Finite-difference time-domain modeling and experimental characterization of planar waveguide fluorescence sensors

The finite-difference time-domain method (FDTD) is a powerful numerical technique for solving Maxwell's equations in a discretized space and time grid. Its applications have up to now been in the analysis of electrically large structures in the microwave domain, and the scope of investigations has been extended to the optical region only recently. Because of computer memory limitations, the method is generally restricted to configurations which extend to the order of tens of wavelengths in three dimensions, or hundreds of wavelengths in two dimensions. Optical sensor structures are therefore of suitable size to be modeled with FDTD, and e. g. fluorescence sensor design can benefit from the use of FDTD in optimization of the waveguide structures. In general, the integration of chemical and optical design is difficult, but FDTD can bring the two design approaches closer together. One of the main advantages of FDTD is its ability to include near-field effects, such as distribution of protein molecules on the active surface of optical sensors in the model, which has been shown to be important in estimating the fluorescent excitation and collection efficiencies of molecules on surfaces. In addition, for planar structures, two-dimensional models are adequate for studying many aspects of sensor design. We applied FDTD to design of planar fluorescence sensors. Excitation and emission models were analyzed for planar waveguide structures with side collection of emitted light in mind. Planar waveguides were fabricated on fused silica substrates, and the characteristics of the waveguides were compared to the model. Good agreement was found with the FDTD modeling to the physical model, and based on this knowledge, an FDTD sensor model was prepared predicting good fluorescence excitation and emission side collection efficiencies.

Finite-difference time-domain modeling and experimental characterization of planar waveguide fiuorescence sensors Jyrki S. Kimmel and Douglas A Christensen* Department of Bioengineering. and *Department of Electrical Engineering University of Utah. Salt Lake City. UT 84112 ABSTRACT The fmite-difIerence time-domain method (FDID) is a powerful numerical technique for solving Maxwell's equations in a discretized space and time grid. Its applications have up to now been in the analysis of electrically large structures in the microwave domain. and the scope of investigations has been extended to the optical region only recently. Because of computer memory limitations. the method is generally restricted to configurations which extend to the order of tens of wavelengths in three dimensions. or hundreds of wavelengths in two dimensions. Optical sensor structures are therefore of suitable siU to be modeled with FDID. and e. g. fluorescence sensor design can benefit from the use of FDID in optimization of the waveguide structures. In general. the integration of chemical and optical design is difficult, but FDTD can bring the two design approaches closer together. One of the main advantages of FDID is its ability to include near-field effects. such as distribution of protein molecules on the active surface of optical sensors in the model. which has been shown to be important in estimating the fluorescent excitation and collection efficiencies of molecules on surfaces. In addition, for planar structures, two-dimensional models are adequate for studying many aspects of sensor design. We applied FDID to design of planar fluorescence sensors. Excitation and emission models were analyzed for planar waveguide structures with side collection of emitted light in mind. Planar waveguides were fabricated on fused silica substrates, and the characteristics of the waveguides were compared to the model. Good agreement was found with the FDlD modeling to the physical model, and based on this knowledge. an FDID sensor model was prepared predicting good fluorescence excitation and emission side collection efficiencies. 1. INTRODUCTION Yee first introduced FDID as a numerical method for solving Maxwell's equations in 19661• and the pioneering works on the application of this technique in the microwave domain were done by Taflove2. Recently, the scope of FDlD bas been extended to the optical range of frequencies, as suitable problems to be analyzed with the method have emerged3. Planar optical sensors are particularly good candidates to be modeled with FDID. Their structure is simple enough for the 136 1 SPIE Vol. 1587 Chemical. Biochemical. and Environmental Fiber Sensors 1/1 (1991) 0-8194-0718-61921$4.00 model to be used effectively, and due to the planarity, only a two-dimensional analysis is required for many cases. With a: modem-day supercomputer, structmes of hundreds of wavelengths long can be analyzed in two dimensions. FDIDis an initial-value problem, where an electromagnetic field evolves, as specified by the sources, in discrete time steps along a lattice including the structure to be analyzed. The propagation of the field is affected by the complex dielectric constants at each lattice point (cell), and at boundaries with differing complex permittivities, reflection, refraction and diffraction can be observed. The time-stepping is carried out to several complete cycles of a sinusoidally varying source, until the source fields have propagated through the whole model space, and maximum values of the field component magnitudes during a half -cycle after the last complete cycle are stored. Thus, a steady-state solution of the field components is achieved for each cell in the model space. In the analysis, great care must be taken in choosing the boundary conditions. Since FDID is an initial-value time­domain method, the propagating wave will reflect at the outer lattice boundaries unless special conditions are imposed on the fields at the boundary cells to make them absorbing while minimizing reflections at the absorber interface. We have used the double precaution of lining the entire area of the lattice with a four-cell layer of an absorber in addition to employing Mur second-order absorbing boundary conditions4. This prevents a significant amount of energy from reflecting at the lattice boundaries. Although the absorbing boundaries extend the limits of the lattice and thus reduce space of the structure to be analyzed, FDID presents savings in memory and execution time. Whereas other methods require storage and computation time on the order of (3N)2 and (3N)3, respectively, where N is the number of cells in the model, FDID requires only N for both. This is a direct consequence of the time-domain aspect of the method. The structure analyzed with FDTD can include features down to one cell in size, which is an important advantage in the analysis of near-field effects. Since the lattice is usually made of cells in the order of a tenth of a wavelength, features of a few tens of nanometers can be included in models in the optical region. These features can be sources, or absorbers and scatterers. We fabricated doped quartz-on-silica waveguides designed according to a previously published FDTD modeling study5, and then compared the waveguides to an FDTD model. The measured physical characteristics of the waveguides were then applied to an FDTD model to predict the performance of the components in fluorescence sensing. 2. THEORY The starting point for the FDTD analysis is the general Yee lattice unit cell as shown in Fig. 1. Maxwell's equations in a rectangular coordinate system, which are SPIE Vol. 1587 Chemical. Biochemical. and Environmental Fiber Sensors III (1991) / 137 - dBx I at= dEzl dy - dEy I az (la) - dBy I at = dEx I az -dEz I ax (Ib) dBz I dt = dEx I dy - dEy I dX (Ic) dDxl dt=dHzl dy - dHyl az -Jx (Id) dDy I at = dHx I dZ - dHz I ax - Jy (Ie) dDz I at = aHy I ax -aHx I dY -Iz (If) can be discretized in a lattice fanned of such cells by finite-difference equations as follows l : [Bxn+ l/2( i. j + I{l.. k + I{l.) - Bxn-I{l.( i. j + I{l.. k + I{l.)] I At = [Eyn( i. j + I{l.. k + I) - Eyn( i. j + I{l.. k)] I Az - [Ezn( i. j + I. k + I{l.) - Ezn( i. j. k + I{l.)] lAy (2) for (la). and [Dxn( i + l{l.. j. k) - Dxn-l ( i + l{l.. j. k)] I At = [Hzn-1/2( i + I{l.. j + l{l.. k) - Hzn-l{l.( i + l{l.. j - l{l.. k)] lAy - [Hyn-l{l.( i + l{l.. j. k + l{l.) - Hyn-l{l.( i + l{l.. j. k - l{l.)] I Az + Ixn-1/2( i + l{l.. j, k) (3) for (ld), where (ij,k) are the spatial indices for the cell, and the superscript n represents the nth time value of the field. Discretization for the other equations (lb-c, e-t) can be obtained similarly to (2-3). A two-dimensional model based on the Yee lattice discretization was used in this study. 3. FABRICATION AND CHARACTERIZATION OF PLANAR WAVEGUIDE COMPONENTS 3.1. Waveguide fabrication Waveguides were·fabricated for the purposes of this study by principles shown in a previous theoretical study5. Fused silica substrates (grade S I-UV) were obtained from Esco Products Inc., precJeaned with acetone, methanol and deionized 138 / SPIE Vol. 1587 Chemical. Biochemical. and Environmental Fiber Sensors 11/ (1991) water. dried with nitrogen gas. and placed in the sputtering chamber of a MRC 822 sputtersphere together with a <111> n­type silicon wafer. A SiOz sputtering target was used with Ar and N2 in the chamber for doping the quartz to attain a higher refractive index than what is possible with plain argon sputtering. The samples were sputtered for 270 minutes with an average RF power of 380 W in a base pressure of 6·10-6 torr. and with argon pressure of 7.8 pmHg. and N2 pressure of 1.2 J.UIlHg. 3.2. Waveguide characterization The waveguides were characterized by measuring the mode angle and loss (see Fig. 2). Also the thickness and refractive index were measured by ellipsometry on the silicon wafer sample. The internal mode angle measurement was done by prism-coupling TM-polarized green HeNe laser light at 534.5 om through a SF-6 prism (Karl Lambrecht Co.) into the waveguiding layer. and then measuring the light power coming out of the waveguide with different internal coupling angles. The maximum outgoing power was obtained at an internal prism coupling angle of 53.52 degrees. as can be seen from Fig. 3. The corresponding mode angle in the waveguide was found to be 85.04 degrees. with waveguide thickness 1.38 pm and refractive index 1.4672 as calculated with programs provided by Jeff Ives6• The ellipsometry study was performed with a Rudolph Research ellipsometer at 65.2 and 67.55 degree incidence angles. The results are in fairly good agreement with the ones obtained from mode angle measurement. giving waveguide refractive index of 1.464 at 632.8 nm. and thickness of 1.42 JUD. At 543.5 nm the refractive index would be 1.467. assuming identical dispersion to the substrate material (see section 4.1). The loss was measured by prism-coupling 534.5 om light into the waveguide and imaging the waveguide with a CCD camera (photometries Series 2(0). as described in detail elsewhere 7. The intensity in the image of the streak of light gave the attenuation as a function of distance traveled by the light in the waveguide. Taking the logarithm of the intensity vs. the distance gives a rather high loss of 6 dB/em (see Fig. 4). 4. FDTD MODELS OF A PLANAR WAVEGUIDE FLUORESCENCE SENSOR We have reported previously on optimization results for planar fluorescence sensor structures5. The strategy in the study was to develop a model for a single-mode waveguide which would be weakly guiding for TM-polarized excitation light. While being cutoff for emission at a longer wavelength. Therefore. no light from fluorescent emission would be guided in the waveguide. and more emission could be collected passing through the substrate of the waveguide. The results of the study gave guidelines for the design of these components. which were fabricated as explained in section 3.1. The waveguides were then characterized experimentally. and physically realistic two-dimensional FDTD models were subsequently developed based on the measurements (see section 3.2.). First. to compare the measured characteristics of the components. a model with identical physical characteristics to the waveguide was made. This model had no sensor-function. as air was used as a superstrate. Then. sensor-related models were developed separately both for excitation and emission of fluorescence. SPIE Vol 1587 Chemical. Biochemical. and Environmental Fiber Sensors III (1991) / 139 4.1 Model to compare FDTD to characterization studies of the waveguides TIle model used in bOth comparing FDTD to the physical characterization of the waveguides and sensor modeling is shown in Fig. 5. For the fmt case, the supersttate was air (refractive index n=1.00 + H)'OO). TIle refractive index of the quartz substrate at 5435 nm was estimated by a third-order polynomial interpolation from the supplier's data sheet8 to be 1.4601 + i·O.OO. The observed waveguide refractive index was 1.4672+ ioO.OO. Based on these values, the boundary conditions were determined for minimum reflectance at 45 degree incidence angle, for the superstrate region and substrate region separately. The analysis gave absoIber conductivities of 2.5457·104 n-Im-1 and 5.4270.104 n-I m-I , respectively. The 1.38 J.Ul1 waveguide thickness was modeled with 30 unit cells, each 46 run in size, and the total dimensions of the model were 9.2 J.Ul1 by 18.4 J.UIl for the 200 by 400 unit cell model space. TIle excitation source was assumed to be a 12 element line source array oscillating sinusoidally at 5.51978.1014 Hz (for free space wavelength of 5435 run). The magnitudes of the Ex fields at the source segments were weighted according to the field distnbution shown in Fig. 6, as determined from the Ives model6. 4.2 Excitation and emission sensor models The physical dimensions and cell spacing of the excitation and emission models were kept as designed in the previous section, and to correspond to the fabricated sensor, fused silica substrates, with nitrous quartz waveguides formed on the sensing surface were assumed. We also included an immobilized 92 run thick protein layer on the waveguide-superstrate interface. The superstrate was assumed to be water, for which the refractive index was determined to be 1.3346 + ioO.OO at 543.5 om by third-order polynomial interpolation of data from the AIP Handbook9• The refractive index of protein was assumed to be 1.497 + i·O.OO. These values were used for excitation modeling. The absorber conductivity for the region adjacent to the superstrate was now changed to 4.5340·104 a-1m-I. The same excitation source as in the previous section was used for this model as well: To analyze the behavior of the emission, the waveguide properties were interpolated to 570 nm giving n = l.4590 + i·O.OO for the substrate, and n = 1.3336 + ioO.OO for the water superstrate. The waveguide refractive index difference an = 0.0071 was maintained, and the refractive index of the protein was also assumed to change by the same percentage as the index of the substrate, giving n = 1.4959. The absorbing boundary conditions were designed for this wavelength again, and the conductivities were assigned at 4.3171.104 n-l m-l and 5.1669.104 n-1m-1 for the superstrate and substrate, respectively. Fluorescent emission with frequency 5.26316.1014 Hz (which corresponds to a 570 nm free space wavelength) was modeled by creating five 25 cell wide random source distributions in the protein layer adjacent to the waveguide. Each cell in the source distribution had an even chance of being a source, and in tum had an even chance of being oriented in the x or y direction. In addition, each source had a uniformly distributed random phase. An example of a source layer is shown in Fig. 7. 140 / SPIE Vol. 1587 Chemical. Biochemical. and Environmental Fiber Sensors III (1991) 4.3. Computer resources aud program features The models were analyzed by an FOID program written in Fortran-77 in the University of Utah Department of Electrical Engineering. The program was run on an HP-9000 Unix-based computer. For the 200 by 400 cell model. the program required approximately one hour of CPU time using 50 source oscillation cycles. The program provides integration of the nonnal component of the Poynting vector at a half-closed rectangular surface extending from the location of the sources to the end of the model space. with an arbilIary z-dimension. This integral multiplied by two gives the total power generated by the sources, assuming symmetry about the sources. The value of the normal component of the Poynting vector can also be integrated across the waveguide and the adjacent protein layer. These values are used to determine the efficiency of excitation and the power carried by the waveguide for the emission and excitation by calculating the ratio of integrated Poynting vector in the waveguide or protein layer and the total integrated Poynting vector in the half-space containing the end of the waveguide. Another important feature of the program is that it averages the dielectric constant of adjacent cells. This is useful because otherwise. as the magnetic field component is always calculated a half cell distance away from where the electric field is detennined. at surfaces with different dielectric constants in adjacent cells. the magnetic field component of a cell may in fact be calculated with an incorrect value of the dielectric constant, which will result in minor errors in analyses. S. RESULTS AND DISCUSSION The sensor components fabricated with the sputtering method were found to be single-mode near-cutoff structures. The losses of 6 dB/em are nearly unacceptable for a direct fluorescence sensor. Scattering seemed to be the dominant loss mechanism. which for energy-transfer based sensing schemes may not be an important factor. The results of the comparison of FOID model to the measured characteristics of the components at the end of the model space (y = 388) are shown in Fig. 8. The FDTD model gives a slightly wider electric field distribution in the x­direction (Ex component) than the physical model, which may be a result of inaccuracies in the measurements. In addition, some kinks are observed near the maximum field values indicating possible back-reflections from the end face of the model space. The ripples at the substrate side of the model are probably due to evolving substrate modes due to the truncation of the model space at 200 cells. At the waveguide-superstrate interface, which is the region of interest, the FDTD model performs very well. The results from sensor modeling are shown in Fig. 9. For excitation modeling, the Ives model6 agrees as well with the FDTD model as in the previous case. and similar characteristics are seen in the excitation FDTD model. The Poynting SPIE Vol 1587 Chemical Biochemical and Environmental Fiber Sensors II/ {1991 J / 141 vector integration shows that 60 % of light energy is propagating in the waveguide. and 3.1 % in the protein layer at Y = 388. This means that fluorescence excitation is possible with this sensor configuration. As for the emission modeling. Fig. 9 shows that the averaged source distributions give a highly radiative Ex component across the componenL Indeed. only 0.88 % of light energy is found to propagate in the waveguide at y =388. which means the original rationale for maximizing side-collection efficiency is supported by this study. In summary. FDTD modeling is shown to produce realistic results in fluorescence sensor design. Further work is needed in the area of assigning optical properties to the protein layers in order to bring the chemical design closer to optical modeling and to predict sensor responses more accurately. 6. ACKNOWLEDGEMENTS The authors wish to thank the personnel of the HEDCO microelectronics laboratory of the University of Utah for valuable help in fabricating the waveguides. Mr. Jinyu Wang for assistance in the ellipsometer measurements. and Ms. Shellee Dyer for her help in waveguide loss measurements. J. S. K. wishes to express his gratitude to the following organizations who have contributed to this research by scholarships. funds and grants: Technical Research Centre ofFinIand. Academy of Finland. Instrumentarium Science Foundation. Finland-U.S. Educational Exchange Commission. Tampere University of Technology. and United States Information Service. 7. REFERENCES 1. K. S. Yee. "Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media." IEEE Transactions on Antennas and Propagation. vol. AP-14. pp. 302-307. May. 1966. 2. A. Taflove. "Application of the finite-difference time-domain method to sinusoidal steady-state electromagnetic­penetration problems." IEEE Transactions of Electromagnetic Compatibility. vol. EMC-22. pp. 191-202. August. 1980. 3. D. A. Christensen and J. Andrade. "Analysis of efficiency of fluorescent coupling in guided-wave immunosensors". presented at Biosensors '90. Singapore. May 2-4. 1990. 4. G. Mor. "Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic field equations". IEEE Transactions of Electromagnetic Compatibility. vol. EMC-23. pp. 377-382. 1981. 142 / SPIE Vol. 1587 Chemical. Biochemical. and Environmental Fiber Sensors 11/ (1991) 5. J. S. Kimmel and D. A. Christensen. "FDTD Modeling in the design of optical chemical sensor structures". presented at Transducers '91, The 6th International Conference on Solid-State Sensors and Actuators. San Francisco. June 23- 27.1991. 6. J. Ives. Optical waveguide sensors: Characterization of evanescent and scatter excitation. PhD dissertation. University of Utah. 1990. 7. J. S. Kimmel. S. Dyer. and D. A. Christensen. "Waveguide Loss Measurement by CCD Imaging". to be published. 1991. 8. ESCQ Product Catalog. ESCO Products. Inc. Oak Ridge. NJ. 1987. 9. D. E. Gray (editor). American Institute of Physics HandboOk. Third Edition. p. 6-105. McGraw Hill Book Company. New York. 1972. z (i,j,k) y Fig. 1. Unit cell in a three-dimensional Yee latticel . 3 Fig. 2. Experimental arrangemenL I: Green HeNe laser (543.5 nm). 2: Waveguide holder with coupling prism. 3: Power meter. 4: CCD camera. 5: XYZ stage. 6: Goniometer base. SPIE Vol. 1587 Chemical Biochemical and Environmental Fiber Sensors 1/1 (1991) / 143 1.5 0.2 .~ ~i 0.0 1.0- 'll~ - -_0 -0.2 == :-::: -:1- C) 0 a. -0.4 0.5 .> -0.6 J ~ ..:. ..:. - ~ -0.8 0.0 .. • . • 0.0 0.2 0.4 0.6 0.8 1.0 1.2 51 52 53 54 z (em) Internal angle, degrees Fig. 3. Wave~ide coupling angle measurement: outgoing power vs. coupling angle at 0.57 mW input power. Fig. 4. Waveguide loss measuremenL The straight line is a least-squares fit to the CCD data giving a loss of 6 dB/em. -x -CD (J) CD e ::J o (J) 400 cells substrate absorbing boundary Fig. S. FDTD model space. The entire 200 by 400 cell (9.2 J.Ull by 18.4 J.Ull) model is lined inside with a 4 cell wide absorber. The source locations are shown for excitation (ex) and emission (em) models. The figure is not to scale. 144 / SPIE Vol. 1587 Chemical. Biochemical. and Environmental Fiber Sensors 11/ (1991 ) 35-44 45-54 55-64 -... 65-74 II» 75-84 .Q E 85-94 ::::I c 95-104 II» 105-114 u - 115-124 ~ 125-134 135-144 145-154 o~:~:~:~~:~:~~~~t~!:{~:~~:~~~:~:~:~~:~~~~~~~~?~::~:~~:~:~~~~:!:::~~~:~:~~::::~~~~~:::~~~~:~:o~:~:W!:}~~~~~~!:!:~~!:t:~:~!:!::~:~:~!:!:~f!:~~::::~~~~:~~~!:~%~!:!::~!::::~~~!:!:~~~~t!:~~!:!:~~~!::~~~~~~tt~:~:~~~:~!:!:~~~%~!:~~:~~~~~~~~~~:~:~~~~~~~~~~~~~~~~tM~H )~~~~~~~~i@ttii::~~i:i:iii:ii::ii~i~:tii:~~::tiiii~ii~~~~~~~:~~it~i~i~~~i~i:~:~iftf~it~@~~:~~~~~~§~i:i~~~~~:i~~:~~~;~;;:i~i~~:~~~~~;~i~:i:~t;~~~~;~i~~~~~~i:i:~:i~;:;::~i:i:~:~~*~~:ttt::tt:iItf r~~I~i:ii~g~:~:~:@:~~::r~i~iii~~~i~i~~~~i~~:~~~;;~~~~ttJ :t~::!:~~~i:~~iii~it~l;:tl r I I • I • • 1 o 200 400 600 800 1000 E (8. u.) X Fig. 6. Excitation model source distribution. superstrate y=16 249.43 50.97 53.10 waveguide 347.97 153.55 • Ex source m Ey source 40 100.69 56.00 294.18 68.89 295.80 Fig. 7. Example of emission model source distribution. The numbers designate the phases of the individual source elements, in degrees. . .. S '5111(1991)/ 145 SPIE Vol 1587 Chemical. BIOchemical. and EnVIronmental fiber ensor 400~---------''----=1E--------------------------~--' 300 wM 200 100 a 50 100 150 200 x (cell number) Fig. 8. Comparison of the characterization results (solid curve) of the waveguide to FDTD modeling (open squares) at y = 388. The vertical lines represent the borders of the waveguide with superstrate on the far left and substrate on the far right ~O~--------Ir------~----------------------------' 400 --:- 300 :J -III w M 200 100 o 50 100 150 200 x (cell number) Fig. 9. Results ofFDTD sensor modeling at y = 388. Solid curve: Ives model6• open squares: excitation model. closed diamonds: averaged emission model. The double vertical line represents the location of protein layer on the waveguide at the superstrate interface. and the single vertical line shows the waveguide­substrate interface. 146 I SPIE Vol 1587 Chemical. Biochemical. and Environmental Fiber Sensors 11/ (1991)