Radiation efficiency analysis of submillimeter-wave receivers based on a modified spectral domain integration technique



[1] Reflector backed submillimeter-wave slot antennas on substrate lenses are promising candidates for novel astronomical applications in integrated receivers for the far infrared region. Using an adjustable reflector, considerable resonance frequency tuning can be achieved, but with the risk of exciting parasitic parallel plate wave modes. Employing a surface/volume integral equation method and advanced spectral domain integration techniques, a complete radiation efficiency analysis of such kind of receivers is performed. Convenient integration path deformations in complex wavenumber planes allow a very fast evaluation of the spectral domain integrals in combination with efficient database and asymptotic extraction techniques. Saddle point and residue calculation techniques are employed to accurately determine the power of each parallel plate mode and the space wave contributions. The simulations reveal an appropriate set of parameters reducing the radiation loss to less than 10 percent for doubleslot and quadratic slot geometries preserving high tuning capability and optimized radiation patterns as well.

1. Introduction

[2] With recent improvements of a surface/volume integral equation approach [Vaupel and Hansen, 2000a, 2000b] a much faster analysis of complex coplanar/microstrip structures could be achieved with increased accuracy especially when applied to structures with negligible dielectric losses. These improvements are now utilized within a detailed radiation efficiency analysis concept particularly well suited for a special class of reflector backed slot antenna receivers for the submillimeter-wave region [Aroudaki et al., 1994]. These receivers are based on the concepts in Zmuidzinas and LeDuc [1992] and Chattopadhyay et al. [2000]. First realizations with measurements are presented by Schäfer et al. [1995, 1997] but without the possibility of a detailed efficiency consideration. The basic concept of such receivers comprises the utilization of a quartz lens with doubleslot or strip dipole antenna as lens feeding structures. Using a quartz lens, a lens matching layer is not absolutely necessary, but the low permittivity of quartz causes a large backward radiation into the air region below the lens. In order to diminish these radiation losses, an additional reflector was attached below the lens. Numerical simulations in Aroudaki et al. [1994] have shown furthermore, that an effective tuning of the antenna resonance frequency can be performed by adjusting the distance of this reflector. Nevertheless, a parallel plate medium is formed by this additional reflector and thus the possible excitation of parasitic parallel plate modes may lead to radiation losses as well. For the detailed analysis of all radiation effects, we have improved the spectral domain integration technique of our surface/volume integral equation approach in Vaupel and Hansen [2000a, 2000b]. Since the parallel plate poles of the corresponding Green's function cause large variations of the integrands, we have consequently used an integration path deformation for both the kx and ky wavenumbers similar as in Yang et al. [1990] and Garino et al. [2000] together with an integration area reduction to the first quadrant of the kx-ky plane. Due to the smooth integrand behavior achieved by these measures, we apply a piecewise uniform integrand sampling by an appropriate subdivision of the kx-ky plane. The optimized row and column arrangement of the sampling points allows the setup of a very efficient database for the basis functions leading to a drastic reduction of computational and storage requirements compared with the hitherto evaluation in polar coordinates [Vaupel and Hansen, 1999a, 1999b; Horng et al., 1992]. The integration path deformation also allows the computation of structures with negligible dielectric losses, a prerequisite for a detailed radiation efficiency analysis. For this analysis, we extract the parallel plate wave contributions using the corresponding scalar Green's functions for the symmetric Sommerfeld integrals, the modified saddle point method and the residue theorem. The space wave power is determined with the standard saddle point method. The reliability of the combined application of the various methods is tested by checking the overall power balance of the structure.

2. Formulation

2.1. Modified Spectral Domain Integration

[3] The basic structure consists of a hyperhemispherical quartz lens with a radius of several wavelengths backed with a metallization sheet carrying the feeding slot antennas (Figure 1). The antennas are excited with a microstrip feeding/mixer network attached below the slot plane. Due to the application of a quartz lens with relatively low permittivity, a considerable part of the input power is radiated into the air below the antennas. To reduce these radiation losses, an additional reflector below the antennas was proposed allowing as well the variation of the resonance frequency by an adjustable distance d [Aroudaki et al., 1994]. Unfortunately, this reflector forms a parallel plate transmission line in combination with the slot antenna metallization sheet. Therefore the performance of such structures is strongly affected by possible losses due to parallel plate mode excitation.

Figure 1.

Setup of a submillimeter-wave receiver with adjustable reflector.

[4] For any power loss analysis, an accurate determination of the structure currents is necessary. For this goal, we utilize a versatile Method of Moments (MoM) procedure with an improved spectral domain integration technique.

[5] The structure description and the used basis functions are the same as in Vaupel and Hansen [1999a, 1999b]; thus only a brief recapitulation is given here. In order to apply Green's functions for layered media with infinite lateral extension, the substrate lens in Figure 1 is considered as an infinite dielectric half-space in positive z-direction; all metallization planes are considered as laterally infinite as well.

[6] Actually, the finite extension of the lens affects, e.g., the input impedance of the slots due to back reflections at the lens surface, but in our case with a quartz lens of relatively low permittivity (ϵr ≈ 4.44) combined with a nonelliptic lens geometry, these effects can be typically neglected.

[7] For the current discretization we use basis functions for electric and magnetic surface currents and polarization volume currents on arbitrary nonuniform rectangular grids leading to the current description

equation image

with Nle, Nlb and Nlv the numbers of the basis functions equation image for electric and magnetic surface currents and volume currents, respectively. The unknown amplitudes of the basis functions are determined by simultaneously fulfilling the impedance boundary condition for the electric surface current, the magnetic field continuity condition within the slot areas and the polarization current conditions in the volume current areas.

[8] Applying the standard Method of Moments, we get a system matrix with impedance and admittance elements for couplings of electric/polarization currents and couplings of magnetic currents. Couplings between electric or polarization currents and magnetic currents lead to matrix entries without dimensions.

[9] For slot antenna structures we mainly have to compute admittance entries given by

equation image

where equation image (kx, ky, zlb, zlb) denotes the composite Green's function of the magnetic fields of the magnetic currents above and below the slot plane at z′ = zlb, equation image (kx, ky) denote the Fourier transforms of the basis functions associated with magnetic currents.

[10] The main problem evaluating equation (2) is related to strong variations of the integrand near parallel plate wave poles and branch points. This behavior is shown in Figures 2b and 2c for the imaginary and real part of the Green's function component GMxxH for a typical half-space/parallel plate medium (solid lines).

Figure 2.

(a) Integration path deformation, (b) imaginary part, and (c) real part behavior of a half-space/parallel plate medium Green's function. Solid lines: real axis integration; dashed lines: with integration path deformation.

[11] The imaginary part of the Green's function shows strong variations especially close to the parallel plate wave poles (real kx-integration, fixed ky = kxm/2). An even more complex behavior we observe for the real part: Here strong negative peaks appear especially near the branch point associated with the radiation into the upper half-space. In case of moderate dielectric losses, the poles are shifted away from the real axis allowing the use of adaptive real axis integration methods [Vaupel and Hansen, 1997a, 1997b]. In the case of very small losses needed for radiation efficiency analysis, especially the peaks in the real part become extremely deep and narrow inhibiting the use of adaptive integration methods. Particularly the real part contributions must be evaluated very carefully, since they comprise the whole information of the space and parallel plate wave power.

[12] To overcome all these problems we apply an integration path deformation for both the kx and ky wavenumbers (Figure 2a), e.g., for kx with typical parameters). Similar ideas are suggested by Yang et al. [1990] and Garino et al. [2000].

[13] The effects of these path deformations are outlined in Figures 2b and 2c, dashed lines, showing now a smooth behavior for both imaginary and real part of the Greens's function nearly independent of number and locations of the singularities. This allows an integration in Cartesian wavenumbers kx, ky for the whole integrand in contrast to former approaches based on integration strategies in polar coordinates [Vaupel and Hansen, 1999a, 1999b; Horng et al., 1992]. Additionally, the evaluation in Cartesian wavenumbers has the advantage of a faster decay of the integrands with a weaker oscillatory behavior, and it allows the application of a very efficient database concept explained as follows.

[14] The overall subdivision of the first quadrant of the kx, ky plane is given in Figure 3. The inner area AIN containing all singularities is outlined together with the sketched integration path deformation. The outer area is subdivided in two narrow areas AOUT1 and AOUT3 and a larger rectangular area AOUT2. With this kind of subdivision a uniform sampling with a very advantageous row and column arrangement can be applied in each area.

Figure 3.

Subdivision and sampling of the first quadrant of the kx, ky-integration plane.

[15] For basis functions on rectangular subdomains, the Fourier transforms read:

equation image

They consist of two factors only depending on one wavenumber kx or ky. This means together with the sampling scheme of Figure 3, that we have to compute and store only one row for Fm (kx) and one column for Fm (ky) in the inner and outer area, respectively, leading to a drastical reduction of computation and storage effort for the generation of the database.

[16] A general matrix entry is computed by

equation image

with equation image.

[17] The dyadic functions equation image in equation (4) are general Green's functions already containing analytical integrations with regard to z and/or z′ in case of volume currents, equation image (kx,ky) are asymptotic representations of these Green's functions, their subtraction leads to an exponential decay of the integrands of the first and second integral in equation (4) [Vaupel and Hansen, 1999a, 1999b].

[18] With the reduction of the integration area to the first quadrant of the kx, ky plane using symmetry properties, we get additional real- and/or imaginary part operations Re{..}, Im{..}, denoted with OP1{..} and OP2{..}. The computation of the real- and imaginary parts must be done very carefully on the deformed complex integration paths: First we use the relations Fm,n*(kx) = Fm,n(−kx), Fm,n*(ky) = Fm,n(−ky), valid for the real valued basis functions used for the slot discretization. Since the real- and imaginary part operations in equation (4), first integral, are nonanalytic functions, they are carried out using the relations

equation image

For example:

equation image

By these measures the analytical properties of the integrands are preserved in the inner area in any case, necessary for the validity of Cauchy's theorem.

[19] The asymptotic parts subtracted in the first and second integral of equation (4) are added by the third integral of equation (4) again, which is formulated in polar coordinates. This polar coordinate formulation allows the complete analytic/closed form evaluation of the third integral with the methods given by Vaupel and Hansen [1998] and [1999a, 1999b] for all combinations of basis functions.

2.2. Power Loss Analysis

[20] After careful determination of the structure current distribution, we determine the overall power radiated by the structure with the complex energy theorem. If we have, e.g., only magnetic currents, we get

equation image

using the spectral domain formulation of the fields combined with the series representation of the magnetic currents.

[21] For the detailed power loss analysis we have to determine the subdivision of the overall power Ptot into space wave power Pspace and the power of all parallel plate modes Pparr. The space wave far fields can efficiently be derived with the saddle point method (e.g., for magnetic currents):

equation image

where ku is the wavenumber in the lens medium. Subsequent integration over the Poynting vector in the upper half space provides the space wave power Pspace.

[22] The parallel plate wave contributions are derived with the help of Sommerfeld integrals, the modified saddle point method and the residue theorem and might be more systematic than the formulations of Harokopus et al. [1991] and Lin and Wu [1999] which are mainly based on a direct evaluation with the Parseval identity. The derivation starts with the Sommerfeld integral formulations:

equation image
equation image

where the Ez and Hz-component arise from an x-oriented magnetic current element in the origin. GMzE,H(kρ) are components of the so-called scalar Green's functions (see Appendix A).

[23] The integral equation (8) is transformed with the well-known substitution kρ = k0 sin(w) mapping the complex kρ plane into the complex w plane. After deforming the Sommerfeld-path into the steepest descend path, the latter is displaced to the saddle point ws = θ = π/2 [Hansen, 1989; Felsen and Marcuvitz, 1994], which describes in our case the propagation plane of the parallel plate waves. During this displacement, the parallel plate wave poles are crossed by the path; consequently, their contributions must be considered by their residues, e.g., for the EMz component:

equation image

where kρp are the pole locations of the scalar Green's component; in this case they correspond with the TEM and TM parallel plate wave poles. Additionally, we have applied asymptotic forms of the Hankel functions for large arguments. The complete solution together with the HMz-component is given in Appendix A. The steepest descent path itself provides no contribution to the parallel plate wave far fields and can be neglected.

[24] Since equation (9) is only valid for a current element in the origin, we have to integrate in the next step over the magnetic current distribution provided by the MoM, substituting the phase term equation image by the phase term equation image Under far-field conditions ρ ≫ λp we can approximate this phase term by

equation image

Thus, we get the radiation function equation image by evaluating the integral

equation image

[25] Employing the series representation of the magnetic currents with asymmetric rooftop functions, equation (10) can be solved analytically; thus all relevant parallel plate mode field components of a given magnetic current distribution can be calculated in closed form. In this context it is worth mentioning that this important field component information is not accessible by a direct power evaluation with the Parseval identity.

[26] In the last step, we form the Poynting vectors for each propagating parallel plate mode and integrate them over a cylinder surface in the far-field region:

equation image

The integration with regard to z is done analytically, whereas the ϕ-integration is performed by a Gauss-Legendre quadrature so far; at this point also a complete analytical treatment may be possible. In any case, the effort for the radiation analysis is always negligible compared with the effort for the MoM.

3. Numerical Results

3.1. Doubleslot Antennas

[27] For the analysis of parallel plate mode losses, the influence of the microstrip feeding network can typically be neglected, because it is mounted on a very thin polyimide layer (≈1–3μm) with negligible coupling to the parallel plate modes. In order to circumvent the consideration of the whole feed network, we apply a low and small airbridge with centered delta gap voltage source (Figure 4a), or we impress a delta current source in the slot center with reduced width (Figure 4b). The vertical currents of the airbridge are modeled with small volume currents. Due to these measures the clamp conditions are fulfilled in any case [Vaupel and Hansen, 2000a, 2000b] leading to very stable results particularly with regard to the imaginary part of the input impedance. Figure 5 shows the computed impedance of a doubleslot antenna excited according to Figure 4a with two different reflector distances d.

Figure 4.

(a) Slot excitation with an airbridge (height ≈ 3μ) and voltage source; (b) current excitation in the slot center with reduced width.

Figure 5.

Slot impedance simulated with the model according to Figure 4a, two different reflector distances d.

[28] Very remarkable are the sharp bends in the impedance curves especially for d = 240μm at about 620 GHz. At these frequencies we observe the onset of the higher TM/TE parallel plate modes, whereas for lower frequencies we have only the TEM mode propagation.

[29] The resonance frequency, which is defined as the frequency with vanishing imaginary part of the input impedance, amounts to about 860 GHz for both reflector distances in the second resonance. The results with the model according to Figure 4b are quite similar.

[30] Figure 6 shows the parallel plate mode radiation patterns for the configuration of Figure 4a, i.e., the far-field distribution over the angle ϕ, for the three parallel plate modes propagating at the frequency 850 GHz with d = 230 μm. For this case the TM mode dominates, followed by the TE and the TEM-mode. Such mode radiation patterns are very important to get a further physical insight of the radiation mechanism of reflector backed slot antenna configurations.

Figure 6.

Parallel plate mode radiation patterns of a typical reflector backed doubleslot antenna. The fields only exist for −d < z < 0.

[31] In Figure 7 we have investigated the normalized radiation losses due to the parallel plate modes in dependence on the slot distance s (fixed frequency and reflector distance). We observe a quite strong dependence of the different mode losses on the distance s of the slots. Unfortunately, the maximum total loss with about 39 percent of the space wave power is given just for s = 100μm, the value for the radiation pattern with best symmetry. A hint to reduce these strong losses is given by the impedance behavior in Figure 5, where we get nearly the same resonance frequency for both reflector distances d = 160μm and d = 240μm. The decisive difference between both cases is given by the fact, that for d = 160μm only the TEM mode can propagate near slot resonance. The determination of the resonance frequencies in dependence on the reflector distance reveals similar relations for other reflector distances as well (Figure 8).

Figure 7.

Normalized parallel plate mode losses in dependence on the slot distance. Other dimensions according to Figure 4a.

Figure 8.

Second resonance frequencies and input impedances (in Ω) in dependence on the reflector distance.

[32] We can distinguish a left and a right branch of the curve, separated by a reflector distance of ≈200μm. If the antenna is operated on the left branch of the curve, only the TEM parallel plate mode exists, whereas on the right branch also higher modes can propagate as in Figures 6 and 7. The total loss due to parallel plate modes is illustrated in Figure 9.

Figure 9.

Normalized total parallel plate mode loss in dependence on the reflector distance (slots in resonance). Other dimensions according to Figure 4a.

[33] If the antenna is operated on the left branch, the loss due to the TEM-mode always remains below 10 percent. On the right branch the losses exceed 60 percent for d = 200μm descending to 25 percent for d = 280μm. Additionally, we have given the loss of about 31 percent due to the backward radiation of the corresponding doubleslot antenna without reflector.

[34] A good validation of the performed computations is given by checking of the total power balance of the structure; i.e., for a structure with, e.g., three propagating parallel plate modes the relation

equation image

must hold. In all computations the error in fulfilling this power balance amounts to only 1–2 percent demonstrating the high reliability and accuracy reached by the combined application of the various methods.

3.2. Quadratic Slot Antennas

[35] A promising antenna geometry is also given by quadratical slot antennas. Figure 10 (top) shows a typical specification with a coplanar stub needed for additional matching purposes and pattern optimization. Figure 10 (middle) shows the impedance behavior by exciting the structure with an impressed electric current at the right end of the stub. When can observe a flat impedance behavior at the second resonance of ≈760 GHz leading to a higher bandwidth than comparable doubleslot antennas. Due to this property, a variation of the reflector distance d has only a negligible effect on the input impedance in the second resonance region (Z ≈ 25Ω) if only the TEM parallel plate wave propagates. The radiation losses due to the TEM parallel plate wave are even lower as for comparable doubleslot antennas. For 755 GHz we get 11 percent loss for d = 100μm decreasing to less than 6 percent for d = 190μm just before the onset of the higher TM/TE modes. Figure 10 (bottom) shows the antenna pattern of the overall structure with a typical hyperhemispherical quartz lens, showing a good symmetry of E and H planes. The influence of the lens is considered by a raytracing algorithm combined with an aperture integration technique [Vaupel and Hansen, 1997a, 1997b]. Figure 11 finally gives the magnetic current distribution at 755 GHz illustrating the good current symmetry on the quadratic slot parts. The quadratic slot structure was also used for a brief comparison of the numerical efficiency of the improved spectral domain integration outlined in the last chapter and former evaluations in polar coordinates [Vaupel and Hansen, 1999a, 1999b]. With 132 basis functions we get the following computation times in milliseconds for the system matrix generation (350 MHz Pentium without code optimization): integration in polar coordinates [Vaupel and Hansen, 1999a, 1999b], 1500 ms; integration in Cartesian wavenumbers, <260 ms.

Figure 10.

Analysis of a quadratic slot antenna. (middle) Impedance behavior. (bottom) Typical overall antenna pattern with a hyperhemispherical quartz lens.

Figure 11.

Magnetic current distribution of the quadratic slot antenna at 755 GHz.

[36] With our new approach, we get the same results as in Vaupel and Hansen [1998, 1999a, 1999b], guaranteeing the high accuracy of our approaches in terms of input impedance and S-parameter determination. For larger structures the acceleration due the new integration and database techniques becomes even higher.

4. Conclusions

[37] The presented surface/volume integral equation concept based on advanced spectral domain integration techniques shows a significantly higher numerical performance in terms of computation times and accuracy as compared with former concepts using integration techniques in polar coordinates. The further combination with saddle point and residue calculation methods allows a very accurate radiation efficiency analysis of antenna structures mounted on substrate lenses. The large backward radiation arising with a quartz lens could be drastically reduced using an additional reflector for both doubleslot and quadratic slot structures. The excitation of higher TM/TE parallel plate modes is prevented by a convenient distance adjustment of this reflector which is utilized as additional tuning instrument as well.

Appendix A

[38] The necessary scalar Green's function components of a parallel plate medium reads (see Figure 2b) for the configuration):

equation image
equation image

[39] Equation (9) leads together with equation (10) to

equation image
equation image

where BTM(kρ), BTE(kρ) are the derivatives of the denominators of equations (A1) and (A2), respectively. The other relevant field components can be derived from equations (A3) and (A4) by

equation image

where all terms with a higher descend than equation image can be neglected.