Finite difference time domain study of highefficiency photonic crystal superprisms
Toshihiko Baba, Takashi Matsumoto and Manabu Echizen
Yokohama National University, Department of Electrical and Computer Engineering79-5 Tokiwadai, Hodogayaku, Yokohama 240-8501, Japanbaba@ynu.ac.jp

Abstract: Input and output boundaries of a photonic crystal (PC) are
optimized so that the superprism exhibits low insertion loss. It is shown thatprojected-airhole and half-circular airhole interfaces achieve transmission
loss of 0.3 dB and 1.0 dB, respectively, for small and large incident anglesof light against normal to boundaries. The finite-difference time-domain
simulation shows that a low loss is essentially realized by a periodic phasemodulation of the incident beam by the interfaces. It also demonstrates theclear steering of collimated light beam with varying wavelength. The
enhancement of angular dispersion is also demonstrated by a PC composedof a dispersive medium.
2004 Optical Society of America
OCIS codes: (230.3990) Microstructure devices; (230.3120) Integrated Optics Device

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The superprism is a diffraction device of defect-free bulk photonic crystals (PCs), which
operates at frequencies of higher order photonic bands [1] It exhibits wide steering ofPoynting vector S in PCs by a small change of frequency and incident angle. Therefore, wecall it the S-vector superprism. In early studies, large angular dispersion of this prism arisingfrom an anomalous dispersion characteristic of PCs was focused, and the realization of highperformance demultiplexer, beam scanner, and so on, was expected [1-6]. However, theanomalous characteristic simultaneously degrades the quality of light beam, so rather reduces#4988 - $15.00 US(C) 2004 OSA

Received 5 August 2004; revised 11 September 2004; accepted 14 September 2004

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the spatial and spectral resolution [7]. In recent studies, we proposed the k-vector superprism[8,9] which utilizes the steering of k vector instead of S vector. The dispersion analysis
showed that the k-vector prism improves the resolution by the refraction of highly collimatedbeam at an angled output end of the PC. In addition, it allows narrow input beam and acompact PC. This point is important not only for the device itself but also for succeeding
researches, since it enables us to use more simply the finite-difference time-domain (FDTD)
simulation, which is a powerful tool for the design of structural details. Now therefore, thenext issue is to find low loss design. Let us consider a standard PC composed of high index
contrast media, e.g. airholes in a semiconductor slab (PC slab). Then, the transmittancethrough the superprism is no higher than 10% due to two reasons. One is a reflection loss at
input and output ends caused by a field mismatch between an incident wave and a Bloch wave
in the PC. The other is many diffraction beams generated at these ends. Because of thesereasons, no studies have succeeded in clear demonstration of the beam steering even innumerical simulations. Some studies theoretically discussed the modification of interfaces toimprove the transmittance against the normal incidence of light [2,10,11]. We proposed a
projected airhole interface, which smoothly modulates the phase of the incident wave tocompensate the field mismatch and theoretically achieves < 0.3 dB total insertion loss [10].
We also showed that diffraction beams are suppressed to some extent by a flat interface [2]. In
this paper, we show in the FDTD simulation that the optimization of these interfaces allows
clear steering of highly collimated beam with low insertion loss. We also discuss theenhancement of beam steering angle with the help of material dispersion.

60.2

70.2

7680.20.20.2

280.

100.

290.2

91030.3301030 30

10001015º

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5º10º(a)15º10010º(b)Fig. 1 Contour plots of eigen frequency (thin black curve), incident angle (thick gray curve)
and resolution parameter (thick black curve) of 2D PC in a triangular lattice rotated by 45°. (a)S-vector prism. Resolution parameter of > 30 is omitted. (b) k-vector prism with 45° output
end. Black region indicates air lightcone when the structure is airbridge PC slab.

This paper assumes a finite-size two-dimensional (2D) PC composed of airholes in asquare lattice rotated by 45° against an input end of the PC. The airhole diameter to lattice
constant ratio 2r/a is 0.61. The background index is initially fixed to 3.065, whichcorresponds to an effective modal index of a semiconductor slab [7]. The PC has an output
end of 0° or 45° against the input end, and is surrounded by the same background medium.
The polarization is assumed to be inside the 2D plane. Figure 1 shows contour plots of eigen
frequency, incident angle of light and resolution parameter (an index of resolution). Theresolution parameter is defined as [∂θ / ∂ (a / λ)] [∂θ / ∂θ in ] , [7,8] where θ is the lightpropagation angle in the PC and outside of the PC for the S-vector and k-vector prisms,
respectively, and θin is the incident angle. They were calculated by the plane wave expansion
method for the second band and plotted in the limited region of the Brillouin zone. Here, thek-vector conservation law was applied for the first and the second Brillouin zones at input andoutput ends, respectively, assuming the rear side light output. We omit to explain the detail of#4988 - $15.00 US(C) 2004 OSA

Received 5 August 2004; revised 11 September 2004; accepted 14 September 2004

20 September 2004 / Vol. 12, No. 19 / OPTICS EXPRESS 4609

the superprisms, the analysis, and the resolution parameter, because their details are fully
explained in Refs. [7-9]. Figure 1 simply indicates an important point for this study, i.e., thebest θin is 10° for the k-vector prism. As shown in this figure, a high resolution parameter of >30, which is 6 times larger than a typical value of a first order diffraction grating, is obtained
as a k-vector prism, when θin of a Gaussian beam is 10° against normal to the input end and1/e2 full angular divergence of the Gaussian beam is within several degrees. Although the best
θin for the S-vector prism is 7°, a standard resolution parameter as a grating is maintained at10°. Therefore, θin is fixed to 10° in this paper.
a/λ = 0.29, θin = 10°
θin2raa/λ = 0.29
θin = 10°86420−10?

θpr = 20°

θprb81020Angle θpr [°]30

θin = 10°
θpr = 20°6420

θpr = 20°010(d)

Transmission Loss [dB]a/λ = 0.29, θin = 10°

(b)

10(c)

Transmission Loss [dB](a)0.5 dB

0.28
0.29
0.30
Normalized Frequency a/λFig. 2 Light transmission characteristics of S-vector prism. (a) Light intensity distribution for
PC without special interfaces. (b) That with projected airhole interfaces. (c) Dependence ofinsertion loss on angle of projections. (d) Dependence on normalized frequency a/λ.

Figure 2(a) shows the light intensity distribution obtained by the FDTD simulation, when
input and output ends have no special interfaces and the Gaussian beam is excited outside of
the PC. The 1/e2 width of the beam is fixed to 20a, which satisfies the above angulardivergence condition. At the input end, light is reflected almost perfectly, and very weakintensity enters the PC, exhibiting the negative refraction of the S-vector prism. Due toadditional reflection at the 0° output end, light output is hardly observed. In many
calculations, we noticed that the result is strongly dependent on θin. For example, for θin = 25°,
the insertion loss decreases up to 2.4 dB. However, light exhibits the positive refraction in thePC, which is not the operation expected as an S-vector prism. This implies that the interface
must be optimized individually for the target incident angle and Bloch wave. In principle,
however, the interface should act as a phase modulator which compensates the fieldmismatch. Based on this concept, we considered a projected airhole interface for θin = 10°, as
shown in Fig. 2(b). Here, projections are oriented in the opposite direction against the incident
beam, i.e. θpr = 20°, and their length b is set to be 1.97a. These are critical conditions underwhich the incident beam uniformly suffers the phase shift from each projection without
duplication. Figure 3 shows magnified field profiles at input interfaces with different θpr. It isclearly observed that the optimum θpr gives an adequate modulation to the phase front so that#4988 - $15.00 US(C) 2004 OSA

Received 5 August 2004; revised 11 September 2004; accepted 14 September 2004

20 September 2004 / Vol. 12, No. 19 / OPTICS EXPRESS 4610

the internal light is smoothly excited. Consequently, the insertion loss is drastically reduced
and the clear negative refraction is observed. The lowest loss is 0.5 dB (0.25 dB/interface)when θpr = 20°, as shown in Fig. 2(c). This loss value is attractive even when the prism isconsidered as a simple diffraction grating. A loss of < 1 dB is maintained for b/a = 1.8 – 2.3
and in a wide spectral range of a/λ = 0.286 – 0.301, as shown in Fig. 2(d). Figure 4(a)demonstrates the beam steering of the S-vector prism with the optimized interface. The beam
divergence at a/λ = 0.275 is caused by a small bend of the frequency contour. Except for this,clear steering of the collimated beam is observed. In the range of a/λ = 0.272 – 0.299, thebeam steering angle ∆θc is 16°, as shown by the solid line in Fig. 4(b). This value almostagrees with that expected from the wavelength sensitivity parameter ∂θc / ∂(a / λ) in thedispersion analysis [7].
As a k-vector prism, a structure and light output are considered, as illustrated in Fig. 5(a).
A 45° output end is placed on the rear side of the PC so that the internal light dominated by
the S-vector prism reaches to this end. The FDTD simulation previously showed that light
output occurs in a direction determined by the second Brillouin zone [9]. Such an arrangementof the output end and a direction of light output correspond to assumptions for Fig. 1. The
output end is largely inclined against the internal light, so the output beam is expanded to 75a– 150a in width and well collimated. However, the large inclination severely restricts theoptimization of the interface. For example, the total insertion loss cannot be lower than 6 dBby the projected airholes, since projections cannot be freely optimized without overlapping.
Through trial calculations of many interfaces, we found that half airholes oriented by 60°
against normal to the output end gives a low insertion loss of 1.3 dB (~1 dB at the output end).
A < 1.5 dB loss is maintained for a/λ = 0.279 – 0.289, and the steering of the collimated beam
is observed outside of the PC, as shown in Fig. 5(a). The beam steering angle θout is smaller
than that of the S-vector prism, and is comparable to that of the first order grating. However,
the expansion of the output beam improves the beam collimation and realizes the higherresolution.

(a)

(b)
Incident WaveFig. 3 Shaded drawings of field intensity distribution (H field in the direction perpendicular tothe 2D plane) of light propagation at input interfaces of PCs. Dark lines denote the shape ofairholes. (a) θpr = −20°. (b) θpr = 20°.

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0.281

0.275

0.299

0.293

0.287

(a)
Beam Steering Angle θc [°]65(b)θin = 10°
θpr = 20°60
θc555045
0.27

0.28
0.29
0.30
Normalized Frequency a/λFig. 4 Beam steering characteristics of S-vector prism. (a) Light intensity distributions forvarious a/λ. (b) Beam steering angle as a function of a/λ. Solid and dashed lines denotebackground material of PC without and with material dispersion.

θin = 10°(a)
θpr = 20°θc
θpc = 45°aλ = 0.281
75%

0.287
73%

(b)

Beam Steering Angle θout [°]θout60°484644420.270.293

θin = 10°
θpr = 20°

0.28
0.29
0.30
Normalized Frequency a/λ53%

Fig. 5 Beam steering characteristics of k-vector prism. (a) Calculation model and light intensity
distributions for various a/λ. (b) Beam steering angle as a function of a/λ. Solid and dashedlines denote background material of PC without and with material dispersion.

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To realize a compact demultiplexer by the superprisms, not only the high beamcollimation but also the wide beam steering is necessary. The beam steering angle can beenhanced by the material dispersion near the electronic band-edge of the background mediumof the PC. Let us consider 1.5-µm-GaInAsP bulk crystal lattice-matched to InP, as anexample. Its refractive index is calculated almost precisely to be 3.676 – 3.565 at λ = 1.503 –1.581, respectively [12]. If an airbridge PC slab of 0.23 µm thickness is assumed, the modal
effective index changes in the range of 3.079 – 2.924, respectively. Beam steering angles θcand θout for this material dispersion and a = 0.52 µm are shown by dashed lines in Figs. 4(b)and 5(b). The angles increase to 1.2 – 1.5 times as large as the case of the fixed refractiveindex. This enhancement is an advantage of superprisms, which cannot be obtained inreflection-type gratings.
In conclusion, projected airhole and half airhole interfaces were assumed, respectively, atinput and output ends of a square lattice 2D PC, and the steering of collimated light beam bysuperprisms was successfully demonstrated in the FDTD simulation. Practical low insertion
loss of 0.5 dB and 1.3 dB was estimated for the S- and k-vector prisms, respectively. Beamsteering angles demonstrated here were not so large, since the prisms were optimized for highresolution. It was also shown that the material dispersion near the electronic band-edge canenhance the steering angle. These results strongly encourage future experimental studies. Thiswork was supported by IT Program and 21st COE Program of MEXT, and by CREST of JST.

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Received 5 August 2004; revised 11 September 2004; accepted 14 September 2004

20 September 2004 / Vol. 12, No. 19 / OPTICS EXPRESS 4613