Experimental results on wavelength-dependent angular dispersion in InGaAsP triangular lattice planar photonic crystals are presented. An abrupt variation of the angular dispersion is observed for TM-polarized waves whose frequencies are comprised between those of the fourth and sixth allowed bands. According to the crystal period, the measured angle of refraction is found to either decrease or increase by 30° within a wavelength range smaller than 30 nm. Experimental results are reproduced well from 2D finite difference time domain calculations. The observed phenomena are interpreted from the coupling of the incident light to different modes of the photonic crystal that travel with different group velocities and propagate in different directions within the crystal. Mode dispersion curves and mode patterns are calculated along with isofrequency curves to support this explanation. The observed discontinuous wavelength super-refraction opens a new approach to the application of superprisms.
© 2006 Optical Society of America
One of the main points of interest of photonic crystals (PhC) stems from their capabilities to handle the flow of light on propagation distances of only several wavelengths [1–5]. The realization on a miniature scale of optical structures performing wavelength multiplexing/demultiplexing or add/drop functions and allowing a tight control of light propagation was, and is still remaining one of the major motivations for the research in this domain.
One particular example of the wavelength demultiplexing devices is the “superprism” . Its operation is based on the group velocity dispersion related effect. A large change in the deflection angle of a light beam within the photonic crystal is achieved by a slight change of the wavelength or of the incident angle. The usual approach to a high wavelength angular dispersion consists in operating within sharp corner regions of the PhC dispersion diagram near the bandgap. This “superprism” effect has been demonstrated by Kosaka et al. in “autocloned” three-dimensional PhCs [7, 8] as well as by Wu [9, 10], Baumberg  and Lupu  in planar photonic microstructures.
The aim of the present work is to investigate the “superprism” effect in a planar 2D semiconductor structure having a low index contrast in the vertical direction and operating on high photonic crystal bands. Our experiments reveal an abrupt variation of the angular dispersion for TM-polarized incident waves whose frequencies are comprised between those of the fourth and sixth allowed bands. Experimental results are well reproduced by 2D finite difference time domain (FDTD) calculations, and are interpreted from the coupling of the incident light to different modes of the photonic crystal. This discontinuous super-refraction effect presents a potential interest for new photonic devices.
2. Experimental configuration
Our material system is a guiding InGaAsP/InP heterostructure perforated by a triangular PhC lattice of circular air holes, as represented in Fig. 1(a). The guide core consists of a 500 nm thick quaternary alloy InGaAsP layer with a composition corresponding to λg=1.18 μm. This layer is surrounded by the InP underclad substrate and a 500 nm thick InP overclad layer on top. Two different PhC structures with the same averaged air filling factor of 19–20% are investigated in this work. One has a lattice parameter a of 672 nm and hole diameter of 334 nm while the other has a lattice period of 744 nm and hole diameter of 372 nm. The structures were fabricated by using electron beam lithography and inductively coupled plasma etching (ICP). As shown in Fig. 1(a), high quality samples with very low sidewall roughness were obtained with these techniques.
The experimental device used to test the “superprism” effect is shown in Fig. 1(b). An input ridge waveguide of either 2 μm or 3 μm width is used to inject light in the PhC area at an incidence angle of 8° with respect to the Γ-K direction of the photonic crystal. The PhC fills a 80 μm wide and around 10 μm thick rectangular area with 14 periods of holes arrays and acts as a parallel optical plate. The light traveling through the PhC crystal area is still guided in the vertical direction by a short slab waveguide region and exits the sample by the cleaved edge. The output facet is antireflection coated to avoid light back reflection into the photonic crystal. The total distance between the end of the input waveguide and the cleaved output facet is 56 μm.
The PhC parallel plate geometry  was presently preferred to the half-circle geometry used in most of the previous works [9, 10, 12]. In the half-circle geometry, the angular resolution is determined by the angular separation of the output waveguides, which are used to collect the light transmitted through the photonic crystal. This typically leads to angular resolutions comprised between 3.5°  and 10° [9, 10]. The advantage of the parallel plate geometry is the absence of any output waveguide. The angular deviation can then be determined by measuring the shift with the wavelength of the transmitted light beam. This allows one to investigate the wavelength-dependent angular dispersion with a much higher resolution, the latter being essentially limited by the optical resolution of the experimental setup.
Another important advantage of our design stems from the fact that it eliminates most of the parasitic diffracted beams due to the small value (~ 18°) of the critical angle for total internal reflection at the slab/air interface. Indeed, most of the diffracted beams are oriented along directions close to the main PhC axes and thus propagate at angles, which are close to a multiple of 30 and 60 degrees with respect to the slab/air interface (see for example [5, 13] and also our results in Figs. 4 and 5(b)). These beams do not exit the slab waveguide unlike the case of the semi-spherical geometry.
3. Experimental results
The experimental set-up used to characterize the devices was based on a tunable semiconductor laser operating in the spectral range from 1520 to 1610 nm. The linearly polarized laser light was coupled into the input waveguide (Fig. 1(b)) using a polarization maintaining lensed fiber. The output light was collected by a 40× objective and was either imaged with an IR vidicon camera or measured with a 1D CCD array high sensitivity IR camera that provided the spatial distribution of the transmitted light. A polarizer was inserted in the optical path between the light collecting objective and the IR camera to control the polarization of the transmitted light.
The experimental set-up then allowed us to measure the wavelength-dependent angular dispersion phenomena for both the TE and TM light polarizations. The evolution of the light intensity distribution measured at the sample output facet is represented in Fig. 2 for the two lattice periods, the two polarizations and an input waveguide of 3 μm width.
The same (reference) level is used for the intensity distributions in Figs. 2(a)–2(d) in order to allow the comparison of the results for different structures and polarizations. The zero distance on the horizontal axes in Figs. 2(a)–2(d) corresponds to the output position at the end slab facet of the beam, which would propagate in the absence of the PhC region. This beam trajectory is represented as a dotted red line in Fig. 1(b).
The results for the TM polarization are reported in Fig. 2(a) (a = 0.672 μm) and Fig. 2(b) (a = 0.744 μm), respectively. Those for TE polarization are reported in Fig. 2(c) (a = 0.672 μm) and Fig. 2(d) (a = 0.744 μm), respectively. This representation allows us to clearly show the variation of the light deflection with the wavelength and polarization. The use of two lattice periods allows us to explore two neighboring regions of normalized frequencies comprised between 0.417<a/λ <0.442 and 0.462 <a/λ < 0.489, respectively.
As seen in Fig. 2, the evolutions are different for the TE and TM polarizations. In the TE case [Figs. 2(c) and 2(d)] and for each lattice period, only one lobe (i.e., one transmitted beam) is observed at the sample output except for the range of frequencies between a/λ = 0.43 and a/λ ≈ 0.442 where the transmission is very weak. In the TM case [Figs. 2(a) and 2(b)], single-lobe regions are found, but clearly there are also intermediate situations where the output intensity distribution is shifted from one lobe to another. These transitions occur within wavelength ranges smaller than 30 nm. Correspondingly, the measured angle of refraction within the photonic crystal is found to change by approximately 30°.
For the PhC lattice with 0.672 μm period and at the wavelength of 1610 nm (i.e., a/λ = 0.417), the energy of the transmitted light is essentially concentrated into one lobe [Fig. 2(a)]. The intensity maximum of this lobe is four times larger than that of the secondary lobe. The calculated refraction angle is -20° [negative refraction, blue crosses in Fig. 3(a)]. The angular dispersion with wavelength is – 0.1°/nm. However, the intensity of this dominant lobe decreases at shorter wavelengths while that of a secondary lobe regularly increases. At 1580 nm (i.e., a/λ = 0.425), the second lobe is dominant : the position of the transmitted beam has thus shifted by more than 3 μm at the output facet.
The refraction angle calculated at the wavelength of 1580 nm is 16° [red circles on Fig. 3(a)]. The transmitted output then remains single lobe at shorter wavelengths down to 1520 nm [a/λ = 0.442, see Fig. 2(a)]. The refraction angle essentially decreases with the wavelength, thus leading to a negative angular dispersion.
A nearly single-lobe transmission with a similar wavelength dispersion is obtained for the PhC lattice with 0.744 μm period [Fig. 2(b), red circles in Fig. 3(b)]. This behavior occurs for wavelengths between 1590 and 1600 nm, i.e. for normalized frequencies varying from 0.462 to 0.468. Starting from 1590 nm, a second maximum appears in the spatial distribution of the transmitted light [green stars in Fig. 3(b)]. Correspondingly, the measured refraction angle within the PhC area shifts by around 30°, this shift being of opposite sign compared to that previously observed for the PhC structure with smaller period [Fig. 3(a)]. As seen from Fig. 2(b), the amplitude of the second maximum continuously grows up when decreasing the wavelength, i.e. when increasing the normalized frequency. At 1520 nm wavelength (a/λ = 0.489), the transmitted beam is mainly single-lobe while some satellite peaks are also present. The calculated propagation angle within the crystal at this wavelength is -2°. The presence of the satellite peaks as well as the small angular deviation shift visible at 1540 nm in Fig. 3(b) are probably related to the excitation of evanescent waves in the photonic crystal.
As seen in Fig. 3(a), the propagation angle essentially decreases with the wavelength. This result is obtained in most of our experiments, and corresponds to a negative wavelength dispersion of the refraction index. In the regime of continuous variations, the measured angular dispersion is typically between -0.07°/nm and -0.37°/nm. These values are quite comparable to those obtained in other superprism experiments [9, 10, 12].
It must be finally noticed that the evolutions reported in Fig. 2 and Fig. 3 were identically reproduced for an input waveguide of 2 μm width (instead of 3 μm). This indicates that diffraction effects associated to the limited aperture size of the input waveguide have just a weak influence, if any, on the PhC dispersion characteristics measured in our experiments.
In brief, two kinds of refraction phenomena have been observed in our experiments. The first one is related to the continuous wavelength-dependent angular dispersion that can be as high as -0.37°/nm. The second one is the abrupt variation of the angular dispersion that is observed for TM-polarized waves.
Let us focus on the second result. First, it is worthwhile noticing that the PhC structures presently operate on high-order allowed photonic bands. For the TM polarization, the domain of investigated frequencies (0.417 < a/ν < 0.489) extends from the fourth to the sixth allowed photonic bands. Each allowed mode follows its own dispersion characteristic and as a consequence, the group velocities and propagation angles of the different modes are in general different.
For a given frequency or wavelength, the light incident onto the PhC structure is expected to couple to several modes. However, our experimental results show that in most cases, one mode is dominant while the coexistence of two modes with similar amplitudes only occurs in a limited spectral range. These experimental findings are confirmed from two-dimensional finite-difference time-domain (FDTD) calculations.
Fig. 4 shows the results of numerical 2D FDTD simulations performed in TM polarization for four frequency values in the range of interest. The angle of incidence (8°) onto the photonic crystal is the same as in the experiments [Fig. 1(b)]. The crystal parameters are also the same as in the experiments except for the air filling factor which is finely adjusted to fit the frequency position of the first discontinuity observed in the crystal dispersion (a/λ ≈ 0.425). This fine adjustment may account for crystal imperfections due to fabrication as well as for the influence of the vertical guiding on the effective index of refraction. As seen, for an air filling factor of 0.183 close to the experimental value of 0.19-0.20, both the sign and amplitude of the beam deflection measured in the experiments (Figs. 2 and 3) are reproduced well by simulations. Abrupt variations of the crystal dispersion are found at the expected frequencies. A single lobe output is essentially obtained in the other frequency regions where the crystal dispersion continuously varies with wavelength. This general agreement between experimental results and 2D- FDTD calculations leads to the intermediate conclusion that out-of-plane (3D) losses do not play an important role in the present situation The PhC insertion loss was measured to be ~ 8dB. This value did not significantly vary with the number of periods (i.e. with the superprism length), which means that most of this loss is due to the reflections at the interfaces.
Let us notice that a similar agreement between experiments and FDTD calculations was found for the TE polarization, a weak transmission of less than - 20 dB being calculated for normalized frequencies between 0.42 and 0.46 [see Figs. 2(c) and 2(d)].
The fact that despite the presence of three photonic bands in the investigated frequency range [Fig. 5(a)], only one PhC mode (rarely two PhC modes) is (are) excited in the photonic crystal, is related to the efficiency of the coupling between the input waveguide mode and each of the PhC modes. This coupling efficiency depends in turn on the PhC mode symmetry and on the group velocity associated to this mode.
Fig. 5(b) shows isofrequency contours calculated from a 2D plane wave model for the fifth and sixth PhC bands and for the normalized frequencies a/λ = 0.415, 0.435, 0.46 and 0.49. The directions of propagation allowed in the crystal are determined by considering that the wave vector component parallel to the interface is conserved. The line indicating the conservation of this component is drawn for each of the graphs represented in Fig. 5(b). Each intersection between this line and the corresponding isofrequency contour defines the extremity of a Bloch wave vector of the field transmitted in the crystal. The normal to the isofrequency contour at the intersection point defines in turn the group velocity associated to the Bloch mode and thus the direction of propagation of light in the crystal. The fourth PhC band is presently out of concern since there is no intersection for the investigated range of normalized frequencies.
At the smallest values of a/λ (≤ 0.42), there is only one solution, which corresponds to the fifth band (Fig. 5(b)). In agreement with experiments (Fig. 3(a)) and FDTD calculations (Fig. 4(a)), the angular deflection of the refracted beam is negative and comprised between -15 and -20°. In the transition region (0.425 ≤ a/λ ≤ 0.435), there are two main intersections with the isofrequency contours associated to the fifth band. One intersection point is found on a flat side of the star-shaped contour in the second Brillouin zone while the other is found near a pronounced dip of the contour in the first Brillouin zone. Because of the spread of wave vectors contained in the incident beam, only the first situation will correspond to the propagation of a nearly collimated beam in the photonic crystal (self-collimation effects ). The angular deviation of the refracted beam is now positive and comprised between 5 and 10°, not far from the experimental values [Fig. 3(a)] and those estimated from FDTD calculations [Fig. 4(b)]. Let us notice that several diffracted waves are also predicted from the graphs associated to the fifth and sixth bands, respectively (see Fig. 5(b), a/λ = 0.43). However, because of their angular deviations, these waves cannot exit the slab waveguide
For normalized frequencies up to ~ 0.46, the previous situation prevails. The main refracted wave is constructed from the intersection with the isofrequency contours associated to the fifth band. As seen in Fig. 5(b), for a/λ = 0.46, another refracted wave is also predicted from the graph associated to the sixth band. However, the intersection point is close to a cusp of the hexagonal contour so that small changes in the incident wave vector produce a high angular deflection in the propagation direction. In other words, the refracted beam is highly divergent . This situation abruptly changes for higher values of the normalized frequency. The small isofrequency contours associated to the fifth band tend to vanish while the intersection point associated to the sixth band moves away from the region of the cusp. This is illustrated in Fig. 5(b) for a/λ = 0.49, where only one solution exists for the refracted wave. The intersection point now is found on a nearly flat side of the hexagonal contour associated to the 6th band. The angular deviation is negative again with values not far from - 15°. This transition from the fifth to the sixth band explains well the experimental results of Fig. 3(b) as well as the results from FDTD calculations in Figs. 4(c) and 4(d). The graph reported in Fig. 5(b) for a/λ = 0.49 also shows that the number of diffracted waves may significantly increase with the frequency.
The mode selection mechanisms proposed from the results of plane wave calculations agree with the results of numerical FDTD simulations, regarding not only the angular deviation of the refracted wave but also the field amplitude distribution associated to this wave. For example, Fig. 6 shows the field amplitude distributions calculated from the two methods for a/λ = 0.415. The mode pattern of Fig. 6(a) corresponds to the unique solution determined for the fifth PhC band at this normalized frequency in Fig. 5(b). The mode pattern of Fig. 6(b) is an enlarged view of the field distribution calculated from the FDTD method (Fig. 4(a)). As seen, the propagating mode in Fig. 6(b) can be unambiguously identified with the fifth PhC mode calculated from the plane wave method. The extrema of the field amplitude are located at the same positions in Figs. 6(a) and 6(b) regarding the holes of the photonic crystal. The small inclination of the fringes in Figs. 6(a) and 6(b) is readily explained by the fact that the wave does not exactly propagate in the ΓK direction.
5. A new approach to superprisms
The observed discontinuous wavelength refraction based on the coupling of the light to different PhC modes opens a new approach to superprisms for integrated optics. As it is was already mentioned in the introduction, the usual way to achieve a high angular wavelength dispersion consists in operating the photonic crystal within sharp corner regions of its dispersion characteristics near bandgap. The present results however show that operating near a PhC gap is not a necessary condition for a high wavelength angular dispersion.
To precisely illustrate this assertion, let us consider our TE results for the PhC lattice period a = 0.672 μm [Fig. 2(c)]. A clear photonic gap is experimentally found for the wavelengths below 1570 nm. However, the angular dispersion measured around this gap is only 0.12°/nm. This value is much lower than the dispersion values obtained in TM experiments where the photonic crystal was not operated in the vicinity of a photonic bandgap.
The advantage of our approach over conventional superprisms stems from the fact it is applicable to the case of divergent input beams and thus tolerates the use of relatively narrow input waveguides. This in turn allows a significant reduction of the device dimensions. For superprisms in planar integrated optics, the incident beam usually includes a certain amount of plane-wave components at different angles. A standard superprism effect (occurring near a cusp of an isofrequency curve) would deviate these components to various directions. As a consequence, the large angular dispersion associated to the small beam waist would considerably reduce the angular resolution of such a superprism . One solution to overcome this drawback would consist in increasing the beamwidth at the superprism entrance. However, this would also result in an increase of the device size to keep the same level of crosstalk between the output channels.
In our approach, the continuous variation of the refraction angle is relatively small compared to the discontinuous angle deviation occurring in a reduced spectral range of ~ 30 nm. When operating in the vicinity of the discontinuity, the shift of the output beam is important. When operating outside the discontinuous region, the angular variation is small. In other words, outside this region, the different plane-wave components included in the input beam only suffer a weak spatial dispersion. We think that this “stepwise” wavelength behavior of our superprism version is preferable to the steep continuous variation of “standard” superprisms.
The great potential of the current approach is related to the compactness of the structure. For less than 10μm crystal length, the transmitted beam can be deflected by several microns. For an input waveguide of 2(3) μm width, the fabricated structure is able to separate two wavelength channels spaced by 30 nm.
The switching amplitude could be further amplified by concatenating PhC structures with properly designed lattice parameters. Such an approach would offer the advantage of tailoring the wavelength range within which the switching phenomenon occurs while optimizing the refraction angle variations.
It is worthwhile noticing that neither the incident beam parameters (incident angle and width) nor those of the PhC structure were optimized in the present experiments. Despite of that, our results already show the possibility of achieving performances compliant with practical device specifications. For the moment, a simple geometry has been used to investigate the reported effect and to get the interpretation of the results the easiest as possible. Clearly, a more sophisticated “mode-matched” PhC geometry must be used to reduce the reflection losses at the interfaces.
In addition to beam deflection, the discontinuous behavior of the crystal refraction with wavelength can also be applied to a new type of filtering function. Wide pass-band filters can be envisaged, for instance, for coarse wavelength demultiplexing (CWDM) applications in the telecommunication access networks.
To summarize, we have studied the angular dispersion of an InGaAsP triangular lattice planar photonic crystal operated in the high photonic bands extending from a/λ = 0.417 to a/λ = 0.49. For the TM polarization, the dispersion was found to exhibit both continuous and discontinuous evolutions with the wavelength. In the continuous regime, a negative angular dispersion was observed with values up to -0.37°/nm. These values are of the same order as those previously reported for planar 2D PhC structures [9, 10, 12].
In the discontinuous regime, the refraction angle was found to abruptly increase or decrease by an amount of 30° for a wavelength variation of less than 30 nm. Experimental results have been reproduced well from 2D-FDTD simulations. The abrupt variations of the refraction angle have been explained from the coupling of the incident light to different photonic crystal modes with different group velocities.
This discontinuous wavelength super-refraction opens a new approach to superprisms in integrated optics. It also leads to the possibility of designing very compact CWDM pass-band filters that could be exploited in telecommunication access networks.
The authors are very grateful to Xavier Le Roux for the scanning electron microscope obervations and to Benoit Douville for the mask coding. The authors are also grateful to the reviewers for their useful comments that contributed to the manuscript improvement. This work was supported by the French RNRT project CRISTEL.
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