Wireless communication systems in the terahertz (THz) frequency range promise to dramatically increase available bandwidth in the electromagnetic spectrum. These wireless systems will require filtering techniques capable of operating in this relatively unused part of the spectrum. Here, we report a versatile technique for designing different classes of THz plasmonic filters based on a k-space methodology, in which the desired frequency response is mapped into two-dimensional (2D) k-space and then inverse Fourier transformed into the spatial domain. We use a recently developed inkjet printing technique to fabricate the spatial patterns allowing for grayscale conductivity variation. In general, any technique that allows for high-fidelity reproduction of the real-space grayscale variation in the fabricated plasmonic structure can be used. We demonstrate the flexibility of this approach by creating several classes of filters that allow for changes in the relative magnitudes in multiresonant filters; the polarization dependence, where the anisotropy can be carefully controlled; and the resonance bandwidth. We further demonstrate that, by cascading or adding filter functions together, even more complex filter designs can be achieved. We expect this approach to dramatically expand the design capabilities for filter technology for THz systems applications, such as THz wireless communications as well as applications in other spectral regions.
© 2015 Optical Society of America
Filter technology is an extremely broad and well-studied field, with numerous applications in science and engineering, from the first power line filters  to the advanced digital filters in modern electronics [2,3] and wavelength division multiplexing filters in fiber optic communication systems . The versatility of a wide range of filters enables the interconnected society we enjoy today through the vast array of digital communication systems that keep the world connected. The effectiveness of these filters is possible through the well-developed design methodologies that allow designers to easily implement a desired frequency response that fits with specified system requirements. While well-defined filter design methodologies exist for much of the electromagnetic spectrum [5,6], the THz region of the spectrum lags behind in filter complexity and lacks the same caliber of design capabilities. Such capabilities will be required in creating large-scale THz communication systems .
The relative lack of complex THz filter designs is consistent with a general scarcity of devices that operate effectively in the THz spectral range from 100 GHz to 10 THz, resulting in the well-known THz gap . Various approaches for creating filters in the THz region have been shown, largely to demonstrate bandpass filtering [9 –13]. Different forms of filtering have also been shown, including band stop , high-pass [15,16], and other filters with the ability to implement an arbitrary frequency response or change their passband characteristics [17,18]. While such filters have been demonstrated, i.e., low-pass, high-pass, bandpass, and band stop, an approach is still needed that can show a versatility in designing arbitrary frequency responses within a single design methodology and workflow. Plasmonic filters, which are geometrically scalable over a wide range of the electromagnetic spectrum, provide a possible pathway for developing such a methodology to create filters with tailored frequency responses in the far infrared.
In this work, we demonstrate a versatile technique for designing different classes of THz plasmonic filters based on a k-space methodology. In this design methodology, the designer is free to define a desired filter frequency response, which is then mapped into k-space using the simple relationships between the wave vector and frequency, , and the desired symmetry of the device. The k-space representation of the filter is then inverse Fourier transformed into the spatial domain. We fabricate the spatial domain design using a recently developed inkjet printing technique that allows for grayscale printing in plasmonic structures, ensuring high-fidelity reproduction of the real space pattern. This technique allows for the design and fabrication of arbitrarily defined filter responses in a single unified design flow, a capability not previously demonstrated at THz frequencies.
2. MATERIALS AND METHODS
The primary difficulty in implementing a k-space design process is associated with the issue of fabricating a plasmonic structure that matches the spatial description obtained from the inverse transformed spatial frequency spectrum. Traditional hole arrays allow for only a binary physical representation with a hole present or not; to create a grayscale physical representation, a different fabrication technique is required. In this work, we employ an inkjet printing technique that allows for the deposition of varying levels of a conductive silver ink or the simultaneous deposition of conductive silver ink and resistive carbon ink . In this latter printing mode, the conductivity can be spatially controlled, which corresponds to grayscale plasmonic response [i.e., the propagation length of surface plasmon polaritons (SPPs) can be varied by changing the local conductivity].
All of the digital images used to create k-space structures were generated with the standard version of MATLAB with no additional toolboxes or add-ons. The digital image in k-space is designed as a array of pixels. Initially the temporal frequency response of the filter is defined to meet desired specifications, such as the type of filter, center frequency, bandwidth, etc. This filter frequency response is then mapped into a two-dimensional k-space image based upon the desired degree of rotational symmetry of the device. See Fig. S1 in Supplement 1 for a more detailed description of the filter design flow. After taking a 2D inverse discrete Fourier transform (IDFT) of the k-space representation, the real part of the transform is retained. It should be noted that the spatial image obtained from the IDFT contains negative values that cannot be represented by a printer system that deposits varying amounts of ink within a range of 0–255. Thus, the spatial image must be thresholded and normalized to the 0–255 range in order to be printed. This thresholding process causes higher-order combinations of reciprocal vectors (RVs) to appear in k-space and in the measured spectrum.
Once the digital filter image has been thresholded and normalized, it can be directly printed using a commercial inkjet printer with a conductive ink cartridge. We used inkjet printable inks from Methode Development Company (9101 conductive silver ink and 3804 resistive carbon ink). The filters were printed using a low cost, commercially available color inkjet printer, Epson Workforce 30. The printed silver ink sinters at room temperature immediately after printing, yielding a final device that requires no additional post-processing steps. We used a combination of silver and carbon ink printed simultaneously to achieve a grayscale conductivity variation. Thus, filters requiring spatially varying conductivities could be printed using a color-coded image that mapped to the silver and carbon ink cartridges.
The transmission spectra for each filter were obtained using standard THz time-domain spectroscopy measurements . We used photoconductive antennas for both THz generation and detection stimulated by an ultrafast Ti:Sapphire laser. The broadband, linearly polarized THz radiation generated by the emitter was collected and collimated by an off-axis parabolic mirror, such that the radiation was normally incident on the filters. A second off-axis parabolic mirror placed after the filter was used to focus the transmitted THz radiation onto the photoconductive detector. To test the polarization-dependent filters, the printed devices were mounted on a rotation stage and measurements were made at 10° increments over a 90° range.
3. RESULTS AND DISCUSSION
To illustrate the possibilities of k-space design, we show an artistic rendering of a plasmonic THz filter in Fig. 1(a), where both carbon and silver ink are used to create a spatially varying conductivity pattern. Here, the transmission of THz radiation is higher if polarized along one axis compared to the other. A photo of an actual printed sample exhibiting the desired anisotropic behavior is shown in Fig. 1(b), along with an accompanying microscope image in Fig. 1(c). The details of this specific filter are discussed later in this work and are summarized in Fig. 4.
A. K-Space-Designed Comb Filter
We begin by demonstrating the use of this approach in controlling the relative amplitudes of resonances in a comb filter. In Fig. 2, we summarize how the amplitudes of two specific resonances can be individually adjusted. In Figs. 2(a)–2(c), we create two four-fold rotationally symmetric resonances, one at 0.2 THz and a second at 0.4 THz, where the relative amplitudes of the corresponding discrete spots vary in k-space (additional information regarding the real-space patterns is given in Supplement 1). Note that the k-space spectrum is restricted to symmetric patterns in order to ensure that the transformed real-space image is composed of only real values. The transform from k-space to real space for the different versions of the comb filter show that all three versions have the same underlying structure, but with different contrast between the peaks and valleys of the sinusoidally varying structures. By applying a thresholding algorithm, the different physical space comb images can be viewed as an aperture array designed for 0.2 THz with additional intermediate holes (corresponding to 0.4 THz) darkened according to the weight defined in k-space.
The corresponding real-space pattern supports not only these two resonances, but also resonances at 0.24 THz ([2,0] metal–substrate resonance), 0.28 THz ([1,1] metal–air resonance), 0.29 THz ([2,1] metal–substrate resonance), and 0.35 THz ([2,2] metal–substrate resonance), as shown in the measured spectral response in Fig. 2(d). Several of these additional peaks are due to the asymmetry of the material structure stack. In general, this can complicate k-space filter design by adding additional resonances that can merge together as they are shifted down in frequency. Nevertheless, these substrate contributions may be beneficial in creating more complex comb filters, as is the case in Fig. 2(d), where the comb of frequencies is extended from two resonant peaks to essentially four resonant peaks. Alternatively, the additional resonances can be minimized by overcoating the structure with a medium that is index-matched with the 150 μm thick plastic substrate . In Fig. 2(e), we show the relationship between the designed k-space ratio of the amplitudes of the air [1,0] and [2,0] resonances to the measured ratio of their magnitudes. We find a linear relationship between the designed ratio and the measured ratio. This approach can be extended to incorporate larger numbers of resonances and may be useful in applications where the relative transmittance of several different frequencies needs to be carefully controlled.
B. K-Space Elliptical Filter
The k-space approach also allows for the design of polarization-dependent filters, where different polarizations of incoming radiation are filtered differently. In Fig. 3, we show two different implementations of this idea. In both cases, we design the k-space pattern such that the RV corresponds to 0.2 THz along the axis and to 0.4 THz along the axis. In the first implementation, shown in Fig. 3(a), this is accomplished using four discrete spots in k-space. The corresponding real-space design is shown in Fig. 3(b). The asymmetry in the locations of the discrete spots creates a stretching effect in the real-space aperture geometry. In Fig. 3(c), we show the measured THz transmittance for each of the two resonance peaks as a function of the incident polarization angle. Both sets of data are fit to an expected sinusoidal variation associated with the projection of incoming radiation onto the and axis, though the variation is somewhat weaker with the higher-frequency resonance. The offset from zero transmittance is associated with that fact that some THz radiation is directly transmitted through the holes, while the asymmetry in the response arises from the fact that the “holes” are a different fractional size for the two different resonance frequencies.
The asymmetry in the response can be reduced by drawing a complete ellipse in k-space, as shown in Fig. 3(d). The resulting real-space geometry, shown in Fig. 3(e), has the appearance of an asymmetric bullseye structure stretched along the axis, although in this pattern there are low-conductivity arcs across the pattern that allow for THz transmission not only through the central aperture, but also through these arcs. The resulting resonance peaks occur at 0.33 and 0.16 THz, as shown in Fig. 3(f). Once again, the magnitude of each resonance peak is characterized by a sinusoidal polarization dependence. It is worth noting that the measured spectra of previous bullseye structures  are fully consistent with the predicted k-space design approach described here.
C. K-Space Anisotropic Filter
We can also demonstrate an anisotropic filter similar to the filters described in Fig. 3. However, instead of shifting the frequencies of the resonant peak or the relative magnitudes of multiple peaks, here we control the magnitude of the sinusoidal variation associated with the transmission peak of a single resonance by changing the polarization of the incoming light. In Fig. 4(a), we show a two-fold symmetric filter that, in k-space, has a magnitude at 0.2 THz that is four times as large along as it is along . This results in a real-space geometry, shown in Fig. 4(b), where the image contrast along one axis of the image is four times greater than that along the orthogonal axis. In Fig. 4(c), we show the magnitude of the transmission resonance peak at 0.18 THz as a function of the incident polarization. Since this approach operates only on a single resonance, the amplitude ratio (maximum/minimum ) is larger than that observed in Fig. 2. The magnitude of this anisotropy can also be varied. In Fig. 4(d), we show the contrast (ratio of the maximum to the minimum amplitude) for anisotropy values ranging from to . This contrast ratio is found to vary linearly with the magnitude of the k-space anisotropy.
This breadth of potential designs for complex plasmonic filters can be significantly expanded if we now consider the possibility of cascading sequential linear filters into a single filter. Such an operation typically requires performing a convolution of two or more filter functions, typically in the real-space domain, or a multiplication of the corresponding filter functions in the conjugate k-space domain. In general, the convolution can take place in either the real-space or k-space domain to create more complex filter functions. Alternatively, complex filter functions can be obtained by simply adding filter functions, yielding a response that may not be straightforward to achieve by other means. We give two examples to demonstrate the utility of this general approach.
D. K-Space Bandpass Filter
In the first example, we demonstrate the ability to control the bandwidth of a bandpass filter. In Fig. 5(a), we show the k-space representation of an eight-fold symmetric bandpass filter, along with its inverse transformed real-space geometry, in Fig. 5(b). To control the bandwidth of the filter, a Gaussian soft aperture was used to weight the k-space image (by convolving filter functions in k-space or performing multiplication in real space), yielding a real-space pattern that included an additional Gaussian spatial modulation (i.e., took on grayscale values). An example of such a structure obtained using a soft aperture with a bandwidth of 33 GHz is shown in Fig. 5(c). A more complex example is discussed in Supplement 1.
In Fig. 5(d), we show the measured spectral response for a filter without a soft aperture. The lowest-order resonance, corresponding to the RV, corresponds to 0.3 THz for the metal–air interface and 0.195 THz for the metal–substrate interface . The lower-frequency resonance arises from the fact that the plastic substrate has a measured THz refractive index of across the THz spectral range of interest. Note that the frequencies associated with the RVs correspond to the amplitude dips on the high-frequency sides of the resonances, and that the resonance peaks occur at somewhat lower frequencies . In Fig. 5(e), we show the measured bandwidth of the metal–air resonance for five different soft-aperture widths, where a wider opening corresponds to a narrower bandwidth. The linear fit to the data shows good agreement between the designed k-space bandwidth and the measured bandwidth. The lack of exact numerical agreement between the two arises from the fact that real metals exhibit loss, which is not incorporated into the design technique. It may be noted in Fig. 5(d) that the roll-off and stop-band extinction of the filter are not particularly high. This is because the frequency response is only a first-order response. Because these plasmonic filters operate in the linear regime, a multilayer stack of such filters can create an overall frequency response that is given simply by the product of the transmittance of each individual filter. Though it is beyond the scope of the present work, this approach has been shown to greatly increase the roll-off and stop-band extinction .
E. K-Space Geometric Filter
In the second example, we demonstrate a filter response that arises from the superposition of two distinct k-space objects. In Fig. 6(a), we show the k-space spectrum of two superposed geometric shapes, a rectangle and a variably weighted circle. Because these two shapes have different symmetry properties, they will have different polarization-dependent contributions to the overall frequency response of the filter. The rectangular shape corresponds to a resonance at 0.2 THz along the direction and a resonance at 0.4 THz along the direction, while the weighted annulus corresponds to a polarization-independent resonance frequency of 0.4 THz.
Figure 6(b) shows the real-space image obtained from an inverse Fourier transform of the k-space rectangle, with the weighting value of the annulus, W, set to 0 (effectively leaving only the rectangle). As the weighting value of the annulus, W, is increased, its effect on the real-space structure can be seen in Figs. 6(c) and 6(d). With increasing values of W, the real-space structure becomes increasingly rotationally symmetric. When W is set to 0 and the filter exhibits only the anisotropic contribution from the rectangle in k-space, the measured ratio of the transmission magnitude at the two resonant frequencies, will vary with the polarization angle as shown in Fig. 6(e). As the weight of the annulus is increased, the ratio is also increased, while maintaining same polarization dependency, creating the ability to control the relative magnitudes of different resonances. More generally, the ability to combine multiple filter functions, where each function may be weighted differently, offers the opportunity to create a range of unique THz filter capabilities.
In summary, we have proposed a novel THz filter design methodology that involves defining the k-space representation of a desired filter and then transforming it to obtain the real-space geometry for the device. Using conventional inkjet printers, we printed the resulting grayscale images using a combination of conductive and resistive inks. While this approach is sufficient for the designs demonstrated here, alternate approaches may be necessary in designs that require higher spatial resolution with high-fidelity grayscale matching. Furthermore, the dielectric losses associated with the printed metals place limits on the minimum bandwidths that may be achieved. However, designs that limit the interaction length of surface plasmons may be able to minimize the effect of these propagation losses. We demonstrated a number of different types of filters in which the resonance frequencies, resonance bandwidths, and polarization properties can be well controlled and cannot easily be designed using other approaches. Additional flexibility and complexity in filter designs can be achieved by multiplying or adding several different 2D filter functions to obtain response functions that cannot be achieved easily through other means. While the filters here are passive, use of inks that respond to external stimuli or the incorporation of other materials that are active should allow for the fabrication of dynamic filters. Further exploration of this approach is expected to lead to new optoelectronic device capabilities that may be useful for THz applications, which are largely devoid of device and systems technology.
National Science Foundation (NSF) (DMR 1121252).
See Supplement 1 for supporting content.
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