Planar photonic crystal gradient index lens, simulated with a finite

PHYSICAL REVIEW B 74, 113103 共2006兲

Planar photonic crystal gradient index lens, simulated with a finite difference time domain method Filippus S. Roux* and Israel De Leon† Department of Electrical, Electronic and Computer Engineering, University of Pretoria, Pretoria 0001, South Africa 共Received 13 April 2006; revised manuscript received 31 July 2006; published 11 September 2006兲 We demonstrate the performance of a planar gradient index 共GRIN兲 lens in a two-dimensional photonic crystal. The gradient index is achieved by spatially varying the air-hole radius in the photonic crystal, resulting in a smooth variation of the effective refractive index. We simulate the GRIN lens using a two-dimensional finite difference time domain method to show that it performs as conventional GRIN lenses do. Some of the differences between conventional GRIN lenses and the photonic crystal GRIN lens are discussed. DOI: 10.1103/PhysRevB.74.113103

PACS number共s兲: 42.70.Qs, 42.25.Bs, 42.30.Va, 42.30.Kq

Integrated photonic systems provide a solution for the interconnection challenge in microelectronic systems. Such integrated photonic systems would necessarily be twodimensional and should ideally be monolithic, manufactured from the same substrate that is used for the microelectronics. It is therefore necessary that diverse optical systems can be designed within these constraints. One of the most powerful processing capabilities of optics is the instantaneous Fourier transforming property of a conventional lens.1 One can exploit this capability in integrated photonic systems if one can produce such a two-dimensional 共2D兲 lens. There are several ways in which one can make a planar 2D lens. For instance, it has been shown that one can construct such a lens using, a slab of a photonic crystal, operated under conditions where it behaves like a negative index material.2 Although it does have imaging capabilities,3 such an imaging device does not have the Fourier transforming capability. 共It cannot focus plane waves; it merely reproduces them.兲 Here we employ the properties of photonic crystals to produce a planar 2D gradient index 共GRIN兲 lens,4 which has the Fourier transforming capability. First we briefly review some pertinent aspects of GRIN lenses in general. To produce a planar 2D GRIN lens 共in any medium兲, one needs a varying refractive index, given by the function n共y兲 = n0共1 − Ay 2兲1/2 ⬇ n0 − 21 n0Ay 2 ,

共1兲

where y represents the transverse direction 共perpendicular to the propagation direction兲, which replaces the transverse radial coordinate r in three-dimensional GRIN lenses; n0 is the maximum refractive index found in the center on the optical axis of the lens; and A is the 共squared兲 gradient constant of the lens. The latter is usually small compared to the square of the width of the lens and it is the dimension parameter that determines the scale for the focal length of the lens. The focal length of such a GRIN lens varies as the inverse of a sine function with respect to the length of the GRIN lens L. This periodic property of a GRIN lens gives rise to the pitch length L p = 2␲ / 冑A, which represents one complete period. The minimum focal length is obtained for a quarterpitch length, in which case the GRIN lens focal length becomes 1098-0121/2006/74共11兲/113103共4兲

f p/4 =

nb

n0冑A sin共L p/4冑A兲

=

nb

n 0 冑A

,

共2兲

where nb is the 共background兲 refractive index of the medium that surrounds the GRIN lens. The refractive index of the medium in front of, and behind, a GRIN lens affects the front and back effective focal lengths of the GRIN lens. The length of a quarter-pitch GRIN lens can be expressed in terms of the focal length and is given by L p/4 =

␲

= 2 冑A

␲n0 f p/4 . 2nb

共3兲

Such a GRIN lens represents a 2-f system 共which has a Fourier relationship between its input and output planes兲 and the front and back surfaces of the GRIN lens, respectively, represent the input and output planes 共front and back focal planes兲 of the 2-f system. An incident Gaussian beam with a waist size of ␻0 in the front focal plane is changed into an output Gaussian beam with a waist size of

␻1 =

␭f , ␲␻0

共4兲

in the back focal plane, where ␭ is the wavelength in the surrounding medium 共with refractive index nb兲 and f is the focal length of the lens in the 2-f system. Note that the relationship in Eq. 共4兲 comes from the Fourier-relationship between a Gaussian function and its Gaussian spectrum. It is therefore valid for any 2-f system and not only for a quarterpitch GRIN lens. A half-pitch GRIN lens represents a 4-f system 共two cascaded 2-f systems兲, where the front and back surfaces of the GRIN lens represent the system’s input and output planes, respectively. This 4-f system is an imaging system with a unity magnification, therefore, the incident Gaussian beam waist size is reproduced at the back surface 共output plane兲 of the GRIN lens. Here we demonstrate that a planar 共2D兲 GRIN lens can be implemented with a 2D photonic crystal by varying the crystal parameters such that the effective refractive index varies in the transverse direction of the crystal structure. Previously, graded photonic crystals have been used to implement tapers for waveguides5,6 and to bend light beams.7 We use a 2D finite difference time domain 共FDTD兲 analysis to simulate the photonic crystal GRIN lens, which con-

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©2006 The American Physical Society


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FIG. 1. 共Color online兲 Band structure for a hexagonal lattice with hole-sizes 0.3a 共solid lines兲 and 0.32a 共dashed lines兲, showing the second band 关green共bottom兲兴, third band 关red共middle兲兴, and fourth band 关blue共top兲兴. The normalized frequency where the device is operated 共0.37兲 is shown as a horizontal black line.

sists of a hexagonal photonic lattice of air-holes in a dielectric medium, with dielectric constant ⑀r = n2b = 12.08. Note that we use the same refractive index nb for the homogeneous medium in front of the photonic crystal and for the background medium of the photonic crystal. This is because in an integrated optical application one would prefer to manufacture these photonic crystals from the same medium that surrounds it. One can define an effective refractive index for a 2D photonic crystal8–12 in frequency ranges where the equifrequency contours 共EFC’s兲 are approximately circular over the width of the paraxial beam. The effective index is positive 共negative兲 if the group velocity vg = ⵜk␻共k兲, which is perpendicular to the EFC, points away from 共toward兲 the center of the EFC. Here we use a positive range of refractive indices. Therefore, we work in the fourth band of the photonic crystal’s band structure at a normalized frequency 共a / ␭0兲 of 0.37. Apart from nb’s effect on the front and back effective focal lengths of the GRIN lens, it also affects the operation of this GRIN lens through its effect on the dispersion curve of the photonic crystal, which in turn determines the effective refractive index inside the crystal. The varying refractive indices are then obtained by varying the radii of the air-holes from 0.3a to 0.32a, where a is the lattice constant. Note that because the hole size varies across the photonic crystal it does not have a unique band structure. The band structures 共second, third, and fourth bands兲 for two of the hole radii 共0.3a and 0.32a兲 are shown in Fig. 1. Note that the EFC’s are approximately circular because the horizontal line at the normalized frequency of 0.37 cuts through the fourth band curves at approximately the same distance from the ⌫-point along both the ⌫-M and the ⌫-K directions. Therefore, one can define effective refractive indices over this range of hole radii. There is also an EFC around the K point for hole radii of 0.3a, but since the beam is launched along the ⌫-M direction it does not affect the beam. The range of effective refractive indices 共0.34–0.4兲 is smaller than 1, which is to be expected because circular EFC’s are mostly found close to a band edge where the radius of the EFC shrinks to zero. This implies that the phase velocity diverges, however the group velocity approaches

FIG. 2. 共Color online兲 FDTD simulation result 共E-field兲 of a Gaussian beam with ␻0 = 2.26 ␮m and ␭0 = ␲ / 2 ␮m propagating through a 2D photonic crystal GRIN lens. The dashed line represents an interface between the homogeneous background medium on the left-hand side and the photonic crystal on the right-hand side.

zero at the band edge. The radii of the air holes vary gradually along the transverse direction as to give a parabolic variation in the effective refractive index. We used an empirically determined relationship between the hole radius and the effective refractive index to obtain the required index variation. The resulting variation of the hole size is given by ␣ = 0.306 192+ 0.000 164y 2, where ␣ is the hole radius as a fraction of the lattice constant a and y is the transverse coordinate measured in ␮m. In our FDTD simulation the front surface of the photonic crystal GRIN lens is located at x = 1.67a, where x is defined along the propagation direction. The length of the photonic crystal in the x direction is 80a cos 30° 共the cos 30° is a result of the crystal orientation兲, which gives us roughly a half-pitch GRIN lens. We do not simulate another homogeneous medium behind the photonic crystal GRIN lens because that would increase reflections from the back surface of the GRIN lens. Such reflections need to be addressed for the practical implementation of such a device, but this is beyond the scope of the present analysis. In the y direction the structure is made wide enough 共60a兲 to avoid reflections from the sides. We use Berenger’s perfectly matched layer absorbing boundary conditions13 on all sides. To be definite we choose the wavelength in air to be ␭0 = ␲ / 2 ␮m, giving a lattice constant of a = 0.58 ␮m, and we use a Gaussian beam with an incident beam waist of ␻0 = 2.26 ␮m in our simulation. The designed GRIN lens parameters are n0 = 0.4 and A = 0.0037 ␮m−2, which would give a quarter-pitch GRIN lens focal length of f = 142 ␮m. The size of the Yee cells are ⌬x = ⌬y = a / 30 and the time increments are ⌬t = ⌬x / Sc, where c is the speed of light in vacuum, and S = 2 is a stability factor. The FDTD result after 20 000 time steps is shown in Fig. 2. The wave propagates from left to right. The fine spotted appearance of the field is a result of the fine periodic structure. One can see that, starting from its initial waist size, the Gaussian beam diverges to reach a maximum width halfway through the lens. Then it converges again to reach another waist of approximately the same size as the incident waist at the back of the lens. This familiar behavior confirms that the photonic crystal structure indeed behaves like a GRIN lens. However, there are a few notable differences between the

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FIG. 3. 共Color online兲 The k-space representation of the E-field in the GRIN lens shown in Fig. 2. The hexagonal grid of dashed lines represent the primitive cell boundaries. The primitive cell in the center is the first Brillouin zone. The inset shows a blown-up version of the peaks in one of the primitive cells, with the dominant peak labeled by A.

conventional GRIN lens and the photonic crystal GRIN lens. Perhaps the most significant difference, which is a benefit of the photonic crystal GRIN lens, is the fact that the effective refractive index is smaller than 1. The implication is that the length of the GRIN lens is much shorter than an equivalent GRIN lens in conventional material would have been. The reason for this can be seen from Eq. 共3兲, which shows that the length of a GRIN lens for a given focal length is proportional to n0. The fact that the effective index is smaller than 1 is a consequence of operating the device in the fourth band. One could also have designed it for the first band where the dispersion curve looks like that of an ordinary homogeneous medium. This would imply that the photonic crystal is used purely as an effective medium,14,15 while the detail of the entire band structure becomes irrelevant. The benefit of working in the fourth band is that the holes are not as small compared to the wavelength as they are in the first band. This makes the fabrication of the device for a particular frequency easier. Another difference comes from the fact that the field inside a photonic crystal exists in the form of a superposition of Bloch modes. This implies that the k-space representation 共or spectrum兲, shown in Fig. 3, consists of several harmonics 共visible as small peaks in Fig. 3兲, located in different primitive cells. The interface between the homogeneous medium and the photonic crystal is oriented vertically with respect to the spectrum in Fig. 3. The peaks in Fig. 3 are very narrow because of the relatively large area over which the field exists in the photonic crystal. For the case of a photonic crystal operated in the fourth band, as considered here, the spectrum is severely suppressed inside the first Brillouin zone,16 which is located in the center of Fig. 3. The dominant part of the spectrum in this case lies in the fourth Brillouin zone, which is located in the ring of primitive cells neigboring the first Brillouin zone. Each of these primitive cells contain a bright spot 共the dominant peak兲 on the right-hand side and a faint spot 共lower peak兲 on the left-hand side. The dominant peaks represent the forward propagating field and the lower peaks represent a small reflected field. The latter is a numerical

FIG. 4. 共Color online兲 Demodulated reconstruction of the forward propagating beam in the primitive cell labeled by A in Fig. 3. The dashed line represents an interface between the homogeneous background medium on the left-hand side and the photonic crystal on the right-hand side.

artifact that results from the absorbing boundary condition at the end of the GRIN lens in the FDTD simulation. In the fourth Brillouin zone the wavelength is smaller than 1.15a, while the free space wavelength at a normalized frequency of 0.37 is 2.7a. So provided that the 2D photonic crystal is sandwiched in the z direction between layers of sufficiently lower index materials, out-of-plane radiation losses can be avoided. Some of the Bloch mode harmonics would give rise to higher diffraction orders behind the GRIN lens if these harmonics can couple to propagating plane waves in the homogeneous medium. These are the harmonics located in primitive cells that lie above or below the horizontal row of primitive cells that include the first Brillouin zone, as shown in Fig. 3. To avoid this situation the medium behind the lens must support propagating modes only for harmonics in the horizontal row of primitive cells that include the first Brillouin zone. If not, the higher orders would couple to propagating orders behind the lens. Then, in addition to the expected desired zeroth order, there would also be higher diffraction orders. Depending on the rest of the system, these may interfere with the desired beam, causing noise and a loss of information. The higher diffraction orders would also carry away some of the optical power giving rise to a loss of optical power in the desired zeroth order. The behavior of the GRIN lens can be seen more clearly by looking at only one of the dominant peaks. For this purpose we use the peak in the primitive cell shown in the inset in Fig. 3, labeled by A. The demodulated16 reconstructed beam of this peak is shown in Fig. 4. To demodulate this peak we first isolated it by multiplying the k-space spectrum by an aperture function that is equal to 1 over the area of the peak and 0 elsewhere. Then we shift the primitive cell in which the peak is located to the location of the first Brillouin zone. Finally, we transform the result back to the spatial domain. One can see that the effective wavelength is relatively large. This is because the effective refractive index is small. One can also see the curvature of the wavefront as it first diverges toward the maximum width and then converges toward its waist at the back, in accordance with the way light propagates in a GRIN lens. Since the first 共or second兲 half of our simulated GRIN

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lens represents a 2-f system, with its associated Fourier transforming property, we expect to see the Gaussian spectrum of the Gaussian input function midway through the photonic crystal in Fig. 4 共or Fig. 2兲, as we indeed do—the waist sizes are related by Eq. 共4兲, which confirms the Fourier relationship between the input function and the Gaussian function with the maximum waist in Fig. 4. In conclusion, we performed an 2D FDTD simulation of the behavior of a photonic crystal GRIN lens. In this way we demonstrated a successful implemention of a 2D GRIN lens in a 2D photonic crystal. The gradient index is achieved by

The authors wish to thank the SiLED group at the Carl and Emily Fuch Institute for Micro-electronics for the use of their computing facilities. One of the authors 共I.D.L.兲 would like to thank the Faculty of Engineering, Built Environment and Information Technology for their hospitality and financial support.

9 S.-Y.

*Electronic address: [email protected] †Electronic

varying the radii of the air holes in the hexagonal photonic lattice. A benefit of a photonic crystal GRIN lens is the fact that the effective index is typically smaller than 1, which implies a shorter length for a given pitch, thus occupying less real estate.

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