plates having a 90° azimuthal rotation of the extraordinary axes; 
     FIG. 9 illustrates two dissimilar birefringent crystals with e-axes orientated 90 degrees from one another; 
     FIG. 10 shows the refraction of an input beam into a birefringent crystal with extraordinary axis direction; 
     FIG. 11 illustrates the cross-section of the two-crystal arrangement of FIG. 9; 
     FIG. 12 shows the refraction and displacement of an input beam after transit through the two stacked birefringent plates of FIG. 9, which leads to Eq. (1); 
     FIG. 13 illustrates a cross-section of three birefringent crystals; 
     FIGS. 14 a  and  b  illustrate the cross-section view of the complete folded birefringent filter utilizing the above-designed three-stage composite birefringent crystal. FIG. 14 a  illustrates a folded filter with uniform principal crystal thicknesses and FIG. 14 b  illustrates a folded filter with multiple principal waveplate thicknesses; 
     FIG. 14 c  illustrates one manner in which polarization diversity can be utilized in the present folded filter designs; 
     FIG. 14 d  illustrates the most general filter as three blocks: a first reflective unit, a composite crystal, and a second reflective unit; 
     FIG. 15 shows a section of a single-order tunable unfolded filter including a variable retarder tuner; 
     FIG. 16 illustrates representative intensity responses (at different orientations) of a folded filter as a function of frequency; 
     FIG. 17 a  illustrates the folded filter architecture of FIG. 14 a  with two additional elements including a fixed QWP located parallel to and behind the composite-crystal principal waveplate and a rotatable QWP located parallel to and behind the fixed QWP; and the composite-crystal principal waveplate and a rotatable QWP located parallel to and behind the fixed QWP; and 
     FIG. 17 b  illustrates the relative e-axis orientations of the elements in the tunable folded filter. 
    
    
     DETAILED DESCRIPTION 
     The elements of a birefringent filter proper are uniaxial birefringent crystals. Typical uniaxial birefringent crystals are mica, calcite, crystalline quartz, and rutile. A uniaxial crystal is characterized by two of three orthogonal crystalline axes possessing the same refractive index and the remaining axis possessing a different refractive index. The two common axes are called the ordinary axes and the dissimilar axis is called the extraordinary axis. A crystal is referred to as positive or negative uniaxial depending on whether the refractive index of the extraordinary axis is greater than or less than the ordinary refractive indices, respectively. 
     The cut of the crystal for the purposes of a birefringent filter element is such that the extraordinary axis, or e-axis, is in the plane of the crystal face onto which optical radiation is incident. This crystal cut is herein referred to as a waveplate. In the waveplate orientation, optical radiation which propagates through the crystal body, is split into two distinct linear polarizations: one which is aligned to the e-axis and one which is aligned to the orthogonal ordinary axis, or o-axis. Since the refractive indices of the e- and o-axes differ, the phase and group velocities of the linearly polarized waves as they travel through the crystal differ. It is common to refer to the axes as fast and slow. However, the correspondence between fast and slow, and ordinary and extraordinary, depends of the sign of the crystal. A positive uniaxial crystal exhibits n e &gt;n o , so the e-axis is the slow axis and the o-axis is the fast axis. A negative uniaxial crystal exhibits n o &gt;n e , so the e-axis is the fast axis and the  6 -axis is the slow axis. 
     A birefringent filter requires the interleaving of two optical effects, FIG. 3 a . First, the optical power on two orthogonal polarization axes must be mixed by projection onto a waveplate face. The azimuth orientation of the e-axis,  302 , with respect to the incident SOP,  300 , dictates the mixture of the optical power. Second, differential group delay (DGD), or the time-of-flight difference, must accrue between the two orthogonal linear polarization axes. DGD must accrue from one SOP projection to the next. Each waveplate  310  in FIG. 3 a  has, for generality, a distinct azimuth orientation of the e-axis,  302 , and has the same crystal length. Each crystal face mixes the incident optical power on the two orthogonal polarization axes and each crystal length generates the requisite DGD. 
     In order to distinguish the various crystal functions in the following, the crystal waveplates  310  illustrated to in FIG. 3 a  will hereafter be referred to as principle birefringent plates, or principle plates for short. A principle plate is a waveplate that is thick enough to produce the desired periodic frequency-response. 
     Unfolded Birefringent Filters 
     FIG. 3 a  illustrates how a typical birefringent filter is constructed. Such construction is potentially simple and thus promises some advantage. However, an alternative and optically equivalent construction is illustrated in FIG. 3 b . In the present case, each principal waveplate  310  has its azimuth orientation of the e-axis  307  rotated to a common position, hereafter referred to as the zero position. Between each principle waveplate and at each end are added half-wave waveplates (HWP) pairs  311 . A single HWP refers to a birefringent waveplate that is very thin—ideally zero. While there is no appreciable accumulation of DGD through a HWP, the input SOP is mirror-imaged about the e-axis of the HWP, which is similar to a SOP rotation. Each HWP pair  311  is composed to two distinct HWPs  312  and  313 . Each HWP  312  in the cascade is rotated to a predetermined azimuth orientation  304  while all HWPs  313  are rotated to zero orientation  306 , which aligns with the zero orientation  307  of principle waveplates  310 . 
     The cascade of HWP pairs and principle waveplates separates out the two component functions of a birefringent filter: the SOP rotation is performed by the lead HWPs  312  and the DGD accumulation is performed by the principle waveplates  310 . The azimuth. orientations  305  of the lead HWPs  312  is calculated from the azimuth orientations  303  of the principal waveplates  310  from FIG. 3 a . The relation between ?  305  and ?  303  is 
     
       
         θ k+1 =(φ k+1 −φ k )/2  (1a) 
       
     
     
       
         φ 0 =0  (1b) 
       
     
     
       
         φ N+1 =0  (1c) 
       
     
     which amounts to simply one-half of the difference between e-axis azimuth angles between adjacent crystals  310  of FIG. 3 a.    
     The purpose of zero-oriented HWPs  313  in FIG. 3 b  is subtle: replacement of rotated principle waveplates  310  of FIG. 3 a  with rotated HWPs  312  of FIG. 3 b  both shifts the mechanics of SOP rotation from rotation to mirror-image, and shifts the frequency response of the following principle waveplate. In order to correct for these effects to establish rigorous equivalence between FIGS. 3 a  and  3   b , additional zero-orientated HWPs  313  must be added to the cascade. In the instance of FIG. 3 b  where all principle waveplates share the same length, HWPs  313  may be systematically “absorbed” by the principle waveplates  310  with the impact of shifting the frequency-response of the birefringent filter proper by one-half of the FSR. 
     The cascade of HWPs and principle waveplates as in FIG. 3 b  is hereafter referred to as an unfolded birefringent filter, equivalent to FIG. 3 a . The construction of the unfolded filter equivalent to FIG. 1 b , where the birefringent crystals are integral multiples of a unit length, requires an additional step. 
     FIG. 4 a  illustrates a birefringent filter proper with multiple principal waveplate thicknesses Lo, 2Lo, and 4Lo. All principle waveplate thicknesses are an integral multiple of a unit thickness Lo. An optical beam  400  is input on first principle waveplate  410  and subsequently transits through principal waveplates  410 ,  410 ′, and  410 ″. Each principle waveplate is rotated along the azimuth direction  402  to the appropriate angle. Angles  403  determine the frequency response of the filter. 
     FIG. 4 b  illustrates a first unfolded construction wherein HWP pairs  411  are utilized to perform the requisite SOP rotation between principle waveplates. HWP pairs are located between each principle waveplate and at either end of the cascade. Moreover, the principle waveplates  410 ,  410 ′,  410 ″ are divided into integral numbers of unit-thickness waveplates; following FIG. 4 b  waveplate  410  is undivided,  410 ′ is divided into two unit waveplates, and  410 ″ is divided into four unit waveplates. All principle waveplates are thus composed of an integral number of principle waveplates  410 . The e-axes  407  of the principle waveplates sections  410  are rotated to the zero position while the e-axes  404  of the lead HWPs  412  are rotated according to Eqs. (1a-c),  405 . The e-axes  406  of the trailing HWPs  413  are rotated to the zero position. 
     Unlike the unfolded cascade of FIG. 3 b , where-trailing HWPs  313  may be excluded with only a consequent shifting of the filter frequency response by one-half of a FSR, the exclusion of trailing HWPs  413  will alter the entire shape of the filter frequency response. Therefore a means to incorporate the trailing HWPs  413  is required. An advantageous manner in which to incorporate HWPs  413  into the unfolded filter cascade is illustrated in FIG. 4 c . Each unit-length principle waveplate that immediately follows each trailing HWP  413  is first split in two equal-length sections  407  and  407 ′. The azimuth e-axis orientation is unaltered. Next, each trailing HWP  413  is relocated to reside between each split section  407  and  407 ′. The composite of first split principle waveplate  407 , first HWP  406 ′, and second split principle waveplate  407 ′ creates a symmetric section  414 . The transformation from FIG. 4 b  to FIG. 4 c  does not alter the azimuth rotations  404 ,  405 , 406 , or  407 . 
     FIGS. 3 b  and  4   c  are unfolded equivalents of the birefringent filters illustrated in FIGS. 3 a  and  4   a . Both of these unfolded equivalents may now be folded. Folding of either filter does not change the sequence of transformations an optical beam experiences during transit of the filter; rather the physical length is reduced and the ease of manufacture is simplified. In the following the various principle waveplates will be substituted with a single, common principle waveplate (CPWP). 
     Folded Filter Basic Architecture 
     FIG. 5 a  illustrates a folded birefringent filter that is equivalent in principle to FIG. 3 b . The filter of FIG. 5 a  contains ten stages rather than four and removes the HWPs  313  with consequent albeit immaterial shifting of the filter frequency response by one-half FSR. A cavity is formed between a CPWP  501  whose length is one-half the length of the principle waveplates of an equivalent unfolded filter, and a “quarter-wave plate array”, or QWP array  502 . The e-axis  550  of the CPWP  501  as illustrated is oriented vertically. An input optical beam  300  is first inclined from the horizontal so that, as the light travels through the filter cavity, a zig-zag pattern is generated. The CPWP has its backside face mirror-coated  507  and the QWP array  502 , too, has its backside mirror coated  508  with the exception of the edge two plates  505  and  506 . Quarter-wave waveplates are substituted for half-wave waveplates because a full half-wave effect is generated via the double-pass through the QWP. To account for the quarter-wave shortfall of edge plates  505  and  506 , external QWPs  503  and  504  are added. Note that the edge plates  505  and  506  may be combined with the QWPs  503  and  504 , respectively, to form half-wave plates. 
     Thus, inclined input beam  300  passed through first QWP  503  and edge QWP  505  and enters the cavity as beam  511 . Cavity beam  511  is inclined with respect to the CPWP by angle  520 . The transit to and from the backside mirror  507  generates the requisite one-pass DGD, equivalent to transit of elements  310  of FIG. 3 b . Emergent beam  512  is then incident on one element of the QWP array, thereby rotating its SOP by a prescribed amount. The backside mirror  508  reflects beam  512  back to the CPWP for additional passes. The double-transit of one QWP element is equivalent to transit of elements  312  of FIG. 3 b . After transiting the cavity as thus and exiting via edge QWP  506  and QWP  504 , the output beam  301  emerges. 
     FIG. 5 b  illustrates the face of a representative QWP array  502 . Several separate quarter-wave plates are cut so that their perimeter is rectangular but their e-axis,  530  and  531 , is orientated according to the prescribed sequence of angles,  305 , necessary to generate the desired filter response. The center-to-center spacing of each plate  540  is determined by the overall geometry of the cavity. The edge plates  505  and  506  do not have backside mirror coating which thief remaining interior plates do. 
     FIG. 6 a  illustrates a folded birefringent filter that is equivalent in principle to FIG. 4 c . A cavity is formed between backside solid QWP  609 , a CPWP  601  (whose length is one-half the length of the thinnest principle waveplate of an equivalent unfolded filter), and a “quarter-wave plate” array  602  (QWP array). The e-axis  615  of the CPWP  601  as illustrated is oriented vertically, as is the e-axis of the backside QWP  609 . The backsides of solid QWP  609  and QWP array  602  are mirror coated with the exception of gaps at edge QWP locations  605  and  606 . An input optical beam  400  is first inclined from the horizontal so that, as the light travels through the filter cavity, a zig-zag pattern is generated. 
     Unlike the folded filter of FIG. 5 a , the present folded filter adds additional, segmented mirrors internal to the cavity. Patterned mirrors  620  and  621  selectively block the cavity optical beam  611  from transit of the respective folded half-wave waveplates. 
     Thus, inclined input beam  400  passed through first QWP  603  and edge QWP  605  and enters the cavity as beam  611 . Cavity beam  611  is inclined with respect to the CPWP by angle  613 . First transit of the CPWP must mimic the waveplate sequence  414  of FIG. 4 c . Thus beam  611  transits the CPWP, solid QWP  609 , is reflected at position  624 , and transits CPWP in one pass, forming beam  612 . This path is equivalent to transit of elements  413  and  410  of FIG. 4 b . Beam  612  is reflected at position  622 , thus transiting the second QWP array element. Element  414  must be produced again and therefore the optical beam transits both the CPWP and backside QWP again. At this point a second transit of the CPWP is required without intermediate SOP rotation. Thus the cavity beam is reflected at point  623 , thereby being blocked from a second transit of the QWP array. The cavity beam is further blocked at point  625  from a second transit of backside QWP. Such selective blocking structure is necessary to construct the equivalence of the folded filter from the unfolded filter illustrated in FIG. 4 c.    
     FIG. 6 b  illustrates the e-axis orientations and reflection patterns located at five planes within the cavity. At plane # 1  QWPs are cut into rectangles with common height but integral widths. The e-axis of each plate,  630  and  631 , is rotated according to sequence  405 . Plates  635 ,  635 ′, and  635 ″ share a common unit width. On plane # 2 , regularly spaced beam positions  640  are indicated. Mirror stripes  641  selectively block the cavity beam from multiple QWP array passes. Plane # 3  illustrates the zero orientation of the e-axis  650  of the CPWP. On plane # 4 , regularly spaced beam positions  660 , offset by one-half beam displacement, are indicated. Mirror stripes  661  selectively block the cavity beam from multiple backside QWP passes. Plane # 5  illustrates the zero orientation of the e-axis  670  of the backside QWP. 
     Composite Crystal Design for Zero Beam Walkoff 
     In accordance with another aspect of my invention, I have determined that the designs of FIGS. 5 a  and  6   a  only work when a composite, rather than single, birefringent crystal is used as the common principal waveplate  501  and  601 . The structure of such a composite crystal is discussed in later paragraphs. The origin of the problem present in FIGS. 5 a  and  6   a  is first described. Without loss of generality, the folded filter of FIG. 5 a  alone will be referred to. 
     The CPWP  501  is a uniaxial birefringent crystal cut so that the e-axis  550  is in the plane of the face through which cavity beam  511  transits and is oriented vertically in the figure. Once the input beam  511  at inclination  520  impinges the surface of the CPWP crystal, the incident SOP is resolved onto the orthogonal crystalline e- and o-axes. As these axes exhibit different refractive indices, the incident beam refracts into the crystal at two distinct angles. FIG. 7 illustrates the transit of input optical beam  701  through birefringent plates  710  and  711  and HWP  712 . Upon incidence, beam  701  is split into two distinct beams  703  that propagate at different angles. Emergence of beams  702  results in two parallel, linearly polarized, and spatially offset beams. Transit of HWP  712  mixes the beam SOPs so that after incidence onto plate  711  each of the two input beams is further split into two beams, yielding four beams total. 
     Each pass through the common crystal (e.g.,  501  of FIG. 5 a ) of a folded birefringent filter, after intermediate SOP rotation, divides each beam, yielding 2 N  beams total after N stages with a binomial distribution in position displacement. Such geometric spatial division of the input beam results in high insertion loss, instability of that loss, and large polarization-dependent loss (PDL). 
     The problem is then how to overcome the detriment of beam walk-off. Certainly the presence of birefringence demands that orthogonal polarizations spatially separate. As part of the invention disclosed herein, the following shows how to engineer a stack of birefringent crystals having a finite FSR and zero net beam displacement. 
     Straightforward Approach 
     Shown in FIG. 8 a  is a straightforward approach that was considered, in which two like birefringent crystals of equal length and cut as waveplates are stacked together with a 90° azimuthal rotation of the extraordinary axes. This construction will not work. Consider, without loss of generality, these crystals to be positive uniaxial. The split of the input beam  801  in the first crystal  802  is corrected by recombination in the. second crystal. However, the fast and slow axes are interchanged from the first to second crystal, resulting in zero net DGD between the beams. 
     In detail, as shown in FIG. 8 b , at the first crystal the input beam  801  with arbitrary SOP is resolved into the ordinary and extraordinary rays that refract at distinct angles  802 . The e-ray  811  propagates along the slow axis, it being a positive uniaxial crystal, while the o-ray  810  propagates along the fast axis. The slow axis has the higher refractive index, therefore the k-vector of the refracted beam is declined towards the horizontal further than the k-vector for the fast axis. Through the length of the first crystal DGD is accrued. At the termination of the first crystal the two beams  802  are spatially separated. 
     Upon entrance to the second crystal with cross e-axis with respect to the first crystal, the beams  802  retain their linear polarization states by exchange their fast and slow designations. The e-ray of the first crystal becomes the o-ray in the second, thereby becoming the fast ray with inclination away from the horizontal. The o-ray of the first crystal becomes the e-ray in the second, thereby becoming the slow ray with declination towards the horizontal. After transit of the same crystal length, the two beams  802  reemerge coincident as beam  803 . However, the DGD accrued in the first crystal was canceled by the second, thereby voiding the purpose of the crossed composite crystal design, when using like crystals. 
     Walk-Off Solution 
     The degree of freedom necessary to create a solution that exhibits zero net beam walkoff which accruing the requisite DGD is to flip the sign of the birefringence from the first crystal to the second. FIG. 9 illustrates two dissimilar birefringent crystals with e-axes orientated 90 degrees from one another. The first crystal  910  is designated positive uniaxial while the second crystal  911  is negative. Exchange of the birefringence sign between the two crystals is possible and does not change the following disclosure. Input beam  901  refracts into two beams  902  upon entrance to first crystal  910 . These divergent beams become convergent upon entrance to the second crystal  911 . However, as in the case of FIG. 8, the exchange of ordinary and extraordinary axes does not exchange the “fast” and “slow” axes because the second crystal  911  has a flipped birefringence sign. Accounting for the birefringent sign change, the e-ray in the second crystal remains “fast” while the o-ray remains “slow”. Thus DGD is accrued throughout transit of the composite crystal. Below is disclosed how to engineer the convergence of the beams in the second crystal  911 . 
     Care must first be taken when considering the propagation of linearly polarized beams through birefringent media. Snell&#39;s Law dictates the refraction of the k-vector; the k-vector being perpendicular to the direction of the phase fronts of the propagating light and proportional to the phase velocity. In addition to the k-vector is the Poynting vector, which points in the direction of energy flow. In a uniaxial birefringent media, the k-vector and Poynting vectors are parallel when the linear polarization of the beam is either parallel or perpendicular to the extraordinary axis of the crystal. However, when the linear polarization of the beam has a component that lies in the direction of the extraordinary axis, the k- and Poynting vectors diverge. The indicatrix is a tool commonly used to represent the relation between the k- and Poynting vectors in birefringent media [Electromagnetic Wave Theory: J. Kong, John Wiley and Sons, 1986]. 
     When linearly polarized light is incident on a birefringent interface where the e-axis is in the plane of the interface, the effective index experienced by the beam as it refracts into the crystal depends on the input SOP and the inclination direction of the beam. Table 1 summarizes the four possible cases: where the input (or incident) beam is inclined towards the o- and e-axis, and where the linear polarization appears as transverse electric (TE) or transverse magnetic (TM). The first three entries in Table 1 exhibit effective refractive indices that are purely ordinary or extraordinary. In these three cases the beam polarization is either parallel or perpendicular to the extraordinary axis. The fourth entry shows an effective index that is a mixture of the ordinary and extraordinary indices, depending on the in-crystal angle at which the beam propagates. The appropriate one of these four effective indices is to be used in Snell&#39;s Law of refraction to determine the k-vector direction. 
     FIG. 10 illustrates the refraction of an incident beam into a birefringent crystal with extraordinary axis direction  1001 . The incident beam inclination is towards the e-axis. One thus expects k-vector and Poynting-vector splitting of the TM polarization component. The ordinary k-vector  1005  refracts at angle  1011  and, being that it&#39;s linearly polarization  1020  is perpendicular to  1001 , the ordinary Poynting vector runs parallel to  1005 . The extraordinary k-vector  1003 , with linear polarization  1021 , refracts at angle  1010 , obeying the mixed effective index equation. Being that the e-ray polarization is partially directed towards the e-axis  1001 , the k- and Poynting vectors split. The Poynting vector  1004  diverges from the k-vector at angle  1013 , given in Table I. It is a rule for this type of k- and Poynting vector split that the extraordinary k- and Poynting vectors always bound the ordinary k- and Poynting vectors, regardless of whether the sign of the birefringence is positive or negative. 
     
       
         
           
               
             
               
                 TABLE I 
               
             
            
               
                   
               
               
                 Effective Index for Inclination and Polarization 
               
            
           
           
               
               
               
            
               
                 Inclination 
                 Polarization 
                 Effective Index 
               
               
                   
               
               
                 o-axis 
                 TM 
                 n o   
               
               
                 o-axis 
                 TE 
                 n e   
               
               
                 e-axis 
                 TE 
                 n o   
               
               
                   
               
               
                 e-axis 
                 TM 
                 
                   
                     
                       
                         
                           
                             n 
                             o 
                           
                            
                           
                             n 
                             e 
                           
                         
                         
                           
                             
                               
                                 n 
                                 e 
                                 2 
                               
                                
                               
                                 sin 
                                 2 
                               
                                
                               ϕ 
                             
                             + 
                             
                               
                                 n 
                                 o 
                                 2 
                               
                                
                               
                                 cos 
                                 2 
                               
                                
                               ϕ 
                             
                           
                         
                       
                     
                     
                     
                         
                     
                   
                 
               
               
                   
               
               
                 Poynting vector inclination from k-vector for e-axis TM:  
               
               
                 
                   
                     
                       
                         
                           tan 
                            
                           
                               
                           
                            
                           γ 
                         
                         = 
                         
                           
                             
                               ( 
                               
                                 
                                   n 
                                   e 
                                   2 
                                 
                                 - 
                                 
                                   n 
                                   o 
                                   2 
                                 
                               
                               ) 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             ϕcos 
                              
                             
                                 
                             
                              
                             ϕ 
                           
                           
                             
                               
                                 ( 
                                 
                                   
                                     n 
                                     e 
                                     2 
                                   
                                   - 
                                   
                                     n 
                                     o 
                                     2 
                                   
                                 
                                 ) 
                               
                                
                               
                                 sin 
                                 2 
                               
                                
                               ϕ 
                             
                             + 
                             
                               n 
                               o 
                               2 
                             
                           
                         
                       
                     
                     
                     
                         
                     
                   
                 
               
            
           
         
       
     
     As the beams that emerge from the crystal are those along the lines of energy flow, it is the e- and o-Poynting vectors, which require ray tracing. 
     FIG. 11 illustrates the cross-section of the two-crystal arrangement of FIG.  9 . There are three rays, identified as: beam b (the k- and Poynting vectors of the ordinary beam, coincident); beam a (the k-vector vector of the e-ray); and beam a-s (the Poynting vector of the e-ray). Such designations are useful as the properties of ordinary and extraordinary change throughout the composite crystal. 
     In FIG. 11, the incident beam  1101  is inclined away from the normal of interface  1110  in the direction of the extraordinary axis  1130  of the first, positive uniaxial, crystal. The beam  1101  is divided in two at the interface  1110 : one k-vector,  1102 , having linear polarization direction  1120  obeys Snell&#39;s Law as an ordinary wave; and the other k-vector,  1103 , having linear polarization direction  1121  obeys Snell&#39;s Law as an extraordinary wave. Being a positive uniaxial crystal, the e-ray k-vector,  1103 , is declined towards the normal more than the o-ray. Now, while the polarization state of the o-ray,  1102 , is perpendicular to the extraordinary axis, the e-ray polarization state is not. The result is that the k-vector and Poynting vectors of the e-ray split. It is the nature of this splitting in a positive uniaxial crystal that the e-ray Poynting vector inclines further from the normal than the o-ray k- and Poynting vectors. The e- and o-ray Poynting vectors diverge, with the e-ray vector “above” the o-ray vector. 
     Entrance of the beams into the second crystal at interface  1111  has two effects. First, the ordinary and extraordinary axes are interchanged. Yet since the second crystal is negative uniaxial, the phase velocity of the original e-ray, now the o-ray, remains the slower. Temporal delay continues. Second, the polarization states of the beams are now either perpendicular or parallel to the extraordinary axis of the second crystal  931 , so the Poynting and k-vectors of each beam run parallel. However, the k- and Poynting vectors of the e-ray from the first crystal are no longer coincident. Consideration of the relative refractive indices in the second crystal shows that the inclination of the original ordinary Poynting vector in the second crystal is greater than the inclination of the original extraordinary Poynting vector. The result is a converging course. The position of terminating interface  1112  is chosen at the point of convergence  1140 . 
     Length Relation for Two-Element Composite Crystal 
     The equation which governs the ratio of lengths for the two birefringent crystals, for small incidence angles, is                  L   2     /     L   1       =         n     o                 2         n     o                 1                (         n     e                 1       /     n     o                 1         -   1     )       (         n     o                 2       /     n     e                 2         -   1     )                 (   1   )                         
     where L 1  and L 2  are the length of the first and second crystals, subscripts e and o refer to the pure ordinary or extraordinary refractive indices, and subscripts 1 and 2 designate the crystal. 
     FIG. 12 illustrates the ray-trace that leads to Eq. (1). Poynting vector beams  1222  and  1223 , and k-vector beam.  1221 , propagate through the first crystal at distinct angles. Over length L 1 ,  1201 , the beams rise to elevations  1211 ,  1212 , and  1213 . These elevations are given as 
     
       
           d   a ( L   1 )=L 1  tan φ a   (1)   (2) 
       
     
     
       
           d   b ( L   1 )=L 1 tan φ b   (1)   (3) 
       
     
     
       
           d   a−s ( L   1 )= L   1  tan(φ a   (1)+γ)   (4) 
       
     
     where d a , d b , and d a−s  refer to elevations  1211 ,  1212 , and  1213 , of beams  1221 ,  1222 ,  1223 , respectively. The refraction angles ? are determined by Snell&#39;s Law by using the corresponding effective index listed in Table I. The displacement angle ? is the splitting between the e-ray k- and Poynting vectors. 
     Upon entrance to the second crystal all beams change propagation angle as dictated by the crystal birefringence and the beam polarization state. Transit of the second crystal with length L 2 ,  1202 , yields a total elevation of Poynting vectors  1215  and of the offset k-vector  1214 . These elevations are given as 
     
       
           d   a ( L   1   +L   2 )= d   a ( L   1 )+ L   2  tan φ a   (2)   (5) 
       
     
     
       
           d   b ( L   1   +L   2 )= d   b ( L   1 )+ L   2  tan φ b   (2)   (6) 
       
     
     
       
           d   a−s ( L   1   +L   2 )= d   a−s ( L   1 )+ L   2  tan φ a   (2)   (7) 
       
     
     The Poynting-vector beams converge when d a−s (L 1 +L 2 )=d b (L 1 +L 2 ). Together the above elevation equations lead to Eq. (1). 
     Finally, note that an effective index can be defined for the overall composite crystal by relating the total accumulated elevation to the sum length of the crystals. The following Eq. (8) is for small inclination angles:                n     eff   -   xtal       =       [         L   1         L   1     +     L   2              (       1     n     o                 1         +         L   2     /     L   1         n     e                 2           )       ]       -   1               (   8   )                         
     First-Order Temperature Independence 
     Two crystals of oppositely signed birefringence are necessary to find a zero net walkoff solution for inclined incident optical beams. Typically these two crystals combined are not temperature compensated to a first-order. To construct a composite crystal that has zero net beam walkoff and is also first-order temperature compensated, a first birefringent crystal, with complimentary properties, must be added. FIG. 13 illustrates a cross-section of three birefringent crystals. The, first crystal has its extraordinary axis  1310  perpendicular to one or both of the remaining two extraordinary axes  1311  and  1312 . The first crystal is also positive uniaxial while one or both of the remaining crystals must be negative uniaxial. Finally, the temperature coefficients of birefringence and expansion must be complimentary in such as manner as to provide a solution. The goal is to find a beam-convergent solution from input beam  1301  to output  1302  that is also first-order temperature compensated. A system of three equations of length L 1 ,  1320 , L 2 ,  1321 , and L 3 ,  1322 , is derived as follows. 
     The free-spectral range (FSR) of the composite crystal is calculated by 
     
       
           FSR[Δn   1   L   1   ±Δn   2   L   2   ±Δn   3   L   3   ]=c   (9) 
       
     
     where the + sign is used when the adjacent extraordinary axes are aligned and the − sign is used otherwise. The constant c is the speed of light. 
     The first-order temperature dependence is governed by                    1     F                 S                 R                   F                   S                 R          T         +         F                 S                 R     c          [         Δ                   n   1          L   1          K   1       ±     Δ                   n   2          L   2          K   2         ±     Δ                   n   3          L   3          K   3         ]         =   0           (   10   )                         
     where T is temperature and the temperature coefficient K is defined as              K   =       1     Δ                 n                 L                   Δ                   n                 L          T                 (   11   )                         
     and where ?n is the birefringence of the crystal. The selection of ± sign follows that of Eq. (9). 
     Finally, the difference of inclination angle of the Poynting vectors within a single crystal is denoted by the symbol ?. The system of equations to find the crystal lengths L 1 , L 2 , and L 3 , assuming a positive then two negative uniaxial crystals, is thus:                  [           Ω   1           -     Ω   2             -     Ω   3                 Δ                   n   1          K   1               -   Δ                     n   2          K   2               -   Δ                     n   3          K   3                 Δ                   n   1               -   Δ                     n   2               -   Δ                     n   3             ]          [           L   1               L   2               L   3           ]       =     [         0           0               c   /   F                   S                 R           ]             (   12   )                         
     As an example, Table 2 provides reported values for birefringence and temperature coefficients for three commonly available birefringent crystals. Note that other materials such as lithium niobate, crystalline quartz, rutile, and mica may also be utilized for the birefringent crystal. Table 3 lists the results of application of Eq. (12) using the values of Table 2. 
     
       
         
           
               
             
               
                 TABLE 2 
               
             
            
               
                   
               
               
                 Reported Values 
               
            
           
           
               
               
               
               
               
            
               
                   
                 Crystal 
                 ?n 
                 K (×10 −6 ) 
                 ? 
               
               
                   
                   
               
            
           
           
               
               
               
               
               
            
               
                   
                 YVO 4   
                 +0.2039 
                 −22.5 
                 1.55 ?m 
               
               
                   
                 Calcite 
                 −0.1744 
                 −37.5 
                 1.55 ?m 
               
               
                   
                 ?-BBO 
                 −0.0733 
                 +104 
                 1.06 ?m 
               
               
                   
                   
               
            
           
         
       
     
     
       
         
           
               
             
               
                 TABLE 3 
               
             
            
               
                   
               
               
                 Crystal Lengths 
               
            
           
           
               
               
               
            
               
                   
                 Crystal 
                 Length (mm) 
               
               
                   
                   
               
            
           
           
               
               
               
            
               
                   
                 YVO 4   
                 8.8 
               
               
                   
                 Calcite 
                 8.3 
               
               
                   
                 ?-BBO 
                 3.4 
               
               
                   
                 Total 
                 20.5 
               
               
                   
                   
               
            
           
         
       
     
     FIGS.  14 ( a-b ) illustrate the cross-section view of the complete folded birefringent filter utilizing the above-designed three-stage composite birefringent crystal of FIG.  13 . FIG. 14 a  illustrates the same folded filter as FIG. 5 a  with the substitution of the single principal waveplate  501  with my composite birefringent crystal  1402 . The composite crystal may be either two or three stages. 
     An input (or incident) optical beam  1401  is inclined from the horizontal so that, as the light travels through the filter cavity, a zig-zag pattern is formed. The CC-CPWP  1402  has its backside face mirror-coated  1404  and the QWP array  1403 , too, has its backside mirror coated  1405  with the exception of the edge two plates  1406  and  1407 . Quarter-wave waveplates are substituted for half-wave waveplates (HWP) because a full half-wave effect is generated via the double-pass through the QWP  1403 . To account for the quarter-wave shortfall of edge plates  1406  and  1407 , external QWPs  1408  and  1409  are added. 
     Thus, inclined input beam  1401  passes through first QWP  1409  and edge QWP  1407  and enters the cavity as beam  1411 . Cavity beam  1411  is inclined with respect to the CPWP  1402  by angle  1420 . The transit to and from the backside mirror  1404  generates the requisite one-pass DGD. Emergent beam  1412  is then incident on one element of the QWP array  1403 , thereby rotating its SOP by a prescribed amount. Note that the length of crystals  1 - 3  is selected (in the manner described in FIG. 13) to produce beam-convergence of beam  1412 . The backside mirror  1405  reflects beam  1412  back to the CPWP  1402  for additional passes. After the zig-zag transiting of the folded filter, the beam exits via edge QWP  1406  and QWP  1408 , as a beam-converged output beam  1410 . 
     The input beam  1401  may originate from an input fiber  1450  wherein the light emergent from the fiber is collimated by a first lens  1451 . After transit of the folded filter, the output beam  1410  may be coupled by into another fiber  1452  by transit of a second lens  1453 . In this manner, a fiber-to-fiber device is created. 
     FIG. 14 b  illustrates the same folded filter as FIG. 6 a  with the substitution of the single principal waveplate  601  with my composite birefringent crystal  1402 . The composite crystal may be either two or three stages. A cavity is formed between backside mirrors  1404  and  1405  with QWP array  1403 , composite crystal  1402 , and monolithic QWP  1426  inside the cavity. Unlike the folded filter of FIG. 14 a , the folded filter of FIG. 14 b  adds additional segmented mirrors internal to the cavity. The segmented mirrors in planes  1424  and  1425  selectively block the optical beam from transit of the respective folded half-wave waveplates  1426  and  1403 , respectively. The patterned mirrors in planes  1424  and  1425  function in the same manner as previously described in FIG.  6 . The incident optical beam  1411  is inclined from the horizontal so that, as the light travels through the filter cavity, a zig-zag pattern is generated. 
     As illustrated in FIG. 2, a polarization diversity scheme is advantageous for the practical application of a birefringent filter. FIG. 14 c  illustrates one manner in which polarization diversity can be utilized in the present folded filter design. Beams  1401  and  1401 ′ are generated by a polarization-diversity element  1432  from input beam  1430 . Likewise output beam  1431  is generated from beams  1410  and  1410 ′ by polarization-diversity element  1433 . The beams  1401  and  1410 , and the zig-zag path that connects said beams, lie in plane  1441 . The complete path between  1401  and  1410  is the second optical path. The beams  1401 ′ and  1410 ′, and the zig-zag path which connects said beams, lie in plane  1440 . The complete path between  1401 ′ and  1410 ′ is the first optical path. The planes  1440  and  1441  are parallel yet vertically offset. The optical beams which travel along the second optical path intersects and transits all elements of the folded filter, in the same order and with the same inclination angles, as does the first optical path. As such, the first and second optical paths are indistinguishable except for spatial displacement. 
     In another application, only one of the two said planes is utilized but where the second optical path is the same as the first optical path but with direction reversed. That is, the second optical path has input at beam  1410  and output at beam  1401 . The order of intersection of the filter elements is reversed, but this is inconsequential. 
     While FIGS. 14 a  and  14   b  illustrate complete folded birefringent filters, variations of my technique may be applied to more simplified designs. Referring to FIG. 14 d , the most general filter scheme is illustrated as three blocks: a first reflective unit  1450 , a composite crystal  1402 , and a second reflective unit  1451 . The composite crystal is located between and oriented parallel with the first and second reflection surfaces. The specific points of reflection within reflective units  1450  and  1451  are intentionally not indicated in this schematic illustration, but reflection does take place somewhere within each reflective unit. Either a reflective unit can have a single surface which reflects an optical beam at all points or a reflective unit can have multiple reflective surfaces wherein at least one of the reflective surfaces may be segmented so as to transmit the optical beam to another reflective surface within the same reflective unit. In the case where multiple reflective surfaces exist, additional optical elements, such as a waveplate, may be added between the reflective surfaces. 
     In general, an input optical beam  1401  impinges on an input window  1452  that provides for transmission of the input optical beam into the cavity. Likewise, the output beam  1410  is transmitted through an output window  1453  which provides for transmission of the optical beam which travels the optical path  1454  from the interior of the cavity to the exterior. 
     Referring to FIG. 14 a , the first and second reflective units correspond to the mirrored backsides  1404  and  1405 , respectively. Referring to FIG. 14 b , the first reflective unit corresponds to surfaces  1404  and  1424 , and the second reflective unit corresponds to surfaces  1405  and  1425 . 
     Continuous Tuning of Folded Filter 
     The optical architecture of the preceding folded crystal designs allows for a simple augmentation to provide for continuous tuning of the frequency response of the filter. Recall that the FSR of the any of the above described filters is determined exclusively by the optical length of the principal waveplate, herein realized by a composite birefringent crystal. The shape of the frequency response of determined exclusively by the sequence of orientations of waveplates located on the waveplate array. What remains to be specified is starting location in frequency of the FSR. The exact length of the composite birefringent crystal typically determines said starting location on the frequency axis. However, in practice one would like to retain control of the starting location. One method to retain such control is to be able to change the optical length of the composite crystal to within one wavelength. 
     Evans [2] described a means to provide for the equivalent of changing the optical length of a crystal to within one wavelength by utilizing a HWP located between a pair of QWPs. The QWPs are prescribed to be rotated 45 degrees about a reference axis perpendicular to the optical beam while the HWP is prescribed to be oriented at −45 degrees plus any offset. When the offset is zero, an optical beam which transits the three waveplates experiences no change. However, with a non-zero offset angle the transiting optical beam is shifted by up to one wavelength. Buhrer [3] demonstrated a configuration for continuously tunable birefringent filters, but in that disclosure each principal waveplate had to have a separate triplet of QWP, HWP, and QWP. 
     As the principal waveplate in the present disclose is the same for each filter stage, only one triplet of QWP, HWP, QWP is needed, thereby substantially simplifying a tunable filter design. 
     FIG. 15 illustrates the location of a variable retarder  1501 , comprising first QWP  1502 , first HWP  1503 , and second QWP  1502 ′, placed at the center of two half-length principal waveplates ( 1500 ,  1500 ′) which, together, form one principal waveplate. External to this assembly are a first and second HWPs  1504  and  1505 . Thus, incident optical beam  1520  transits first HWP  1504 , first half-length principal waveplate  1500 , variable retarder  1501  comprising of first QWP  1502 , first HWP  1503 , and second QWP  1502 ′, second half-length principal waveplate  1500 ′, and second HWP  1505 , finally to form output beam  1521 . The orientation of the e-axis  1510  of the half-length principal waveplates is a reference and, without loss of generality, is drawn horizontal. The e-axes  1511 ,  1511 ′ of the two QWPs  1502 ,  1502 ′ are oriented at 45 degrees to  1510 . The HWP  1503  with zero offset is oriented at −45 degrees to  1510 . Finally, the e-axis orientations of HWPs  1504  and  1505 ,  1513  and  1514 , are determined by the filter synthesis method. Rotation of the e-axis  1512  of HWP  1503  away from −45 degrees effectively changes the optical length of the variable retarder to within one. wavelength. Since the rotation may be performed in a continuous manner, the tuning of the filter response is continuous. 
     FIG. 16 illustrates a representative intensity response  1600  of a folded filter as a function of frequency. The intensity response is periodic with period length of one FSR. The FSR starting point, illustrated as point  1601 , is determined by the composite crystal length. However, with the addition of the variable retarder is a manner described in the following, the intensity response  1600  shifts position on the frequency axis without changing either the FSR or the response shape. As HWP e-axis  1512  is. rotated, the-intensity response shifts; e.g. for orientation  1611  the FSR starting frequency is  1601 , for orientation  1612  the FSR starting frequency is  1602 , and for orientation  1613  the FSR starting frequency is  1603 . 
     To realize the continuous FSR starting frequency tuning as illustrated in FIG.  16 . FIG. 17 a  illustrates the inventive architecture. The folded filter architecture is that of FIG. 14 a  with two additional elements. A fixed QWP  1502  is located parallel to and behind the composite-crystal principal waveplate and a rotatable QWP  1700  is located parallel to and behind the fixed QWP  1502 . The backside of the rotatable QWP  1700  is mirror-coated, thereby replacing the backside mirror coating of the composite crystal as illustrated in FIG. 14 a . Since the backside of the rotatable QWP is mirror-coated, the optical beam double-passes said element, generating a half-waveplate equivalent. Moreover, since the optical beam double-passes the fixed QWP  1502 , only one QWP is required rather than the two of FIG. 15,  1502  and  1502 ′. 
     FIG. 17 b  illustrates the relative e-axis orientations of the elements in the tunable folded filter. The principal waveplate  1710  has its e-axis at zero degrees,  1510 . The fixed QWP  1711  has its e-axis at 45 degrees. The rotatable QWP  1712  has its e-axis at −45 degrees plus an offset. Finally, the waveplate array  1713  has various e-axis orientation as determined by the filter synthesis method. 
     APPENDIX 
     References 
     [1] B. Loyt, Comptes Rendus vol. 197, pp. 1593, 1933. 
     [2] J. Evans, J. Opt. Soc. Amer., vol. 39, no. 3, pp. 229, 1949. 
     [3] C. Buhrer, Applied Optics, vol. 33, no. 12, pp. 2249, 1994. 
     [4] U. S. Pat. No. 4,987,567, issued to C. Buhrer on Jan. 22, 1991. 
     [5] C. Buhrer, Applied Optics, vol. 27, no. 15, pp. 3166, 1988. 
     [6] S. Harris, J. Opt. Soc. Amer., vol. 54, no. 10, pp. 1267, 1964 
     [7] J. A. Kong, “Electromagnetic Wave Theory,” John Wiley &amp; Sons, 1986.