Fourier transform infrared transmissionmicrospectroscopy of photonic
crystal structures
Gregory R. Kilby1,* and Thomas K. Gaylord2
1Department of Electrical Engineering and Computer Science and Photonics Research Center,United States Military Academy, West Point, New York 10996-5000, USA
2School of Electrical and Computer Engineering, Georgia Institute of Technology,777 Atlantic Drive NW, Atlanta, Georgia 30332-0250, USA
Photonic crystal (PC) technology offers unprece-dented capabilities to control the propagation of light[1,2]. Theoretically, devices designed using PCtechnology offer complete, lossless control of photonpropagation and a corresponding unmatched perfor-mance capability. The physical size of PC structuresis on the order of several wavelengths of light, mak-ing feature sizes and device periods a fraction of thiswavelength. The potential benefits of lossless controlof light and the small-size advantage offered in thesedevices makes this technology attractive for many
optoelectronic applications. Indeed, researchershave demonstrated impressive and efficient devicessuch as filters, resonators, and waveguides [3–7].However, few complete PC-based systems or func-tional devices have been commercialized. The rela-tively slow development of this technology iscaused in part by challenges associated with thefabrication and characterization of PC structures.Structures designed to operate in a particular man-ner often do not achieve the predicted performancebenchmarks. Observed macroscopic performance israrely related to microscopic characteristics. To fullyunderstand and to exploit this technology, one mustbe able to relate these performance differences tovariations in fabricated structures. Characterizationtechniques capable of such reconciliation are needed.
One such characterization parameter is the single-incident-angle transmittance of the fabricatedstructure. This quantity, if available, would enableresearchers to measure the performance of severalstructures and to compare the single-incident-angletransmittance of one structure with another.Furthermore, the transmittance of the structure toany incident beam composed of combinations of theseplane waves could be computed. Through the trans-mittance, the performance of a PC structure can berelated to variations associated with fabrication andchanges in the incident beam, as well as establishingtolerances for a particular device.In this paper, an experimentally based technique
to determine the single-incident-angle, plane-wavetransmittance or reflectance of PC structures isdemonstrated. The spectral transmittance measure-ments of fabricated structures provide the single-angle plane-wave transmittance of a structure as afunction of wavelength. Individual measurementswith the demonstrated apparatus represent the PCstructure transmittance of a composite beam consist-ing of superimposed plane waves incident over arange of incident angles. By processing these spectraltransmittance measurements, the single-incident-angle plane-wave transmittances of fabricated struc-tures can be determined. For the purposes of thispaper, transmittance measurements are described.However, the technique is similarly applicable toreflectance measurements.
2. Experimental Configuration
In the apparatus, a Fourier transform infrared(FTIR) spectrometer coupled to an infrared micro-scope is used to measure the transmittance of aPC structure. In this paper, a Bruker IFS 66/Sresearch-grade FTIR spectrometer was coupled toa Bruker Hyperion 1000 infrared microscope. Thesystem is controlled by a personal computer andBruker control software (Optical User Software(OPUS)). In the FTIR system, infrared light isfocused by a Schwarzschild reflecting microscope ob-jective as shown in Fig. 1. In this objective design,light passes through an aperture in the large pri-mary mirror and reflects from the small secondarymirror back to the primary mirror and finally tothe focal point. The focusing light is incident overa range of incident angles ranging from θob;min toθob;max, determined by the geometric constructionof the objective and the position of the mirrors. Arectangular air-slit aperture is mounted in the en-trance aperture of the objective to restrict incidentlight predominantly to the plane of the PC structure.The intensity of the focusing beam as a function ofthe incident angle is measured by placing a model71961 Thermo-Oriel Miniature Thermopile powerdetector at a measurement plane a selected distancefrom the focal point. The beam power is measured asthe detector is scanned across the beam to constructthe beam profile. Since the measurement plane is aknown distance from the focal point and the mini-
mum and maximum objective angles are specified,the measured beam profile can be divided into sec-tors where the area of each sector of the beam profilerepresents the intensity coefficient for light at thecenter incident angle of the sector. The angular widthof the sector affects the resolution to which thesingle-angle plane-wave transmittances can be com-puted. The sector width is the smallest incrementthat can be computed between adjacent single-angleplane-wave transmittances.
The sample is positioned at the focal point of thebeam at a known angular orientation of the objectiveaxis with respect to the sample surface normal (θS).The PC sample is mounted onto a computer-controlled rotation stage to facilitate positioning ofthe PC sample at various objective-axis-to-surface-normal angular orientations. Figure 2 illustratesthe function of the horizontal slit in restrictinglight to the PC plane and the positioning of the PCsample at multiple objective-axis-to-surface-normalorientations.
In this experimental configuration, the probebeam propagates through the sample, and the trans-mitted power is measured as a function of the wave-length. A single transmittance measurement recordsthe response of the PC structure to the incidentbeam. This measurement can be mathematicallydescribed as
Tob;θS ¼ ½a1 a2 a3 a4 �
2664tθk1tθk2tθk3tθk4
3775; ð1Þ
where Tobj;θS is the transmittance measurementwhen the angle between the objective axis and the
Fig. 1. (a) Optical configuration of a Schwarzschild reflectingfocusing objective. Light passes through an aperture in the largeouter mirror and is reflected by the smaller mirror back to thelarge mirror. This mirror reflects light to the focal point. (b) Lightrays from a Schwarzschild objective are focused onto a photoniccrystal sample. The objective axis is at an angle θS with respectto the normal of the sample. The minimum and maximum rayangles of the objective are given by θob;min and θob;max, respectively.An example single-angle plane-wave incident at an angle of θkwith respect to the sample normal is shown.
PC surface normal is θS, the ½a� row vector representsthe weighting coefficients for each incident angleplane wave present in the composite beam, andthe ½t� column vector represents the single-angleplane waves θk that form the composite, multiple-incident angle beam. By varying the angle betweenthe objective axis and the PC surface normal, thismeasurement can be repeated and recorded for arange of angles θS. For each angle, a unique set ofcomponent plane waves will comprise the incidentbeam. The multiple measurements can be formu-lated as a matrix algebra problem [8].For illustrative purposes, the matrices for a set of
measurements are constructed using a Schwarzs-child objective with θob;min ¼ 10° and θob;max ¼ 30°.Additionally, a large angular spacing of 10° is usedfor both the objective axis position Tob;θS and the sin-gle plane wave angles tθk . A maximum objective axisangle of �30° is used. When the objective axis is at30°, the maximum plane wave angle selected by theobjective is 60°. With these conventions, the mea-sured composite transmittances Tob;θS, in terms ofthe single-angle plane-wave transmittances, tθk ,are given by
The single-angle plane-wave transmittances tθk canbe determined by using the measured compositetransmittances Tob;θS and inverting the coefficientmatrix ½a�.However, the coefficient matrix in the single-angle
plane-wave computation discussed above is ill condi-tioned. Even thoughmore measurements (equations)are recorded than the number of desired single-angleplane-wave transmittances, the ill-conditioned sys-tem is effectively underdetermined due to redundantinformation in adjacent measurements. Inversion ofthe poorly conditioned coefficient matrix produces anunstable solution, wherein small changes in the com-posite multiple-incident-angle measurements can re-sult in large variation in the computed solution for
the single-angle plane waves. Errors caused by mea-surement noise or rounding errors in the discretiza-tion of data are amplified and dominate the solution.To compute a meaningful solution for the single-angle plane-wave transmittance, the problem mustbe regularized by including additional informationto assist in solving the problem [9,10].
In this case, a singular value decomposition of thecoefficient matrix shows that the singular valuesgradually decay toward zero. This suggests thatthe problem is defined as an ill-posed problem andcan indeed be regularized using Tikhonov regulari-zation [11]. In this regularization method, theunstable parts of the solution associated with thesmallest singular values are stabilized by using aside constraint. The side constraint forces a stablesolution in the computation of the single-angleplane-wave transmittances. In this case, the minimi-zation of the residual norm of the first derivative ofthe solution was used to stabilize the single-angleplane-wave computation.
To test the measurement noise tolerance of theregularization method, a theoretical compositetransmittance matrix Tob;θS was constructed using
theoretical single-angle plane-wave transmittances,computed using the transfer matrix method and acoefficient matrix. The ideal composite transmit-tances were then perturbed by adding a uniformlydistributed random noise variation to each valuein the composite transmittance matrix. Finally, theregularization method was applied to the noisytransmittances in an effort to recover the theoreticalsingle-angle plane-wave transmittances. The addednoise variation was tested for random noise varia-tions from 0% (no variation) to 20% of the theoreticaltransmittance values. Stable solutions for the single-angle plane-wave transmittances were recovered forall values of added noise in this range. A qualitativecomparison between the recovered and ideal single-
angle plane-wave transmittances showed that theregularization method produced an acceptable solu-tion for the single-angle plane-wave transmittancesin instances when the composite transmittances in-cluded up to 5% additive random noise variation [12].
3. Transmittances from Multiple-Angle Measurements
For the measurements discussed in this paper, aThermo-Oriel reflecting Schwarzschild objectivewith θob;min ¼ 10° and θob;max ¼ 24° was used in theFTIR system. Intensity coefficients were computedfor 2° sectors of the incident beam. Composite trans-mittance measurements were recorded in 2° incre-ments, while the objective axis to PC surfacenormal was rotated over a range from θS ¼ þ20°to θS ¼ −20°. These measurements included planewaves incident at angles ranging from θk ¼ þ44°to θk ¼ −44°. The PC test structure was fabricatedusing standard photolithographic techniques and aSurface Technology Systems Advanced Silicon Etch(STS-ASE) system employing a modified Bosch pro-cess. A 22-period, one-dimensional PC structure thatconsists of alternating layers of air and silicon withthe thickness of the silicon regions equal to 3:15 μmand the air region thickness equal to 4:55 μm wasproduced to test the characterization method.Figure 3 shows a fabricated one-dimensional photo-nic crystal similar to the measured structure. Theinset shows the surface roughness characteristic ofthe Bosch etch process.The measured composite transmittances were
compared to theoretical transmittance calculations.The theoretical calculations were computed usinga transfer-matrix method modeling of the layers ofthe PC structure. For the calculations, the measuredbeam profile and computed beam coefficients wereused to determine the theoretical single-angle
plane-wave transmittances. The simulations in-cluded a 2% uniformly distributed, random deviationin the fill factor to account for the variations in the fillfactor at various depths of the structure due to side-wall taper and ripple. Figure 4 shows the normalizedcomposite measured transmittances compared to thetheoretical calculations at several orientations of theobjective axis with respect to the sample normal.
The measured and theoretical composite transmit-tances show very good agreement for the position ofthe transmission bands and the photonic bandgapson the wavelength spectrum over the full angularorientation range of the objective. Additionally, var-iations in spectral features across this rotation rangeare reproduced in the measurements as the objectiveaxis rotates. There are significant differencesbetween the magnitude of the theoretical and
Fig. 2. (Color online) As the objective axis is rotated with respect to the sample normal in the half-plane in front of the sample, varioussets of incident angle plane waves are selected by the objective. By rotating and measuring the transmittance or reflectance of the com-posite beam in discrete increments, new angles are added while some are removed in each subsequent measurement. If θS is rotated tosufficient limits, all angles in the half-plane before the sample can be selected. The increment from one axis angular orientation to the nextis given as ΔθS.
Fig. 3. Scanning electron microscope image showing one-dimensional photonic crystal structure on an SOI substrate.The one-dimensional PC was etched to a depth of approximately40 μm. The inset shows the roughness of the PC surface caused bythe etch and passivation cycling inherent to the Bosch process.
measured normalized transmittances throughoutthe objective rotation range. Several sources contri-bute to these differences.Magnitude variations between theoretical and
measured transmittances can be attributed to inter-face irregularities and distortions at the PC surface.Losses as high as 60% are reported from both fabri-cation process effects and the methods used toremove the PC structure from the substrate [13].The inset in Fig. 3 shows the irregularities causedby the fabrication process on the sidewalls of thePC structure. When the device is separated fromthe substrate, additional surface irregularities atthe PC entrance plane can be expected. While a uni-formly distributed random variation to the fill factorwas included in the theoretical calculations in aneffort to account for these variances, significantvariation between the theoretical and measuredtransmittances can be expected. Subsequent normal-ization of the measured data results in the mismatchbetween the theoretical and measured transmissionmaximums. For instance, in the theoretical computa-tion, the transmittance maximum corresponds to thetransmission band centered at a wavelength ofλ ¼ 11:5 μm. In the measurements, the transmit-tance maximum is typically associated with thetransmission band centered at λ ¼ 6:8 μm at positiveand near normal objective axis orientations and thetransmission band centered at λ ¼ 8:5 μm at theextreme negative objective axis orientations.Additionally, there are other subtle differences
between the theoretical computation and the mea-surements. In the theoretical computation, a discrete
set of theoretical single-angle plane-wave transmit-tances and the measured beam coefficients are usedto compute the theoretical composite response. Themeasured coefficients are multiplied by the theoreti-cal single-angle plane-wave transmittance asso-ciated with the center of the coefficient sector. Incontrast, the composite measurement includes inci-dent light spanning a small angular range associatedwith the coefficient sector width, and the coefficientis distributed across this incident light.
Furthermore, the discrete set of plane-wave trans-mittances used for the theoretical calculation are in asingle plane normal to the entrance plane of the PCstructure. In contrast, while light is predominantlyrestricted to this plane by the inclusion of the air-slitaperture in the experimental apparatus, some out-of-plane light is incident on the structure. This can beexpected to predominantly affect band edges [13] butmay also explain some of the magnitude variances.
Despite the magnitude differences over the broadspectral range, transmittance details over localwavelength ranges are reproduced by the method.For example, in Fig. 4 the opening of a transmittancenull at a wavelength of approximately 7:5 μm is evi-dent as the objective is rotated from θS ¼ 20°to θS ¼ 0°.
4. Single-Angle Plane-Wave Transmittances
The single-angle plane-wave transmittances weredetermined over the range from θk ¼ þ44° to θk ¼−44° using the measured composite transmittances,the measured beam coefficients, and the regulariza-tion method. The single-angle plane-wave transmit-tances for representative plane-wave angles fromwithin this range are shown in Fig. 5. The measuredtransmittances show generally good agreement withtheory. Since there was a large difference in trans-mittance magnitude observed in the measuredvalues, the large differences in magnitude are ex-pected in the measured single-angle plane-wavetransmittances. Oscillations due to the theoreticalboundary interface are evident in the theoretical cal-culations. The multiple-component incident lightpresent in a single sector represented by one coeffi-cient in the method removes these oscillations in themeasured data. The transmittances for incident an-gles in the center of the measurement range showgenerally more agreement with theory than the lar-ger incident angle plane waves at the edge of themeasurement range. Also, at these larger incidentangles, the transmission bands in the theoretical cal-culation tend to shift slightly to shorter wavelengths.This detail is not reproduced in the measuredsingle-angle plane-wave transmittances. A possibleexplanation is the increased number of larger-errormeasurements.
5. Summary and Discussion
A method to determine the single-angle plane-wave transmittances of PC structures has beendemonstrated. The method requires a careful
Fig. 4. (Color online) Composite normalized FTIR microspectro-scopy transmittance measurements and normalized theoreticaltransmittance calculations of a PC structure. The measured andtheoretical transmittances are shown for objective axis positionsof θS ¼ 20°, θS ¼ 12°, θS ¼ 8°, and θS ¼ 0° with respect to thesample normal.
characterization of the incident light to determinethecomposition of the incident beam. Multiple measure-ments at various orientations of the objective axiswith respect to the surface normal measure thetransmittance of the structure for various subsetsof incident plane waves. A set of measurementscan be formulated as a matrix algebra problem thatcan be solved to determine the single-angle plane-wave transmittances of the fabricated device.When the computations included measurements
that exceeded the identified noise threshold, theagreement between the theoretical and measuredtransmittance degraded. However, over the rangeof measurements, when the measurement noisewas within the established threshold, the spectralcharacteristics, including rapid variations in thespectra from wavelength to adjacent wavelength,were recovered. Magnitude differences between themeasured and theoretical transmittance measure-ments can be attributed to a variety of factors,primarily the surface roughness of the fabricatedstructures.The regularization and deconvolution of the single-
angle spectral transmittances has potential applica-tion in other areas of research. These transmittancescan be used to characterize design variations and
process changes in fabricated devices. The single-angle plane-wave transmittance can be used to relateobserved macroscopic performance to microscopiccharacteristics, thus potentially increasing the paceof the integration of PC devices into commercialsystems. The presented results demonstrate thefeasibility of the characterization method. Furtherrefinements in the methodology are needed in orderto realize the full potential of the approach.
This work was supported by the Army ResearchOffice under grant DAAD19-03-1-0286.
References
1. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062(1987).
2. S. John, “Strong localization of photons in certain disordereddielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489(1987).
3. M. Imada,S.Noda,A.Chutinan,M.Mochizuiki, andT.Tanaka,“Channel drop filter using a single defect in a 2-d photonic crys-tal slab waveguide,” J. Lightwave Technol. 20, 873–878 (2002).
4. A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonatoroptical waveguide: a proposal and analysis,” Opt. Lett. 24,711–713 (1999).
5. A. Mekis, J. Chen, I. Kurland, S. Fan, P. Villeneuve, and J. D.Joannopoulos, “High transmission through sharp bends inphotonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790(1996).
6. E. Miyai, M. Okano, M. Mochizuki, and S. Noda, “Analysis ofcoupling between two-dimensional photonic crystal wave-guide and external waveguide,” Appl. Phys. Lett. 81, 3729–3731 (2002).
7. S. Olivier, C. J. M. Smith, H. Benisty, C. Weisbuch, T. Krauss,R. Houdré, and U. Oesterle, “Cascaded photonic crystal guidesand cavities: spectral studies and their impact on integratedoptics design,” IEEE J. Quantum Electron. 38, 816–824 (2002).
8. T. K. Gaylord and G. R. Kilby, “Optical single-angle plane-wave transmittances/reflectances from Schwarzschild objec-tive variable-angle measurements,” Rev. Sci. Instrum. 75,317–323 (2004).
9. P. C. Hansen, Rank-Deficient and Discrete Ill Posed ProblemsNumerical Aspects of Linear Inversion (Society for Industrialand Applied Mathematics, 1998).
10. Department of Mathematical Modeling, “Regularization tools:a Matlab package for analysis and solution of discreteill-posed problems. Version 3.1 for Matlab 6.0,” TechnicalUniversity of Denmark, Building 305, DK-2800 Lyngby,Denmark, 2001.
11. A. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems(Winston and Sons, 1977).
12. G. R. Kilby, Infrared Methods Applied to Photonic CrystalDevice Development (Georgia Institute of Technology, 2005).
13. S. Rowson, A. Chelnokov, C. Cuisin, and J.-M. Lourtioz,“Three-dimensional characterization of a two-dimensionalphotonic bandgap reflector at mid-infrared wavelengths,”IEE Proc. Optelectron. 145, 403–408 (1998).
Fig. 5. (Color online) FTIR system transmission-based single-angle plane-wave normalized transmittance and theoreticalsingle-angle plane-wave ideal normalized transmittance of a PCstructure. The measured and theoretical transmittances areshown for objective axis positions of θk ¼ 1°, θk ¼ 9°, θk ¼ 21°,and θk ¼ 25°.
