IEICE TRANS. ELECTRON., VOL.E87–C, NO.3 MARCH 2004

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PAPER

Special Section on Photonic Crystals and Their Device Applications

Design and FDTD Simulation of Photonic Crystal k-Vector
Superprism
Takashi MATSUMOTO† , Nonmember and Toshihiko BABA†a) , Member

SUMMARY
We theoretically investigated the resolution of the photonic crystal (PC) k-vector superprism, which utilized the wavelengthdependent refraction of light at an angled output end as a narrow band filter
at 1.55 µm wavelength range. Similarly to the case of the conventional Svector prism, we defined the equi-incident-angle curve against the dispersion surface, and calculated the beam collimation, wavelength sensitivity
and resolution parameters for light propagation in the PC. We estimatedthat the resolution of the k-vector prism is the same as or higher than thatof the S-vector prism and the PC can be significantly miniaturized. In addition, we clarified the relation of the S-vector prism phenomenon and theposition of the output end in the k-vector prism, and diﬀerent results forthe reduced and repeated zone schemes, which are important for the detailed design. We also confirmed that the light propagation simulated by
the FDTD method well agreed with the results of the dispersion surface
analysis.
key words: photonic crystal (PC), superprism, filter, FDTD, WDM

1.

Introduction

The photonic crystal (PC) is a complex multi-dimensional
periodic structure accurately designed by the photonic band
theory. It is expected to realize various novel micro-optic
devices. Thus far, ultrasmall lasers [1], ultrasmall waveguides [2] and resonant-type narrow band filters [3], all ofwhich utilize the photonic bandgap (PBG), have been reported. On the other hand, an ultralow group velocity andan anomalous dispersion occur at the frequency range higher
than the PBG. Especially, the superprism phenomenon has
attracted much attention, because it allows wide-angle deflection of light beam by a slight change of the wavelengthand the incident angle. It has been theoretically discussed
and experimentally demonstrated by Russel [4] and Kosaka,et at. [5]. As applications, a dispersion compensation device [6] and a light deflection device [7] were discussed,
but the most fundamental application is a diﬀraction-type
narrow band wavelength filter. In future, a compact, lowcost and high performance filter will be desired to realizea wavelength division multiplexing (WDM) network up to
end users. But widely used arrayed waveguide grating filter
based on silica waveguides is essentially diﬃcult to miniaturize to less than 1 cm2 . The superprism is expected tosolve this problem. However, there were no reports on thequantitative estimation of the wavelength resolution, whichis the most important performance of such a filter. Previously, we reported the first theoretical calculation of theManuscript received October 10, 2003.
Manuscript revised November 27, 2003.
†
The authors are with Yokohama National University,
Yokohama-shi, 240-8501 Japan.
a) E-mail: baba@ynu.ac.jp

wavelength resolution of the superprism. We used the dispersion surface analysis for the beam collimation parameter
1/p, wavelength sensitivity parameter q, and the wavelength
resolution parameter q/p in the PC [8]. The result showedthat a superprism can realize a suﬃciently high resolution
usable for the dense WDM, but that the PC cannot still be
miniaturized to less than 1 cm2 .
In this paper, we discuss a novel superprism, which we
named the k-vector superprism. Diﬀerent from the conventional prism (we named it the S-vector prism, since it utilizes the deflection of S vector), the k-vector prism utilizes
the deflection of the k vector and the refraction of light at
an angled output end of a PC. We investigated the resolution of the k-vector prism using a similar dispersion surface
analysis. Figure 1 shows schematics of a square lattice PC
and a corresponding dispersion surface. Here, (a) emphasizes the deflection of S vector, while (b) that of k vector.
In (a), the deflection of k vector for diﬀerent wavelengths issmaller than that of S. In this case, the change of the refraction angle will be small, even if some output end is placedin the PC. Therefore, it is necessary to completely separate
the beams inside the PC for the wavelength resolution. This

Fig. 1 Two diﬀerent types of superprisms in real and k spaces. 2-D PC
is composed of airholes with square lattice rotated by 45◦ .

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is an essential reason that disturbs the miniaturization of the
PC in the S-vector prism. On the other hand, the k vector is
widely changing in (b). At the angled output end of the PC,
diﬀerent k vectors lead to diﬀerent refraction angles. This
allows the beam separation through the free space propagation outside the PC, so allows the drastic miniaturization of
the PC to less than (100 µm)2 . By integrating a beam focusing lens at the output end or outside of the PC, the lengthof the free space propagation will be further reduced, so thetotal filter system will also be miniaturized.
We have already discussed the theoretical calculationof fundamental characteristics of the k-vector prism in [9].
In this paper, we first present a brief summary of it, andthen discuss a relation between the S vector and the position of the output end, which is important for the designof a k-vector prism. We also discuss the second Brillouinzone in the repeated zone scheme for the estimation of actual light propagation. Finally, we present the simulation oflight propagation against diﬀerent characteristic parametersby the finite diﬀerence time domain (FDTD) method andconfirm the correspondence to the dispersion surface analysis.
2.

Resolution of S-Vector and k-Vector Prisms

We consider a finite width Gaussian beam incident to a
two-dimensional PC with a square lattice rotated by 45◦ , asshown in Fig. 1(a). The deflection angle of the light beam
can be analyzed from an equi-frequency curve of a dispersion surface in the Brillouin zone. The wave vector in thePC, kc , is determined so that the tangential component of
the incident wave vector kin against the input end of the PC
is conserved. The direction of the S vector is determined by
the gradient of the dispersion surface at a point indicated by
kc . In addition, we consider an equi-incident-angle curve,
which is a locus of the point against a constant incident angle
and varying frequency of light. Figure 2 shows the disper-

Fig. 2 Schematics of dispersion surfaces of lower three bands with equiincident-angle curves (upper) and their calculated examples (lower). (a)First band, (b) second band and (c) third band. Equi-incident-angle curveis drawn with an interval of 5◦ .

sion surface with equi-incident angle curves for the lowest
three photonic bands. For the first band, equi-incident-angle
curves shown by gray lines exhibit a radial pattern near theΓ point. For higher bands, which are more important forsuperprisms, they look like vertical lines. If one aims at awavelength filter for a certain incident angle, one has to use
the dispersion characteristic along the corresponding equiincident-angle curve.
As shown in Fig. 3, we investigated derivative parameters p ≡ (∂θc /θin ), q ≡ ∂θc /∂(a/λ) and q/p for the S-vector
prism by calculating dispersion surfaces. Here, θin and θcare beam angles in the incident medium and in the PC, respectively, a is the lattice constant, and λ is the wavelength.
We assumed the background index to be equal to the incident medium index, i.e., 3.065, the index of airholes to be
1.0, and the size of airhole diameter to be 0.624a. The number of plane waves was 37 [8]. In general, 1/p is low and q
is high in S-vector prisms. This means that the light beam iseasily diverged and its propagation is sensitive to the wavelength. The wavelength resolution parameter q/p exhibits ahigh value in a limited region along an equi-incident-angle
curve. However, this region only allows very narrow latitude of the incident angle; the incident beam must be wideand highly collimated. This is the reason the device size will
be of cm order.
We can discuss similar derivative parameters for the
k-vector prism [9]. We calculated the beam collimation
parameter 1/p ≡ (∂θout /∂θin )−1 , the wavelength sensitivity
parameter q ≡ ∂θout /∂ (a/λ) and the resolution parameterq/p. Here, θout is the beam angle outside the PC, which is
calculated by the conservation law of the tangential component of the k-vector at the output end. Figure 4 shows
1/p, q and q/p for the second band of the PC with 45◦ output end. Calculation parameters are the same as for the Svector prism. Compared with those of S-vector prism, 1/p ishigher and q is lower. Note that q/p exhibits a higher value
and much wider latitude of the incident angle. This owes
to a wide output beam expanded by the angled output end,
which improves the beam collimation outside the PC, andconsequently enhances q/p. At λ ∼ 1.55 µm, q/p ≥ 100
is achieved in a ∼100 nm wavelength range by an incident
beam width of 13 µm. This almost covers C and L bandsof optical fiber communications and allows the butt-joint ofFig. 3 Calculated derivative parameters of S-vector prism. (a) Beam collimation parameter 1/p, (b) wavelength sensitivity parameter q, and (c) resolution parameter q/p.

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Fig. 4 Calculated derivative parameters of k-vector prism with 45◦
output end. (a) 1/p, (b) q and (c) q/p.

a singlemode fiber to the prism. To convert q/p of the kvector prism into an absolute value of the wavelength resolution, the free space propagation has to be calculated withthe modeling of the modulated wave front at the output end
and an integrated condenser lens. Even without such calculation, we can expect a high resolution for the k-vector
prism, when we consider a fact that the S-vector prism canachieve a wavelength resolution of 0.4 nm with a resolution
parameter of 75.
3.

Fig. 5 Relation of S-vector prism phenomenon and output end in
k-vector prism. (a) Front side and (b) rear side.

Detailed Design and Characteristics

In this section, we first discuss the relation of the beam propagation in the PC and the position of the output end of the
k-vector prism. Although the k-vector prism utilizes the deflection of the k vector, the deflection of the S vector always
exists simultaneously. The beam propagation in the PC isdetermined by the S vector, so it is necessary to place theoutput end suitably. Figures 5(a) and (b) show schematics
of light propagation in a PC with 45◦ output end. Light does
not necessarily propagate toward the output end, but sometimes toward the other side. For case (b), the output end
should be placed at the other side. In the following, we call(a) the front side and (b) the rear side.
Next, we discuss the estimation of light propagation
outside the PC using the second Brillouin zone, since thenormalized frequency we mainly consider in this study liesin the second Brillouin zone in the extended zone scheme.
Figure 6 shows the second dispersion surface in the repeatedzone scheme, where both the first and the second Brillouinzones are displayed. The angle of the output end is assumed
to be 45◦ . Directions of light propagation for the front andrear sides are predicted as (a) and (c), respectively, for the
first Brillouin zone. Even for the consideration with the second Brillouin zone, we start from kc in the first Brillouinzone to discuss the same wave propagating in the PC. Figure 6 describes that kc is shifted to the second Brillouin zoneby the reciprocal lattice vector.
By adapting the conservation law to the shifted kc , direction of light propagation for the front and rear sides are
predicted as (b) and (d), respectively. Figure 7 shows q/p
for these four directions, where unusable conditions for eachfront and rear side are shadowed. The characteristic of q/p
is almost the same between the two sides. The result for

Fig. 6 Analysis of light propagation by the dispersion surface with thefirst and the second Brillouin zones. (a) and (b) are results for the front
side, and (c) and (d) for the rear side. (a) and (c) are obtained with the first
Brillouin zone, and (b) and (d) with the second Brillouin zone.

Fig. 7 Resolution parameter q/p against the propagation direction of(a)–(d) in Fig. 6. Shadowed regions are those which cannot be used as a
k-vector prism.

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the second Brillouin zone showed a slightly narrower wavelength range, but almost the same value of q/p. Thus, thefilter performance is equivalent for the four directions. The
resolution parameter particularly increases near the boundary of the shadowed region, where the light beam is expanded at the output end so that the collimation parameter
is improved.
4.

FDTD Analysis

In this analysis, we assumed the same structural parameters as for the analysis in previous sections. The calculation
condition is the same as that describe in [7] except that theYee cell size in this study was a/13. As an incident wave,
a continuous Gaussian beam with a spot diameter of 250a
was excited for the magnetic field normal to the 2-D plane,
Hz . This wave was incident to the PC with θin = 10◦ . Projected airholes were placed at the input end to improve thecoupling eﬃciency [10]. On the other hand, half airholes,
which are eﬀective for extracting collimated output beam,
were placed at the 45◦ output end at the rear side. Figure 8
shows an example of field profiles for a high resolution parameter q/p = 57 (1/p = 15 and q = 3.8). The dispersion
surface analysis predicts that, for a 1.0% frequency change,
the output beam is highly collimated with a beam divergence angle of ∆θ p < 0.2◦ and a deflection angle ∆θq =2.2◦ . Drawing lines at 1/e2 of the beam maximum in Fig. 8,∆θ p and ∆θq were estimated to be ∆θ p < 0.2◦ and 2.3◦ , respectively. Thus, the FDTD calculation well agreed with thedispersion surface analysis. One may think that ∆θq is small,but ∆θ p is so small that the beams can be separated by thefree space propagation.
In the previous study, we observed in an FDTD simulation that light output from the PC shows multiple beams,
which can be predicted with the first Brillouin zone andmany second Brillouin zones [7]. Thus, the direction of
output beam is not determined uniquely. In the above re-sult, however, the optimum condition leads a single output
beam, which agrees with Fig. 6(d) for a second Brillouinzone. The result itself is reasonable, when considering thenormalized frequency assumed in this simulation. But thereason why the single output beam is obtained is still unclear. It is thought to be due to the airhole shape at the output end and/or due to the condition of refraction. It should
be considered in the next study with the detailed consideration of boundary conditions. In addition, as shown in Fig. 8,
some amount of reflection is observed at input and output
ends. At the final stage of the superprism research, the complete suppression of the reflection by an optimization of theinterface structure will be required [10], [11].
5.

Conclusion

For the 2-D PC with an angled output end, we discussed
the k-vector superprism instead of the conventional S-vector
prism. From the dispersion surface analysis, we estimated 1)
a resolution parameter comparable to or higher than that of
the S-vector prism, 2) over 100 nm usable wavelength range
at 1.55 µm wavelength range, and 3) a large latitude of theincident angle, which allows the butt-joint of a singlemodefiber to the prism. Even in the k-vector prism, the S-vector
prism phenomenon restricts the arrangement of the output
end. But it is possible to utilize all conditions by preparing front and rear side output ends. The resolution does notstrongly depend on the arrangement of the output end and on
the Brillouin zone which is used for the analysis. However,
we observed in the FDTD simulation that the beam direction was explained by the analysis with the second Brillouinzone. We also confirmed a highly collimated beam and theirangular dispersion.
Thus, the fundamental design theory of the k-vector
prism was almost established. Next issues are to investigate
the mechanism that determines the output beam, reduce thereflection loss at interfaces, design a condenser lens for the
reduction of free space propagation, and demonstrate a clear
filter function in an experiment.
Acknowledgements
This work was supported by CREST #530-13 of JST, and by
Nano-Photonic and Electron Devices Technology Project of
the IT Program, and by 21st COE Program, both from the
MEXT.
References

Fig. 8 Simulated field profiles for a high resolution condition. (a) Calculation model, and (b) profile of Hz at a/λ = 0.290 (upper) and 0.293(lower).

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Takashi Matsumoto
was born in Shizuoka, Japan, on September 4, 1979. He received
the B.E. degree from the Division of Electricaland Computer Engineering, Yokohama National
University, Japan, in 2002. Now, he is studying
photonic crystal superprism toward the M.E. degree in Yokohama National University.

Toshihiko Baba
received B.E., M.E. and
Ph.D. degrees all from Division of Electricaland Computer Engineering, Yokohama National
University (YNU), Japan, in 1985, 1987, and1990, respectively. During his Ph.D. work, he
had been engaged in antiresonant reflecting optical waveguides and lightwave circuits. From
1990, he joined Tokyo Institute of Technology as a research associate. From 1991–1993,
he discussed the spontaneous emission control
in vertical cavity surface emitting lasers, andachieved the first room temperature continuous wave operation of a longwavelength device. In 1994, he became an associate professor of YNU,
and started the research on photonic crystals (PCs) and microdisk lasers(MDLs). Regarding PCs, he reported the first fabrication and characterization of an InP-based light emitters and a line defect waveguide at lightwave
frequencies. He also studied a PC light extractor in LEDs. His recent interests are nano lasers and functional devices. Regarding MDLs, he demonstrated the smallest cavity and the lowest threshold, a modified structure
named microgear, and near field sensor application. Besides these, he isalso active on a deep grating distributed Bragg reflector laser and Si photonic wire waveguides. Dr. Baba is a member of the IEEE/LEOS, JSAP,
and American Physics Society. He received the Niwa Memorial Prize in
1991, the Best Paper Award of Micro-Optic Conference in 1993 and 1999,
Paper Award and Academic Encouragement Award from IEICE in 1994,
and Marubun Research Encouragement Award in 2000.