1. Introduction

Currently terahertz (THz) spectroscopy and imaging is widely used in many research labs and industry contexts for material characterization [1–3], industrial inspection [4–6], pharmaceutical science [7], biomedical applications [8], communications [9, 10], plant physiology [11] and art conservation [12]. Extending the applications of THz technology requires a variety of quasi-optical devices to manipulate, guide, and direct THz waves. This not only includes waveguides [13, 14], lenses [15, 16], beam splitters [17, 18], wave plates [19], and reflectors [20] but also diffractive elements such as gratings [21], which can be used to spatially disperse THz waves of different frequency.

Diffractive elements for manipulating THz waves at different frequencies have been discussed in literature (e.g [22].). One missing component is a dielectric prism that operates at THz frequencies. This is because materials that are highly transparent to THz waves, such as non-polar polymers, exhibit no or low dispersion. Hence, it is difficult to build efficient THz prisms in the conventional optical sense. Two approaches, one based on the super-prism effect in photonic crystals [23] the other based on an artificial-dielectric using a parallel-plate waveguide [24] was reported by the Mittleman group. Furthermore a dielectric ribbon waveguide has been presented for the THz-regime [25], but it was not used to spatially separate the frequencies of the THz beam.

Here we present a THz prism based on the pronounced frequency dependency of the effective refractive index of a dielectric waveguide leading to significant spatial dispersion of an input broadband THz beam. This type of prism has several potential advantages: ease of design and manufacture, efficiency and effectiveness over a broad frequency range. Such a prism can be used in THz systems, for example, it could be used in a prism spectrometer eliminating the need for a moving delay line in the laser beam path in a time domain set-up.

2. Theory

2.1 Waveguide

Here we consider a dielectric waveguide where the electromagnetic waves propagate by total internal reflection. The angle θ for propagation through the waveguide has to be greater than the critical angle, θ_c, for total internal reflection. If the waveguide material has a refractive index of n and the surrounding medium is air, then the reflectivity (R) of the waveguide-air interface is 1 and the critical angle is given by:

As the wave is reflected back and forth between the upper and lower waveguide-air interfaces, it interferes with itself. A guided mode can exist only when a transverse resonance condition is satisfied i.e. the repeatedly reflected wave has constructive interference with itself. There is also a phase shift ϕ associated with the internal reflection from the waveguide–air interface. The transverse resonance condition for constructive interference is given by the following characteristic equation [26]:

Where k is the wavenumber, d the thickness of the waveguide, $\varphi$ the phase shift of the total reflection at the waveguide-air interface and m a positive integer describing the mode. Because m can assume only integral values only certain discrete values of θ can satisfy the transverse resonance condition. With the relation k = 2πf/c, where f is the frequency and c the speed of light in a vacuum, it becomes clear that θ depends on the frequency.

Although the critical angle, θ_c does not depend on the polarization of the wave, the phase shift $\varphi$ caused by the internal reflection at a given angle θ does depend on the polarization. Thus the phase shift $\varphi$ as given in Ref [26] is:

The guided wave has a frequency-dependent distribution of wave vectors, whereas a plain wave has only a single wave vector, which points exactly in the propagation direction. Therefore, the THz wave travelling through the waveguide experiences dispersive phase shifts and so the total dispersion is the combination of material dispersion and waveguide dispersion. This results in an effective refractive index which changes with frequency. For the frequency dependent incident angle θ the effective refractive index is given by:

The characteristic Eq. (2) can only be solved numerically. Therefore we performed simulations using MATLAB (Mathworks Inc); using the material properties of polypropylene (PP), n ≈1.5 for the three modes over the frequency range 0.1 to 1 THz for two different prism thicknesses (700 and 910 µm). The results are displayed in Fig. 1.

 

Fig. 1 The effective refractive index of PP for p-polarized THz waves for three different modes. The solid line describes the data for a 700 µm thick waveguide and the dashed line for the 910 µm thick waveguide.

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The results of the simulations given in Fig. 1 clearly show how a dielectric material with the appropriate properties can be used as a dispersive element at THz frequencies due to the frequency dependent effective refractive index.

2.2 Prism design

In order to utilize the frequency dependent effective refractive index for spatial dispersion of THz frequencies we designed a prism shaped waveguide. Figure 2 shows a plan view of such a prism and the light path is shown with the coupling and decoupling of the electromagnetic wave in and out of the prism, respectively.

 

Fig. 2 The electromagnetic wave beam path through the waveguide prism.

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For a given angle of incidence α and prism angle γ the exit angle δ, which represents the deflection of the beam by the prism relative to the input direction of the incident beam is given by:

3. Materials and methods

To experimentally verify our calculations we examined two different prism waveguides made of polypropylene (PP) which has a refractive index of n = 1.5. One waveguide had a prism angle of 34° and a thickness of 700 µm, the other one had a prism angle of 33.8° and a thickness of 910 µm. The angular dispersion was measured on a goniometer based THz time-domain spectroscopy setup with fiber-coupled THz antennas shown in Fig. 3 [21]. At one end of the goniometer was a fiber coupled photoconductive THz emitter and the other end was a fiber coupled photoconductive THz detector. The angular positions of the emitter and detector could be moved –80° to + 80° to the normal (defined as when the emitter and detector are exactly opposite each other). The polarization of the THz wave could be changed by rotating the antennas. A pair of lenses in front of the emitter collimated and then focused the THz beam to couple it into the waveguide prism; another pair of lenses collected the THz beam emitted from the waveguide prism and fed the beam onto the detector. The waveguide prism was mounted between two thin metal blades to minimize contact with the top and bottom surfaces, which also conveniently filtered out any surface wave propagation. THz waveforms were recorded over a number of incident and exit angles for each of the prisms.

 

Fig. 3 Experimental setup of the THz path on a goniometer. The THz emitter and detector are fiber coupled. The THz beam is collimated and then focused onto the thin edge of the waveguide prism.

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4. Results

Example THz pulses and frequency spectra are given in Figs. 4(a) and 4(b), respectively, for the 700 µm thick sample. The incident angle was set at −10° and the exit angle varied from 0° to 40°. Figures 4(a) and 4(b) show THz time domain pulses and spectra at exit angles 18° and 28°.

 

Fig. 4 Comparison of THz data for two different exit angles of 18° and 28° with an incident angle of −10°, a prism thickness of 700 μm and p-polarized waves. (a) shows the time domain pulses (b) is a comparison of two frequency spectra, the reference spectrum is included to show the fall off in amplitude with increasing frequency.

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Comparison of the time domain data for two exit angles in Fig. 4(a) shows that the amplitude of the THz pulse decreases with increasing angle. The fast Fourier transform of the time domain pulses gives the frequency spectra in Fig. 4(b) for the same two exit angles of 18° and 28°. It can be clearly seen in Fig. 4(b) that the peak of the frequency spectrum shifts to higher frequencies with larger exit angles.

The spectrum for the exit angle of 18° peaks at 0.15 THz and the spectrum for 28° peaks at 0.25 THz, clearly indicating that the waveguide prism does separate frequency components.

Figure 5 shows the dispersion curves for different angles of incidence of the THz wave for the 700 µm thick sample; the measurement confirms the good agreement between the simulation and experiment.

 

Fig. 5 Dispersion curves for the prism with a thickness of 700 μm for five different angles of incidence. The lines show the theoretical values while points show the frequency maxima of the experimental data.

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The spatial dispersion of frequency components increases with larger negative angles of incidence. However, there is clearly a limit to the incident angle and in Fig. 5 one can see that for −15° (green line), the higher frequencies are heavily deflected by the first interface that the beam at the output surface undergoes total internal reflection and thus does not emerge. From the measurements it is clear that the most pronounced dispersion was achieved with an angle of incidence set at −10°.

Figure 6 shows all frequencies in a range from 0.1 to 0.7 THz for the whole range of the exit angles for the 700 µm sample with an incident angle of −10°.

 

Fig. 6 Two-dimensional representation of the dependence of the exit angle of the THz frequency for an incident angle of −10° with a prism thickness of 700 μm and p-polarized waves. The color bar indicates the amplitude ratio for the reference measurement in dB. All ratios beyond −25dB were set to black color and all ratios above 5 dB were set to white.

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The white areas represent the high amplitude frequencies, while the dark red areas represent very low amplitude frequencies. The water vapor absorption line at 0.55 THz can be clearly seen. Most obvious is the dispersion curve for the zero order mode. The splitting is carried out over a wide range of angles above 15°. From about 0.4 THz the weaker curve of the second mode can be seen.

Figure 7 shows the effect of varying the angle of incidence for the 910 µm thick sample.

 

Fig. 7 Dispersion curves for 910 μm thick sample for five different angles of incidence. The lines describing the theoretical values while the frequency maxima points are derived from the measurement data.

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It can be seen from the comparison of Figs. 5 and 7 that the thicker sample leads to a smaller angle splitting. That is, the thinner the dielectric waveguide is, the more the individual THz frequencies are separated.

As predicted by Eqs. (3) and (4) dispersion is stronger for p-polarized waves than for s-polarized waves. As an example, given an incident angle of 0° the angular difference between 0.11 THz and 0.52 THz for s-polarized waves is 7.98°; while for p-polarized waves we obtained a difference of 16.23°.

Also from the experimental measurements shown in Fig. 6 one can see the second maxima emerging at about 0.4 THz and it is suspected that this is a higher mode. Figure 8 shows the frequency maxima for the 0 mode (red dots) and the frequency of secondary maxima (black dots), taken from the data for the 700 µm thick sample using p-polarized waves and an incidence angle of 0°. The solid lines in Fig. 8 show the simulation data for the zero order mode, the first mode and the second mode.

 

Fig. 8 Dispersion of different modes in a 700 µm thick waveguide for an incident angle of 0° with p-polarized waves. The lines describe the theoretical data, the spots the frequency maxima of the experimental results. Only the zero and second order mode propagate through the wave-guide.

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Figure 8 clearly shows that experimental and theoretical values for each mode correspond well. The data for secondary maxima in Fig. 6 correlates well with the simulated data for the second mode in Fig. 8. Given the onset of the secondary maxima at a frequency just below 0.4 THz suggests that this is not the first mode. If it were then a signal would be seen at frequencies from 0.2 THz and at a much smaller angular displacement relative to the zero order mode. This pattern was observed both in the 700 µm and the 910 µm sample as well as on the frequency spectra.

5. Discussion

It has been shown that a prism-shaped dielectric waveguide leads to an efficient division of the frequency components of a broadband THz pulse. The basic idea behind this type of prism is to use the frequency-dependent effective refractive index of guided modes in the waveguide to create an artificial, high dispersive medium. This dispersive prism-shaped waveguide showed a pronounced frequency dependence of the optical refraction at interfaces as predicted by theory.

The preferred polarizing orientation is p-polarized THz waves, since these are dispersed more than s-polarized waves. The thickness of the waveguide depends on the target application. If a wider frequency range is to be separated then a thinner waveguide is recommended, however, with decreasing thickness coupling the THz wave into the waveguide gets more difficult. For the prism geometry used here the largest angle of incidence that provides maximum frequency separation is −10°. If the incident angle exceeds this value the THz waves experience total internal reflection at the exit boundary surface and hence do not emerge from the waveguide. This critical angle depends on the exact design of the prism and the target spectrum. Designs of prisms with specific dispersive properties for particular applications can be achieved by optimizing the prism geometry. Furthermore, it was found that only even modes are relevant, because the odd modes are not efficiently coupled into the waveguide. This is because the beam profile of the input beam is approximately a Gaussian.

For all measurements there were slight deviations between the theoretical and experimental values. There could be a number of reasons for these differences. The prism was fitted by eye into the measurement setup resulting in a small positioning error and hence in the angle of incidence might be slightly inaccurate. The emitter, optics and detector may not have been perfectly aligned. The edges of the waveguide into which the THz waves were coupled were not perfectly flat or smooth reducing coupling efficiency into and out of the waveguide. The waveguide was held in place by two metal blades which distorted the waveguide slightly which could lead to top and bottom surfaces not been entirely parallel across the whole waveguide. As an example of this are the narrow peaks seen between the zero and second order modes in Fig. 6.

For future investigations using other materials and various thicknesses could be studied. As greater spatial dispersion is predicted with thinner waveguides the limit on the parameters of THz prism needs to be further explored.

Finally, the combination of the spatially spread wavelengths components with a THz-camera would be desirable. A THz prism in conjunction with a THz camera [27, 28] would enable the construction of a THz spectrometer similar to a prism spectrometer at optical frequencies. This would obviate the need for a delay line and in turn could significantly reduce the measurement time and make systems more rugged.

6. Conclusion

We have presented a THz waveguide prism that can easily be fabricated and characterized the angle-dependent properties of the prism by simulation and by experimental measurements in a THz-TDS system. The very good agreement of the results demonstrates the ability of the prism for spatial dispersion of THz waves. Therefore, waveguide prisms can be used as dispersive elements for applications such as spectrometers or novel THz imaging systems, hopefully further enhancing the potential of THz technology and widening its applications.