Abstract:
A two-dimensional photonic crystal slab apparatus having a waveguiding capability is provided. Noncircular holes are introduced to replace the circular holes in the two-dimensional lattice of the photonic crystal to provide waveguiding capability. High guiding efficiency is achieved over a wide frequency region within the photonic bandgap.

Description:
FIELD OF INVENTION  
         [0001]    The present invention relates generally to the field of photonic crystals and more particularly to two-dimensional photonic crystal apparatus.  
         BACKGROUND OF INVENTION  
         [0002]    Photonic crystals (PC) are periodic dielectric structures which can prohibit the propagation of light in certain frequency ranges. Photonic crystals have spatially periodic variations in refractive index and with a sufficiently high contrast in refractive index, photonic bandgaps can be opened in the structure&#39;s optical spectrum. The “photonic bandgap” is the frequency range within which propagation of light through the photonic crystal is prevented. A photonic crystal that has spatial periodicity in three dimensions can prevent light having a frequency within the crystal&#39;s photonic bandgap from propogating in any direction. However, fabrication of such a structure is technically challenging. A more attractive alternative is to utilize photonic crystal slabs that are two-dimensionally periodic dielectric structures of finite height that have a band gap for propagation in the plane and use index-confinement in the third dimension. In addition to being easier to fabricate, two-dimensional photonic crystal slabs provide the advantage that they are compatible with the planar technologies of standard semiconductor processing.  
           [0003]    An example of a two-dimensional photonic crystal structure periodic in two dimensions and homogeneous in the third may be fabricated from a bulk material having a periodic lattice of circular air filled columns extending through the bulk material in the height direction and periodic in the planar direction. The propagation of light in two-dimensional photonic crystals is determined by a number of parameters, including radius of the cylindrical columns, the lattice spacing, the symmetry of the lattice and the refractive indices of the bulk and column material.  
           [0004]    Introducing defects in the periodic structure of a photonic crystal allows the existence of localized electromagnetic states that are trapped at the defect site and that have resonant frequencies within the bandgap of the surrounding photonic crystal material. By providing a line of such defects in the photonic crystal, a waveguiding structure is created that can be used in the control and guiding of light (see, for example, J. D. Joannopoulos, R. D. Meade, and J. N. Winn, “Photonic Crystals”, Princeton University Press, Princeton, N.J., 1995). Light of a given frequency that is prevented from propagating in the photonic crystal may propagate in the defect region.  
           [0005]    A two-dimensional photonic crystal slab waveguide usually comprises a two-dimensional periodic lattice in the form of an array of dielectric rods or air holes incorporated in a slab body. High guiding efficiency can be achieved only in a narrow frequency region close to the upper or lower edge (for dielectric rods or air holes, respectively) of the waveguide band, where there are no leaky modes. Typically, high guiding efficiency is achieved only in a narrow frequency region that is only a few percent of the center frequency of the waveguide band and existing configurations suffer from low group velocities in the allowed waveguide band. Low group velocity increases the unwanted effects of disorder and absorption. (see S. G. Johnson, S. Fan, P. R. Villeneuve, L. Kolodziejski and J. D. Joannopoulos, Phys. Rev. B 60, 5751, 1999 and S. G. Johnson, P. R. Villeneuve, S. Fan and J. D. Joannopoulos, Phys. Rev. B 62, 8212, 2000).  
           [0006]    [0006]FIG. 1 shows an xy view of prior art two-dimensional photonic crystal slab apparatus  100 . Photonic crystal slab  115  has circular holes  110  arranged to from a periodic triangular lattice with a lattice spacing equal to a. Circular holes  110  are filled with air. Region of defects  125  is created by replacing circular holes  110  of the lattice with larger circular holes  120  along a line in the x direction. Ridge waveguide  175  couples light into photonic crystal slab apparatus  100  that may have its edge at line A′, line B′ or line C′ in FIG. 1.  
           [0007]    [0007]FIG. 2 shows the transmission coefficient for two-dimensional crystal slab apparatus  100  as a function of frequency expressed in fractions of c/a—where c —is the speed of light—and a is the lattice spacing. The radius for circular holes  120  is about  0 . 45   a  and the radius for circular holes  110  is about  0 . 3   a . Curve  210  represents the unguided case which has low transmission in the bandgap and high transmission in the allow band. Curve  201  represents the case where ridge waveguide  175  is attached to photonic crystal slab  115  at the edge defined by line A in FIG. 1. Curve  202  represents the case where ridge waveguide  175  is connected to photonic crystal slab  115  at the edge defined by line B in FIG. 1. Curve  203  represents the case where ridge waveguide  175  is connected to photonic crystal slab  115  at the edge defined by line C′ in FIG. 2. The transmission for curve  203  is a maximum for a frequency of about 0.253 c/a and the waveguide band is narrow. Increasing the radius of circular holes  120  to 0.5a causes circular holes  120  to touch and start to overlap. This results in rapid deterioration of the transmission properties of two-dimensional crystal slab apparatus  100  as the light wave becomes less confined due to the decrease of the average dielectric constant of two-dimensional crystal slab  100 .  
         SUMMARY OF INVENTION  
         [0008]    In accordance with the invention, noncircular holes such as elliptical holes or rectangular holes are introduced as defects in the guiding direction of the photonic-crystal slab to create wide wave guiding bands covering more than 10% of the center frequency portion of the waveguide band. The elliptical or rectangular holes form a line of defects in the photonic crystal slab. Because low group velocities occur at the edges of the waveguide bands where the band becomes flat there is a wider range of frequencies with high group velocities available. Elliptical and rectangular holes provide significantly wider waveguide bandwidth and higher group velocity than circular holes. Over 10% of guiding bandwidth is achieved for a wide range of elliptical and rectangular shapes. The presence of a wider range of operating frequencies gives more forgiving fabrication tolerance for practical waveguide and allows more design flexibility when stub tuners, add-drop filters, bends and splitters are added. Higher group velocity will also lower the propagation loss of the waveguide. 
       
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0009]    [0009]FIG. 1 shows a view of a prior art two-dimensional photonic crystal slab apparatus.  
         [0010]    [0010]FIG. 2 shows a transmission versus frequency graph for the prior art apparatus of FIG. 1.  
         [0011]    [0011]FIG. 3 shows a view of a two-dimensional photonic crystal slab apparatus in accordance with the invention.  
         [0012]    [0012]FIG. 4 shows a side view of a two-dimensional photonic crystal slab apparatus in accordance with the invention.  
         [0013]    [0013]FIG. 5 shows a transmission versus frequency graph for a two-dimensional photonic crystal slab apparatus in accordance with the invention.  
         [0014]    [0014]FIG. 6 a  shows a transmission versus frequency graph for a two-dimensional photonic crystal slab apparatus in accordance with the invention.  
         [0015]    [0015]FIG. 6 b  shows a transmission versus frequency graph for a two-dimensional photonic crystal slab apparatus in accordance with the invention.  
         [0016]    [0016]FIG. 6 c  shows a band over midband versus semiminor axis graph for an embodiment in accordance with the invention.  
         [0017]    [0017]FIG. 6 d  shows a band over midband versus major axis to minor axis ratio for an embodiment in accordance with the invention.  
         [0018]    [0018]FIG. 7 shows a view of a two-dimensional photonic crystal slab apparatus in accordance with the invention.  
         [0019]    [0019]FIG. 8 a  shows a transmission versus frequency diagram for a two-dimensional photonic crystal slab apparatus in accordance with the invention.  
         [0020]    [0020]FIG. 8 b  shows a band over midband versus halfwidth graph for an embodiment in accordance with the invention.  
         [0021]    [0021]FIG. 8 c  shows a band over midband versus length to width ratio for an embodiment in accordance with the invention.  
         [0022]    [0022]FIG. 9 a  shows the band structure for a two-dimensional photonic crystal slab apparatus in accordance with the invention.  
         [0023]    [0023]FIG. 9 b  shows the band structure for a two-dimensional photonic crystal slab apparatus in accordance with the invention.  
         [0024]    [0024]FIG. 9 c  shows the band structure for a two-dimensional photonic crystal slab apparatus in accordance with the invention.  
         [0025]    [0025]FIG. 10 a  show a side view of the initial structure for making an embodiment in accordance with the invention using a silicon on insulator wafer.  
         [0026]    [0026]FIG. 10 b  shows an e-beam resist mask layer applied to the initial structure of FIG. 10 a.    
         [0027]    [0027]FIG. 10 c  shows the structure after patterning of the e-beam resist layer.  
         [0028]    [0028]FIG. 10 d  shows the structure after etching of the SiO 2  layer to form a mask for subsequent etching of the silicon layer.  
         [0029]    [0029]FIG. 10 e  shows the completed structure after etching in accordance with the invention.  
         [0030]    [0030]FIG. 11 a  shows a side view of the initial structure for making an embodiment in accordance with the invention using a GaAs substrate.  
         [0031]    [0031]FIG. 11 b  shows an e-beam resist layer applied to the initial structure of FIG. 10 a.    
         [0032]    [0032]FIG. 11 c  shows the structure after patterning of the e-beam resist layer.  
         [0033]    [0033]FIG. 11 d  shows the structure after etching of the SiO 2  layer to form a mask for subsequent etching of the GaAs layer.  
         [0034]    [0034]FIG. 11 e  shows the structure after etching of the GaAs layer.  
         [0035]    [0035]FIG. 11 f  shows the completed structure after oxidation of the aluminum containing layer. 
     
    
     DETAILED DESCRIPTION OF THE INVENTION  
       [0036]    [0036]FIG. 3 shows the xy view of typical two-dimensional photonic crystal slab apparatus  300  in an embodiment in accordance with the invention. Photonic crystal slab  315  has circular holes  310  arranged to form a periodic triangular lattice. A periodic honeycomb lattice may also be used. Circular holes  310  are etched through photonic crystal slab  315  and are typically filled with a low dielectric constant material such as air. A high dielectric contrast is typically required to open a bandgap in the xy plane. In accordance with the invention, region of defects  325  is created by replacing circular holes  310  of the lattice with elliptical holes  320  along a line, for example, in the x direction. Elliptical holes  320  are typically filled with the same dielectric material as circular holes  110 . Ridge waveguide  375  couples light into photonic crystal slab apparatus  300  that may have its edge at line A, line B or line C in FIG. 3.  
         [0037]    [0037]FIG. 4 shows an xz cross-sectional view of photonic crystal slab apparatus  300 . Photonic crystal slab  315  is sandwiched between cladding layer  422  and cladding layer  424 . Typically, cladding layer  422  is positioned between substrate layer  410  and photonic crystal slab  315  (see FIG. 2). Typically, photonic crystal slab  315  is made from a material having a high dielectric constant compared to air such as silicon or a III-V based semiconductor. To provide index confinement in the z direction, cladding layers  422  and  424  are typically SiO 2  or other material having a lower dielectric than the material of photonic crystal slab  315 . Substrate layer  410  is typically of the same material as photonic crystal slab  315  to provide mechanical support but may also be air. Layer  412  positioned over cladding layer  424  is typically air.  
         [0038]    Photonic crystal slab apparatus  300  is capable of transmitting light having a frequency that lies within the bandgap of photonic crystal slab  315  in a straight line. The waveguide band for photonic crystal slab apparatus  300  depends on a number of factors. Increasing the thickness of photonic crystal slab  315  while keeping all other parameters constant increases the effective dielectric constant and shifts the waveguide band of photonic crystal slab apparatus  300  to lower frequencies. Increasing the cross-section of circular holes  310  while keeping all other parameters constant decreases the effective dielectric constant and shifts the waveguide band of photonic crystal slab apparatus  300  to higher frequencies.  
         [0039]    Increasing the thickness of cladding layers  422  and  424  shifts the position of the waveguide band. If layers  412  and  410  are air, increasing the thickness of cladding layers  422  and  424  slowly moves the position of the waveguide band to lower frequencies and saturates where further increases in thickness produce no further shift in waveguide band position. On the other hand, if layer  410  is not air but, for example, silicon, the position of the waveguide band moves to higher frequencies as cladding layers  422  are increased in thickness and saturates where further increases in thickness produce no further shift in waveguide band position.  
         [0040]    A finite difference time domain method is used to simulate the performance of photonic crystal slab apparatus  300 . In the simulation, ridge waveguide  375  couples the light into photonic crystal slab  315 . Photonic crystal slab is taken to have a typical thickness of about 0.6a. A complete line of circular holes  310  is removed along the x direction and replaced by elliptical holes  320 . For calculations, the lattice constant is fixed. For example, if a waveguide band is theoretically indicated to exist at a value of about a/λ=0.26 and the wavelength to be transmitted is about 1.55 microns then the lattice constant a is chosen to be about 0.4 microns. In the calculations, a dielectric slab of thickness of 0.6 a and dielectric constant of 12.96 is used. The slab is placed on top of a semi-infinite thick material of dielectric constant of 2. The ridge waveguide used to couple light to the photonic crystal is 2.28 a wide.  
         [0041]    For the plots shown in FIG. 5, elliptical holes  320  have semiminor axis of about 0.33a and a semimajor axis of about 0.74a where a is the lattice constant. Circular holes  310  have a radius of about 0.3a. Curve  501  shows transmission versus frequency for the case where ridge waveguide  375  contacts photonic crystal slab  315  at the left edge defined by line A (see FIG. 3). Curve  502  shows transmission versus frequency for the case where ridge waveguide  375  contacts photonic crystal slab  315  at a left edge defined by line B (see FIG. 3). Curve  503  shows transmission versus frequency for the case where ridge waveguide  375  contacts photonic crystal slab  315  at a left edge defined by line C (see FIG. 3). It is apparent from the plots in FIG. 5 that it is not desirable to couple to photonic crystal slab  315  along an edge defined by line C for the case of elliptical holes having a minor axis of about 0.33a. The choice of whether to couple ridge waveguide  375  at line A or line B depends on the frequency of the light to be transmitted. For frequencies in a narrow band less than about 0.245 c/a, curve  502  indicates bettertransmission, while curve  501  indicates better transmission for frequencies between about 0.25 c/a to about 0.275 c/a. For all the cases in FIG. 5, left and right interface between the ridge waveguide and the photonic crystal are the same.  
         [0042]    For the plots shown in FIG. 6 a , elliptical holes  320  have a semiminor axis of about 0.37a and a semimajor axis of about 0.738a where a is the lattice constant. Circular holes  310  have a radius of about 0.3a. The width of the waveguide band is about 0.176 of the mid-band frequency. Curve  601  shows transmission versus frequency for the case where ridge waveguide  375  contacts photonic crystal slab  315  at the left edge defined by line A (see FIG. 3) and provides a maximum transmission of about 0.89 at the lower edge of the waveguide band. Curve  602  shows transmission versus frequency for the case where ridge waveguide  375  contacts photonic crystal slab  315  at a left edge defined by line B (see FIG. 3) and provides a maximum transmission of about 0.89 at the upper edge of the waveguide band. Curve  603  shows transmission versus frequency for the case where ridge waveguide  375  contacts photonic crystal slab  315  at a left edge defined by line C (see FIG. 3) and provides a maximum transmission of about 0.5. It is apparent from the plots in FIG. 6 a  that it is not advantageous to couple to photonic crystal slab  315  along an edge defined by line C for the case of elliptical holes  320  having a minor axis of about 0.37a as this typically provides low transmission.  
         [0043]    Coupling ridge waveguide  375  at line B provides a transmission coefficient greater than about 0.5 in the frequency band from about 0.24 c/a to about 0.28 c/a. Coupling ridge waveguide  375  at line A provides a transmission coefficient greater than about 0.5 in the frequency band from about 0.26 c/a to about 0.285 c/a. Coupling ridge waveguide  375  at line C provides a tranmission coefficient of about 0.5 or less for all frequencies of interest as shown by curve  603 .  
         [0044]    [0044]FIG. 6 b  shows the negative effect on transmission that occurs when elliptical holes  320  contact and overlap circular holes  310 . Curve  651  shows the transmission for elliptical holes  320  with a semiminor axis of about 0.39a and a semimajor axis of about 0.872a. Curve  652  shows the transmission for elliptical holes  320  with a semiminor axis of about 0.41a and a semimajor axis of about 0.917a. In both cases, elliptical holes  320  contact and overlap circular holes  310 . As the overlap between elliptical holes  320  and circular holes  310  increases (as the semiminor axis increases) it is apparent that transmission drops off rapidly due to decreased confinement of the wave.  
         [0045]    [0045]FIG. 6 c  shows the width of the waveguide band over the mid-band frequency versus the semiminor axis in units of the lattice constant, a with the ratio of the major axis to the minor axis fixed at about 2.236 for elliptical holes  320 . Curve  654  shows that the maximum width of the mid-band frequency is about 0.176.  
         [0046]    [0046]FIG. 6 d  shows the width of the waveguide band over mid-band frequency versus the ratio of the major axis to the minor axis where the semiminor axis is fixed at about 0.37a. Increasing the ratio of the major axis to the minor axis increases the bandwidth as shown by curve  656 . In both FIGS. 6 c  and  6   d , after the maximum bandwidth shown is reached, the transmission of photonic crystal slab apparatus  300  will decrease rapidly as elliptical holes  320  begin to overlap with circular holes  310  as indicated, for example, in FIG. 6 b.    
         [0047]    The present invention is not limited to using elliptically shaped holes. For example, in accordance with an embodiment of the invention, elliptical holes  320  may be replaced by rectangles  720  to make photonic crystal slab apparatus  700  as shown in FIG. 7. Circular holes  710  have a radius of 0.3a. FIG. 8 a  shows transmission versus frequency for ridge waveguide  375  coupled to photonic crystal slab  715  along the edge defined by line B″ in FIG. 8 a  for both curves  812  and  815 . Curve  812  shows transmission versus frequency for rectangles  720  having a short side of length of about 0.58a and a long side of length about 1.3a. Curve  815  shows transmission versus frequency for rectangles  720  having a short side of about 0.62a and a long side of about 1.38a. Transmission for curve  815  is worse because rectangles  720  start to touch and overlap with circular holes  710 .  
         [0048]    [0048]FIG. 8 b  shows the width of the waveguide band over mid-frequency versus the half-width of rectangle  720  in units of the lattice constant, a with the ratio of rectangle length to width fixed at about 2.236. As curve  821  shows, the maximum bandwidth is about 0.164.  
         [0049]    [0049]FIG. 8 c  shows the width of the waveguide band over mid-frequency versus the ratio of the length to the width for rectangle  720  and a rectangle halfwidth of about 0.29a.  
         [0050]    [0050]FIG. 9 a  shows the band structure for photonic crystal slab apparatus  300  shown in FIG. 3. Solid lines  910  and  920  denote the band edges while line  905  marks the boundary of the lightcone. For the elliptical line defect in FIG. 3 there are three even modes  938 ,  939  and  940 . Even mode  939  has low group velocities, however even modes  940  and  939  have higher group velocities.  
         [0051]    [0051]FIG. 9 b  shows the band structure for photonic crystal slab apparatus  700  for rectangles  720  in FIG. 7 having a short side of about  0 . 58   a  and a long side of about  1 . 3   a . For the rectangle line defect, even modes  948  and  949  have an overlap near the frequency of about 0.27 c/a which indicates mode mixing is present which is not desirable in single mode applications. Even mode  950  is comparable to even mode  940  in FIG. 9 a.    
         [0052]    [0052]FIG. 9 c  shows the band structure for photonic crystal slab apparatus  700  with rectangles  720  in FIG. 7 having a short side of about  0 . 62   a  and a long side of about 1.38a. Again three even modes  960 ,  959  and  958  are present. However, transmission is less effective for the configuration in FIG. 9 c.    
         [0053]    In accordance with the invention, elliptical holes  320  and rectangular holes  720  provide much wider waveguide bands than do circular holes  120 . In all cases, transmission decreases rapidly when there is an overlap between noncircular holes such as elliptical holes  320  or rectangular holes  720 . Transmission also decreases rapidly when there is an overlap between noncircular holes and circular holes  310  or  710 . However, over 10% guiding bandwidth is achieved for a wide range of elliptical and rectangular shapes. Rectangle-like holes with rounded shapes close to neighboring holes also results in wide waveguide bands.  
         [0054]    Photonic crystal slab apparatus  300  or photonic crystal slab apparatus  700  may be fabricated in accordance with an embodiment of the invention as shown in FIGS. 10 a - 10   e . The initial structure is a silicon on insulator (SOI) structure having silicon layer  1010  and SiO 2  layer  1012 . A photonic crystal structure is fabricated in Si layer  1010 . Typically, thin SiO 2  layer  1015  is deposited over Si layer  1010  to serve as a mask layer for subsequent etching of Si layer  1010  as shown in FIG. 10 a . E-beam resist layer  1020  is typically deposited over thin SiO 2  layer  1015  to a typical thickness of about 400 nm as shown in FIG. 10 b . Resist layer  1020  is patterned to the desired lattice hole pattern using e-beam lithography as shown in FIG. 10 c . Then, thin SiO 2  layer  1015  is etched using reactive ion etching to obtain the desired lattice hole pattern as shown in FIG. 10 d . Following creation of the desired mask pattern, the lattice pattern etched in SiO 2  layer  1015  is transferred to Si layer  1010  by a controlled etch typically using HBr shown in  10   e . Note that Si layer  1010  is overetched resulting in penetration into SiO 2  layer  1020 . The completed two-dimensional photonic crystal slab apparatus after the controlled etch is shown in FIG. 10 e  in a side view.  
         [0055]    Photonic crystal slab apparatus  300  or photonic crystal slab apparatus  700  may be fabricated in accordance with an embodiment of the invention as shown in FIGS. 11 a - 11   f . Typically, thin SiO 2  layer  1115  is deposited over GaAs layer  1110  to a typical thickness of about 200 nm to serve as a mask layer for subsequent etching of GaAs layer  1110  as shown in FIG. 11 a . GaAs layer  1110  is attached to AlGaAs layer  1112 . E-beam resist layer  1120  is deposited over thin SiO 2  layer  1115  to a typical thickness of about 400 nm as shown in FIG. 11 b . Resist layer  1120  is patterned to the desired lattice hole pattern using e-beam lithography as shown in FIG. 10 c . Then, thin SiO 2  layer  1115  is etched using reactive ion etching with CHF 3  to obtain the desired lattice hole pattern as shown in FIG. 11 d . Following creation of the desired mask pattern, the lattice pattern etched in SiO 2  layer  1115  is transferred to GaAs layer  1110  by a reactive ion etch typically using Cl 2  as shown in FIG. 1 e . A steam oxidation process is then performed of AlGaAs layer  1112  to convert layer  1112  to AlO 2  to obtain the proper refractive index ˜1.5  
         [0056]    While the invention has been described in conjunction with specific embodiments, it is evident to those skilled in the art that many alternatives, modifications, and variations will be apparent in light of the foregoing description. Accordingly, the invention is intended to embrace all other such alternatives, modifications, and variations that fall within the spirit and scope of the appended claims.