Ring-type Fabry-Pérot filter based on the self-collimation effect in a 2 D photonic crystal

We propose a ring-type Fabry-Pérot filter (RFPF) based on the self-collimation effect in photonic crystals. The transmission characteristics of self-collimated beams are experimentally measured in this structure and compared with the results obtained with the simulations. Bending and splitting mechanisms of light beams by the line defects introduced into the RFPF are used to control the self-collimated beam. Antireflection structures are also employed at the input and output photonic crystal interfaces in order to minimize the coupling loss. Reflectance of the line-defect beam splitters can be controlled by adjusting the radius of defect rods. As the reflectance of the line-defect beam splitters increases, the transmission peaks become sharper and the filter provides a Q-factor as high as 1037. Proposed RFPF can be used as a sharply tuned optical filter or as a spectrum analyzer based on the self-collimation phenomena of photonic crystals. Furthermore, it is suitable for a building block of photonic integrated circuits, as it does not back reflect any of the incoming self-collimated beams owing to the antireflection structure applied. ©2010 Optical Society of America OCIS codes: (230.0230) Optical devices; (230.5298) Photonic crystals; (250.5300) Photonic integrated circuits; (120.1680) Collimation; (310.1210) Antireflection coatings; (070.2615) Frequency filtering. References and links 1. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, NJ, 1995). 2. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58(23), 2486–2489 (1987). 3. T. F. Krauss, and R. M. De La Rue, “Photonic crystals in the optical regime-past, present and future,” Prog. Quantum Electron. 23(2), 51–96 (1999). 4. E. Centeno, B. Guizal, and D. Felbacq, “Multiplexing & demultiplexing with photonic crystal,” J. Opt. A, Pure Appl. 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Introduction
Photonic crystals (PCs) [1][2][3], composed of periodic dielectric materials, have inspired great interest because they possess many unique properties to control the light propagation in PCs at the optical wavelength scale.One of the most promising elements of PCs is the photonic crystal waveguide (PCW) which is constructed by introducing a line defect into a PC to create a band of conduction inside the photonic bandgaps (PBGs).Wavelength-division multiplexing (WDM) has been a widely used technique to enhance the capacity of the optical communication systems.The possibilities of implementing PC based wavelength filters into WDM systems have been discussed [4][5][6][7][8][9][10][11].Recent studies on the self-collimation effect in PCs show that the self-collimated beams propagate without diffraction and it can be well guided without the use of any physical boundaries [12,13].It is well known that the selfcollimated beams can be effectively routed by employing the bends and splitters [14][15][16].Moreover, optical devices utilizing the self-collimated beams have intrinsic potential for high density photonic integrated circuits (PICs), since it enables beam crossing without any crosstalk [17].Various photonic devices based on this unique phenomenon have been investigated including a Mach-Zehnder interferometer and an asymmetric Mach-Zehnder filter [18,19].Very recently, Chen et al theoretically suggested a polarization-independent drop filter based on a PC ring resonator in a hole-type silicon photonic crystal [20].The reduction of unwanted reflection at the interfaces between a PC and uniform dielectrics is an important subject to improve the performance of PC devices.There have been many solutions to reduce the reflection at the ends of a PC [21][22][23][24].
In this paper, we propose a 2D PC ring-type Fabry-Pérot filter (RFPF) based on the selfcollimation phenomenon of light beam in microwave region and the experimentally measured transmission properties are presented.In this PC-RFPF, the self-collimated beams can be effectively controlled by employing line-defect beam splitters and mirrors.Antireflection structures (ARSs) with optimized parameters are introduced at the input and output PC interfaces to minimize the unwanted reflections at the interfaces between a uniform dielectric and the 2D PC [23,24].Numerical calculations for the performance of the PC-RFPF are also carried out by using the finite-difference time-domain (FDTD) technique and excellent agreement exists between the results obtained by the experiment and the simulation.

Design and operation principle
The equifrequency contours (EFCs) are calculated using the plane-wave expansion method [25] to find the propagation direction of the self-collimated light beam.As can be seen in Fig. 1(a), the EFCs in the vicinity of 12.50 GHz are close to straight lines normal to the ГM direction.It is well known that the light can propagate with almost no diffraction as the direction normal to the flat EFCs in a PC [12,13].The 2D PC-RFPF shown in Fig. 1(b) is composed of 29 2 29 2 a a × square lattice of rods in air with two line-defect beam splitters (BS 1 and BS 2 ) and two mirrors (M 1 and M 2 ).The zoom-ins of the input and output ports of the 2D PC-RFPF is shown in Fig. 1(c).The radius of alumina rods r = 2 mm, the lattice constant a = 5 mm, and the dielectric constant ε ≈9.7.From our previous work it was found that electromagnetic waves of frequencies around 12.50 GHz propagate along the ΓM direction without diffraction in this PC structure [24].In order to reduce the unwanted reflections between the PC and the homogeneous background material, the ARSs composed of 11 alumina rods with the radius R ARS = 1.15 mm are introduced at the input and output interfaces of the PC-RFPF.When the distance between the ARS and the PC-RFPF truncation d ARS = 3.60 mm, the coupling loss is minimized.As depicted in Fig. 1(d), each mirror is made by removing two rows of 17 rods along the ΓΧ direction.The beam splitter is a line defect composed of one-row of 17 rods in the ΓΧ direction and the radius of the defect rod r d in each beam splitter is varied from 1.00 mm to 1.75 mm in the step of 0.25 mm.
When a light beam of intensity I 0 is launched into this 2D PC-RFPF structure, a part of the incident self-collimated beam is reflected in the direction to the through port and the other part maintains its original propagation direction at the splitter BS 1 .The transmitted beam is again split into two self-collimated beams at the splitter BS 2 , and the reflected beam comes back to the splitter BS 1 , after undergoing the total internal reflections by the mirrors M 1 and M 2 .The process of splitting and bending of the self-collimated beams remaining inside the resonator continues repeatedly, and the normalized output intensities at the drop port and the through port are given by the well known Airy formulae [26]: (1 ) , ( 1) 4 sin ( / 2) and where R S is the reflectance of each line-defect beam splitter, R M the reflectance of the mirror, and δ the total phase change occurring over one circulation inside the resonator.One can easily obtain Eqs. ( 1) and ( 2) with the method generally used to derive the transmission and reflection of a conventional Fabry-Pérot resonator.The normalized transmission I 1 /I 0 at the drop port periodically oscillates between the minimum value 2 2 (1 and the maximum value of unity as a function of δ.The perfect transmission takes place whenever the phase change δ is an integral multiple of 2π, irrespective of the value of R S .At the frequencies on either side of the frequencies at which R S is maximum the transmitted intensity I 1 /I 0 becomes very small and I 2 /I 0 approaches unity.Thus, it is reasonably predicted that the proposed structure with beam splitters of high reflectance can act as a wavelength filter for self-collimated beams.The reflectance can be controlled by varying radii of rods to form the beam splitters.Therefore, the performance mechanism of our RFPF is very different from that of the polarization-independent drop filter [20] based on the phase delays of the TM and TE beams after one loop propagation in the resonator, even though their structures look very similar.

Calculation and experiment details
In order to investigate the transmission characteristics of the RFPF, we performed FDTD simulations using a freely available software package MEEP [27].The 2D-RFPF sample is composed of alumina rods of which the dielectric constant is approximately 9.7 in the microwave regime.The computational geometry of the 2D PC-RFPF is shown in Fig. 2(a).The perfectly matched layer absorbing boundary conditions [28] are used in the x-and ydirections to remove unwanted reflections from the calculation boundaries.In the simulations, a TM polarized Gaussian pulse with a waist of w = 3a is launched into the input interface of the RFPF.The experimental setup is shown in Fig. 2(b).Transmission measurements are performed using a network analyzer (Agilent HP8720B) and two horn antennas (ETS Lindgren 3160-08), which are used as a source and a receiver, respectively, and the 2D PC sample is placed between two antennas.The size of horns is 50 × 40 mm 2 each and it covers 10 -18 GHz frequency range.The two antennas are arranged to be in the TM polarized microwaves.Two parallel aluminum plates with periodically bored holes are placed at the top and bottom of the structure to hold the alumina rods vertically.These two parallel aluminum plates approximate our PC structure to 2D.In order to eliminate any loss caused by the cables or connecting parts of the setup, the resulting frequency-dependent data are normalized with the transmittance measured without the PC sample.

Result and discussion
We first calculated the transmitted power T S and the reflected power R S of the line defect beam splitter for various values of r d , the radius of rods in the line defect beam splitter.The steady-state electric field of TM polarized mode is shown in Fig. 3(a) and it clearly shows that an incoming self-collimated beam is split into the transmitted and reflected beams.Obtained results from the calculations, depicted in Fig. 3(b), confirm that the reflected power R S at the beam splitter increases as the defect radius r d decreases.As shown in Fig. 4, there are four transmission peaks (dips) in the transmission spectra calculated at the drop port (through port) in the frequency range of 11.8 to 12.8 GHz.Two solid lines represent the experimentally measured values at the drop (black line) and through (red line) port, respectively, while the dashed lines correspond to the simulated values.One can see that a good agreement exists between the FDTD simulations and experimentally measured results.When the value of r d decreases from 1.75 mm to 1.00 mm, the transmission peaks become sharper.When r d = 1.00 mm (R S = 0.88), described in Fig. 4(d), the maximum value of the measured transmission at the drop port are 80%, 86%, 95%, and 62% at the four resonant frequencies 11.93 GHz, 12.19 GHz, 12.44 GHz, and 12.67 GHz, respectively.It is also shown that the self-collimation effect becomes weaker as the frequency of the light beam gets away from the center frequency of the self-collimation frequency f s = 12.50 GHz.In the case of r d = 1.0 mm, the experimentally measured free spectral range (FSR) is 0.25 GHz and a 3 dB bandwidth ∆f is 12 MHz at the center of the resonance frequency f r = 12.44 GHz.A conventional way to show the selectivity of a filter is given by the quality factor Q ( = f r / ∆f).In this case, the Q-factor of 1382 is obtained from the simulations and of 1037 from the experimental measurement.This difference between the simulation and experimental results may be due to the fact that the simulation doesn't include the loss tangent for the alumina at microwave frequencies.When the self-collimated beam circulates through the PC-RFPF, the circulating light beam would experience the loss tangent of the alumina rods repeatedly resulting in the experimental losses.However, the measured Q-factor value is quite high for a photonic microwave filter.Propagation of two monochromatic self-collimated beams: one with the resonance frequency f r = 12.44 GHz and the other with the frequency f = 12.30GHz is investigated by observing the spatial electric field distributions as a function of time when r d = 1.00 mm. Figure 5 shows that (a) the light beam of the frequency f r = 12.44 GHz comes out of the drop port and (b) that of the non-resonance frequency f = 12.30GHz out of the through port.Thus, it demonstrates that the proposed PC-RFPF can act as a wavelength filter for the selfcollimated beams.In the case of conventional 2D PC-RFPFs without any ARSs the back reflection of incoming beam is almost 50% at the resonance frequency, but in our PC-RFPF with ARSs, back reflection of the incident beam into the input port is not detected at all.Since a PIC is a collection of multiple photonic functional devices, the back reflected light may ruin the performance of other functionalities.Moreover, this RFPF can be used not only to separate, but also to combine multiple self-collimated beams.For example, if the light beam of the resonance frequency f r = 12.44 GHz is injected into an additional input port as shown in Fig. 5(c), this self-collimated beam should be emitted out of the through port.As a result, the beam of the resonance frequency can be combined with a light of non-resonance frequencies which are injected into the original input port.Therefore, the results shown in Fig. 5 verify that our proposed PC-RFPF can act as a microwave add-drop filter.It will be interesting to realize the ring type Fabry-Pérot filter to work in an optical communication range.However, in an infrared range, it has been well-known that intrinsic radiation losses due to the scattering from imperfect structures are considerable for propagating modes through 2D or slab photonic crystals.The losses usually reduce the performance of the filter.However, recent improvement of fabrication quality of slab photonic crystals [13] suggests that it may be possible to implement Fabry-Pérot filters in the infrared range.

Conclusion
In conclusion, we propose a design of the RFPF based on the self-collimation effect in a 2D PC and experimentally present its transmission properties in microwave region.It is demonstrated that the proposed RFPF with two line-defect beam splitters and two mirrors can act as a microwave add-drop filter by using the FDTD simulations and microwave transmission measurements.With excellent agreements between the simulation and the measurement, the proposed RFPF has a Q-factor as high as 1037 at the resonance frequency f r = 12.44 GHz.Due to its geometric configuration with the ARSs applied, the RFPF does not back reflect any of the incoming self-collimated beams, and therefore the proposed RFPF is considered to be very suitable as a building block of PICs.

Fig. 1 .
Fig. 1.(a) Equifrequency contours of the 2D PC.The inset represents the 2D square lattice PC composed of alumina rods (ε = 9.7) in air.(b) Schematic diagram of a ring-type Fabry-Pérot filter composed of two line-defect beam splitters (BS1, BS2) and two perfect mirrors (M1, M2).(c) The ARSs are introduced at the input and two output ports.(d) Zoom-ins of the beam splitters and mirrors.Arrows indicate the direction of light propagation.

Fig. 2 .
Fig. 2. (a) Configuration of the FDTD simulations.Perfectly matched layers (PMLs) are placed at the ends of computational domain in the x-and y-directions.A Gaussian beam of width 3a is launched along the ΓM direction.(b) Schematic diagram of transmission measurement setup for the microwave.Two aluminum plates hold alumina rods vertically.

Fig. 3 .
Fig. 3. (a) Simulated spatial distribution of the steady-state electric field of the self-collimated beams of f = 12.50 GHZ at the line-defect beam splitter.(b) Simulated reflection and transmission powers at the beam splitter which are normalized with respect to the input power as a function of the radius of defect rods rd in the line-defect.

Fig. 5 .
Fig. 5. Simulated spatial distributions of the steady-state electric fields for the self-collimated beams at frequencies (a) 12.44 GHz for the drop port and (b) 12.30 GHz for the through port.(c) The proposed PC-RFPF combines the light beam of the resonance frequency fr = 12.44 GHz which is inserted into the additional input port and a light beam of non-resonance frequencies which are injected into the original input port.Calculations are performed for the case of rd = 1.00 mm.