Design and optimization of space-variant photonic crystalfilters
Raymond C. Rumpf,1 Alok Mehta,2 Pradeep Srinivasan,2 and Eric G. Johnson3,*1Prime Research, Blacksburg, Virginia 24060, USA
2College of Optics and Photonics, University of Central Florida, Orlando, Florida 32816, USA3Center for Optoelectronics and Optical Communications, University of North Carolina at Charlotte, Charlotte,
There is a need for easily fabricated devices thatcan be placed over a detector array to perform fil-tering functions uniquely tuned for each pixel. Thisconcept is illustrated in Fig. 1, where a wavelength-discriminating imaging array is formed by fabricat-ing a space-variant photonic crystal on top of anIR-detecting charge-coupled device (CCD) array [1].Although the CCD itself is wavelength insensitve,fabricating a filter over the array enables adjacentpixels to detect different colors. A narrow transmis-sion band of the photonic crystal is tuned across thedevice aperture by adjusting the diameter of holesetched through the volume. Spacing of the holes(lattice constant) is held constant. This technologycould be used in highly compact IR imaging systemsand even incorporate polarization [2,3].
Realizing this concept with conventional multi-layer films [4] would require film thickness to bevaried across the array. This could only be achieved
using a complicated fabrication process with manysteps and difficult alignments. A photonic crystal ap-proach has been conceived that greatly simplifies fab-rication. Filter response at each pixel may be customtuned by adjusting the hole size in the vicinity of eachpixel. This only requires adjustment of transversedimensions and can be realized with a single maskprocess. Although not addressed in this work, an ad-ditional advantage of a photonic crystal is the poten-tial for better performance at oblique angles ofincidence due to modulation of the refractive index inthree dimensions. Achieving angular tolerance in amultilayer film requires adding additional layers.Thicker films introduce more image distortion, aber-rations, and defocusing of the image. Photonic crys-tals promise easier fabrication, thinner devices, andbetter performance at oblique angles.
This paper investigates the design and optimiza-tion of space-variant photonic crystal filters based ona simple fabrication approach. Two separate materialsystems will be addressed. These are SiO�SiN andGaAs�AlGaAs. To realize a sufficiently wide photonicstop band, it will be shown that the GaAs�AlGaAsmaterial system requires an extra undercutting step
where GaAs layers are preferentially etched. All sim-ulations were performed using rigorous coupled-waveanalysis [5,6] or the plane-wave expansion method[7,8].
2. Design and Optimization
To determine a good starting point for device geome-try, properties of the ideal bulk photonic crystals wereoptimized before incorporating defects or consideringfabrication. Properties included lattice symmetry, holesize, duty cycle of the layers, degree of undercutting,and minimum number of dielectric layers for sufficientsuppression in the bandgap. Next, a defect was incor-porated by modifying the thickness of an interior layer.Device parameters were optimized with the defect in-corporated for maximum tuning range while main-taining acceptable background suppression. Finally,fabrication was considered and the taper in the holeradius was found to be the critical limiting factor. Twomethods for compensation will be identified.
A. Square Versus Hexagonal Arrays
To produce strong suppression in the bandgap andoffer better performance at oblique angles of incidence,lattices should have a high degree of symmetry sowaves see essentially the same perturbations regard-less of their direction or polarization. Cubic and hex-agonal lattices are well known for high degrees ofsymmetry.
To initially assess the differences between squareand hexagonal arrays, their photonic band structureswere calculated and are shown in Fig. 2. Both sym-metries provide similar performance in terms of abandgap and tolerance to oblique angles of incidence,but dimensions of the hexagonal lattice may benearly twice as large. Neither lattice provides a com-
plete photonic bandgap but both show a partial gapfor light propagating near parallel to the vertical axis.The width of the bandgap is reduced if oblique anglesare considered where the upper half of the bandgap ismaintained over a cone of angles roughly 45° wide.After observing similar behavior from both symme-tries, it was reasoned the structures were behavingmuch like a multilayer film where the hole radius wasused to control the effective refractive index of eachlayer.
B. Optimization of the Partial Photonic Bandgap
To optimize layer duty cycle and hole radius, the datadepicted in Fig. 3 was generated for both SiO�SiNand GaAs�AlGaAs material systems. This figureshows the width of the photonic bandgap as a func-tion of the hole radius and the layer duty cycle forboth square and hexagonal arrays. To maximize thewidth of the bandgap, the optimum layer duty cyclefor the SiO�SiN-based photonic crystal was found tobe 0.58, but decreasing slightly for increasing holeradius. For the GaAs�AlGaAs-based photonic crystal,the optimum duty cycle was found to be 0.54, decreas-ing only slightly with increasing hole radius. Tomaintain a sufficient photonic bandgap using SiO�SiN, the hole radius should be made less than 0.4aand the duty cycle should be kept between 0.3 and0.8. Using GaAs�AlGaAs without an undercut yieldsa similar conclusion, but the width of the bandgapwas around half that of the SiO�SiN system. Refrac-tive indices were assumed to be SiO �n � 1.4458�,SiN �n � 1.9767�, GaAs �n � 3.5�, and AlGaAs
Fig. 1. Concept for a highly compact color imaging system. Aspace-variant photonic crystal is formed over any array of detectorsso that adjacent pixels can detect different colors.
Fig. 2. Photonic band diagrams for square and hexagonal latticesymmetries. Both provide similar optical performance, but thedimensions of the hexagonal array can be made larger.
�n � 3.0�. Material dispersion was not considered herebecause the devices of interest were only required tooperate over a limited range of wavelengths around1550 nm.
For the GaAs�AlGaAs-based photonic crystal,GaAs layers may be preferentially etched over theAlGaAs layers to form an undercut [9]. This raisesindex contrast, potentially leading to a wider pho-tonic bandgap. The data provided in Fig. 4 was cal-culated to assess the optimum hole radius for eachlayer. The hole radius in the GaAs layers must begreater than in the AlGaAs layers because it is theGaAs layers that are selectively etched. A dashed lineindicates zero undercut. Below this line is not a re-alizable geometry using the prescribed process. It canbe seen that for small degrees of undercut, the widthof the bandgap actually decreases. It is not until theundercut exceeds some breakeven value that under-cutting widens the bandgap. The ideal case was found
to be very small holes in the AlGaAs layers with verylarge holes in the GaAs layers. Given sufficient un-dercutting, it appears more critical to have a largehole radius in the GaAs layers than it is to have asmall hole radius in the AlGaAs layers.
Analysis to this point assumed an infinitely peri-odic photonic crystal. In practice, photonic crystalsare of finite size. To estimate how many layers areneeded to provide sufficient suppression within thebandgap, transmission through a photonic crystalslab with an increasing number of lattice periods wascalculated for both candidate lattices. It was con-cluded that both material systems require approxi-mately 10 periods (i.e., 20 film layers) to achieve atleast 20 dB suppression within the bandgap. Basedon previous analysis, hole radii in the SiO�SiN struc-ture were chosen to be r1 � r2 � 0.3a and the holeradii in the GaAs�AlGaAs were chosen to be r1� 0.2a and r2 � 0.4a. For both material systems, the
Fig. 3. Optimization of lattice parameters for the widest partial photonic bandgap.
duty cycle of the layer thicknesses was chosen to bef � 0.55. In this diagram, two film layers compriseone longitudinal period of the photonic crystal.
C. Incorporation of a Defect
To construct a device that functions in the mannerdepicted in Fig. 1, a defect must be incorporated toproduce a narrow transmission spike within the par-tial photonic bandgap [10]. It was assumed sufficientenergy was incident on the array to be detected afterfiltering. The width of the defect was found to affect theposition of the transmission notch, but with a thick-ness of around ��2 the notch was approximately cen-tered within the bandgap. When the defect in bothsystems was made thicker, the transmission peakmoved to a longer wavelength. For very thick defectlayers, the position of the transmission peaks trans-lated more slowly with respect to a changing wave-length, leading to multiple peaks present within thebandgap. Rigorous simulations assuming an ideal ge-ometry showed the defect layer should be around 1.17ain the GaAs�AlGaAs device and 1.0a in the SiO�SiNdevice.
Although the position of the transmission spike canbe controlled by the thickness of the defect layer, thehole radius is the mechanism that should be used totune its position across the device aperture. The po-sition of the bandgap and the transmission spike donot inherently shift evenly. To compensate for un-even shift, the thickness of the defect should be fur-ther optimized to achieve the maximum tuningrange. For this reason, the defect thickness of theGaAs�AlGaAs device was set to d � 1.3a instead ofd � 1.17a as prescribed previously.
Tuning curves for adjusting the hole radius inboth material systems are depicted in Fig. 5. Thesediagrams show that the position of the bandgapshifts in addition to the position of the transmissionnotch, effectively reducing the operating bandwidthof the device. Inspection of Fig. 5 shows the op-erational bandwidth of the SiO�SiN device to be�0.6a ��600 nm�, whereas for the GaAs�AlGaAs de-vice it was �0.5a ��500 nm� for a � 1 �m. Also, the
line shape of the transmission notch was more uni-form in the SiO�SiN material system than it was inthe GaAs�AlGaAs material system.
3. Hole Taper
As a result of the physics inherent in the etchingprocessing, the hole radius can vary with depth, form-ing essentially a chirped photonic crystal. This hasthe effect of detuning the Bragg mirrors above andbelow the defect, reducing the efficiency of the deviceand affecting the bandgap. The effect of a taperedhole radius is summarized in Fig. 6, where transmit-tance through a 20 layer slab with a defect was cal-culated for both material systems. In these diagrams,three curves are shown. The solid line representstransmission through an idea device. The dashedcurve represents transmission through a device witha small degree of hole taper. In this case, the positionof the transmission notch shifts to a significinatlylonger wavelength. The dotted curve shows transmis-sion through a device with the same hole taper that iscompensated using a method described later.
As taper angle � increases, the bandgap in bothmaterial systems shifts to longer wavelengths be-cause the photonic crystal contains more dielectric.As may be expected, the GaAs�AlGaAs system ismore sensitive because of its higher refractive index
Fig. 4. Optimization of GaAs undercut. The width of the bandgapis decreased until a significant breakeven undercut is achieved.
Fig. 5. Position of the transmission notch as a function of theetched hole radius for a hexagonal array.
and stronger resonance. In fact, when the taper angleapproaches just 0.5° in this system, the spectral re-sponse becomes distorted well beyond being a usefuldevice. In both systems, the position of the transmis-sion notch shifts to longer wavelengths and peaktransmission is reduced. The shift in position can beattributed to a higher fill factor caused by reducingthe hole radius. The decrease in peak efficiency iscaused by reflection bands of the top and bottomBragg mirrors being misaligned and a reduction intotal reflection from the mirrors.
If the hole taper cannot be prevented, there are twopotential solutions. First, the initial hole size can beincreased to reposition the bandgap and transmissionspike. Second, the film thickness can be modified ineach layer such that the position of the bandgap ismaintained throughout the crystal. To investigatethe second approach, the longitudinal period was ad-justed for varying hole radius to maintain the posi-tion of the bandgap in both material systems. Theseresults are shown in Fig. 7. Shaded regions surround-ing the lines show how the bandgap width is affectedby the compensation. Wider shaded bands indicate
wider bandgap. The horizontal dashed line repre-sents the case in which the hole radius is ideal and nocompensation is used. The width of the bandgap inthe SiO�SiN material system decreases with decreas-
Fig. 6. Comparison of transmission through an ideal device, a device with a hole taper, and a tapered device with compensation and usinga hexagonal array of holes. The compensated device maintains the position of the transmission notch.
Fig. 7. Longitudinal period as a function of hole radius to main-tain the position of the partial photonic bandgap using a hexagonalarray of holes.
ing hole radius, whereas for GaAs�AlGaAs it in-creases with decreasing hole radius.
To illustrate the compensation technique, datacalculated for Fig. 7 was imported into the model tocalculate the transmission spectrum through com-pensated devices. As depicted by the dotted curve inFig. 6, the position of the transmission notch wasmaintained for both material systems even in thepresence of hole taper. This is only possible by ad-justing the film thicknesses and the defect thick-ness appropriately. Compensation is better in theSiO�SiN device due to lower sensitivity to structuraldeformations. Differences in layer thicknesses are con-veyed in the leftmost figures, but are barely perceiv-able because the diagrams were drawn to scale.
4. Device Performance at Oblique Incidence
It is important to understand how real devices performat oblique angles of incidence. Inspired by photonicband diagrams, Fig. 8 was constructed to show theposition of the transmission notch as a function ofthe angle of incidence for both material systems. Thepoints marked X along the horizontal axes correspondto light incident near parallel to the x axis. The pointsmarked Z correspond to light at normal incidence. Thepoints marked D indicate light that is incident at near90° aligned along the diagonal of the hexagonal unit
cell as depicted in the rightmost diagram. Intermedi-ate points are marked by their angle of incidence.
The data show that the response is highly symmet-ric along the X and D directions. In general, responsein these directions can be very different. The appar-ent symmetry is another indication that the device isbehaving much like a multilayer film where hole sizecontrols the effective refractive indices of the layers.
Both devices are more tolerant to oblique angles forTE-polarized light than TM, but it is the GaAs�AlGaAs device that is most tolerant. At the sametime, it is the GaAs�AlGaAs that is most sensitive inTM polarization. It can be concluded from this datathat both devices offer good performance over a coneof angles extending �5°. This performance parame-ter is considerably less than predicted in discussionregarding Fig. 2 because of the incorporation of adefect and the device having finite dimensions in-stead of being an infinitely periodic lattice.
5. Conclusion
Space-variant filters have been designed and opti-mized based on an easily fabricated photonic crystalstructures. Two material systems have been consid-ered. These are SiO�SiN and GaAs�AlGaAs. Thecrystals were comprised of an array of holes etchedinto heterostructured substrates. A defect layer was
Fig. 8. Performance of the real device with a hexagonal array of holes at oblique angles of incidence. The diagram at right shows howangles of incidence are defined along the horizontal axes of the plots at left.
incorporated to produce a transmission notch in thecenter of the transmission stop band. The position ofthe notch was tuned by adjusting the hole radiusacross the device aperture to form a space-variantfilter. It was found that a hexagonal array enableddimensions to be larger than a square array so it wasthe preferred geometry. Otherwise, it offered similarperformance to a square array.
The material system using SiO�SiN was a simplerdevice to fabricate because it did not require undercut-ting to produce a sufficient bandgap. This system wasmost limited in tuning range, but the spectral responsewas more robust to the tuning mechanism. The GaAs�AlGaAs material system required the GaAs layers tobe undercut to produce a sufficient bandgap. It wasfound that a minimum amount of undercut was nec-essary for the width of the bandgap to be improved.While more sensitive to structural dimensions, thissystem provided the greatest tuning range. Both ma-terial systems provided fair tolerance to oblique anglesof incidence within a cone of angles extending �5°.
This work was funded in part by the National Sci-ence Foundation CAREER Grant ECS 0348280.
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