## Abstract

Typical lenses suffer from Fresnel reflections at their surfaces, reducing the transmitted power and leading to interference phenomena. While antireflection coatings can efficiently suppress these reflections for a small frequency window, broadband antireflection coatings remain challenging. In this paper, we report on the simulation and experimental investigation of Brewster lenses in the THz-range. These lenses can be operated under the Brewster angle, ensuring reflection-free transmission of p-polarized light in an extremely broad spectral range. Experimental proof of the excellent focusing capabilities of the Brewster lenses is given by frequency and spatially resolved focus plane measurements using a fiber-coupled THz-TDS system.

© 2011 OSA

## 1. Introduction

For many centuries, lens design has been a core discipline of optics, enabling the guiding and shaping of light beams. Yet, even in modern optics, conventional lenses suffer from Fresnel reflections at their lens surfaces. While anti-reflection coatings can reduce this problem, typical coatings only operate in a rather narrow frequency window limiting their applicability in broadband systems. Such coatings usually comprise a single dielectric film or a stack of multiple dielectric layers [1,2]. To achieve a good anti-reflection performance over a broad frequency range, graded index coatings have been developed, which successively match the refractive index of the lens material to the one of free space [3]. Recently, advances in the field of nanotechnology allowed for the fabrication of antireflection surface structures, e.g. in form of an aperiodic array of nanotips [4]. In addition, by varying the density of nanorod ensembles the refractive index of individual nanorod layers can be custom tailored so that highly efficient graded index structures become available [5]. However, the fabrication of such coatings is still quite elaborate and time-consuming so that only few niche applications exist which can afford this technology. In addition to the high fabrication effort, the performance of such designs degrades towards long wavelengths.

Terahertz time domain spectroscopy (THz-TDS) systems rely on octave spanning picoseconds pulses, so that conventional antireflection coatings cannot be applied. While matching layers operate well in a narrow frequency range as has been shown in [6–13], broadband coatings remain a challenge. First steps have been made in the direction of graded index coatings using lithographically microstructured semiconductor surfaces [14,15] as proposed by Jepsen et al. and Zhang et al., but the additional fabrication effort would again limit their applicability.

In this paper we report on the design, simulation and experimental investigation of Brewster lenses in the THz range, which are inherently reflection free for p-polarized light without using any impedance matching layers. We will show that Brewster lenses enable broadband operation with a similar imaging performance as conventional lenses and could become a key component both in terahertz as well as in optical systems.

A conventional lens consists of two refracting surfaces. The exact shape of the surface depends on the type of lens. While a classical lens exhibits a perfect or at least approximate rotational symmetry, a Brewster lens, as first suggested by Hertz and Minkwitz, breaks with this paradigm [16]. Instead, it consists of a spherical and a cylindrical surface which are tilted with regards to the optical axes of the impinging beam. The challenge in the Brewster lens design lies in the complete compensation of the astigmatism, caused by the strong asymmetry of light propagation with regards to the optical axis. Yet, the compensation can be achieved by careful choice of the surface curvatures. If the angle between the surfaces of the Brewster lens and the optical axis equals the characteristic Brewster angle of the lens material and the light is p-polarized, then the wave is fully transmitted without any Fabry-Perot reflections occurring at the lens surfaces. This antireflection mechanism is extremely broadband and only limited by the flatness of the dispersion curve of the lens material. However, as low dispersion is also crucial to minimize chromatic aberrations in conventional lenses, this requirement is usually inherently fulfilled by typical terahertz lens materials.

In the following, we will first discuss the design and the operation principle of Brewster lenses. Additionally, we will provide numerical calculations of the focusing properties of high density polyethylene (HDPE) Brewster lenses for THz frequencies, based on ray-tracing and physical optics simulations. Subsequently, we will experimentally investigate an HDPE-Brewster lens by frequency dependent, spatially resolved focus plane measurements using a fiber-coupled THz-TDS system and will demonstrate that a high focus spot quality can be obtained using this approach. The measured data is compared to the theoretical predictions and the antireflection capabilities are evaluated.

## 2. Design of Brewster lenses

The challenge in the design of a Brewster lens can be summarized by the following two points:

- • The electromagnetic wave has to be incident under the Brewster angle on both the front and the back surface of the lens
- • The lens has to compensate for the strong astigmatism caused by the oblique incidence of light

In a German publication from 1969 [16], Hertz and Minkwitz show that such functionality can be achieved by using the combination of a spherical surface with the curvature radius *r _{s}* on the front side and a cylindrical surface with the curvature radius

*r*on the back side. As the majority of readers will not be familiar with the German language and as the derivation given in [16] is very concise, this section provides a detailed derivation of the analytic expressions of the curvature radii

_{c}*r*and

_{c}*r*. Figure 1 depicts a sketch of the Brewster lens geometry:

_{s}To understand the design methodology and derive analytic expressions for the curvature radii *r _{s}* and

*r*, we will first consider the Coddington equations [17]:

_{c}*i*= 1,2 denotes the first and the second lens interface, respectively. The Coddington equations in general describe the imaging behavior of a small bundle of light rays impinging on a curved dielectric interface, where the rays originate from medium 1 with refractive index ${n}_{1}^{(i)}$ under an angle ${\beta}_{1}^{(i)}$. The curved dielectric boundary is characterized by the meridional radius ${r}_{mer}^{(i)}$ and the sagittal radius ${r}_{sag}^{(i)}$. The refractive index and the propagation angle inside medium 2 are given by ${n}_{2}^{(i)}$ and ${\beta}_{2}^{(i)}$, respectively. The distance between the dielectric surface and the meridional and sagittal object and imaging point is determined by ${\tau}_{1,mer}^{(i)}$ and ${\tau}_{1,sag}^{(i)}$, as well as ${\tau}_{2,mer}^{(i)}$and ${\tau}_{2,sag}^{(i)}$, respectively.

To realize the Brewster lens concept, the angle of incidence on each interface has to equal the Brewster angle given by:

Inserting the Brewster condition into Eqs. (1) and (2) and correctly choosing the refractive indices of the individual layers (n_{air} = 1, n_{lens} = n) leads to the following equations that govern the imaging characteristics from air into the lens (Eqs. (4) and (7)) and from the lens to air (Eq. (5) and (6)) for the meridional,

To fulfill the Brewster angle condition given by Eq. (3) for both lens surfaces, the rays inside the lens have to travel in parallel to each other so that

Furthermore, the distance of the object point to the lens has to be the same for the meridional and sagittal plane:

In our case, this distance would approach infinity for a collimated impinging beam. Inserting Eqs. (8) to (10) in Eqs. (4) to (7) and recombining the resulting expressions leads to

Using Eq. (6) to Eq. (9) the following expression can be derived:

*f*indicates the focal length of the lens. With Eqs. (11) and (12), the Brewster lens can now be designed. To do so, one side of the lens is considered to be spherical with

*r*

_{s}

*= ${r}_{mer}^{(1)}$*= ${r}_{sag}^{(1)}$, and the other side is assumed to be of cylindrical shape with

*r*

_{c}= ${r}_{sag}^{(2)}$ and ${r}_{mer}^{(2)}\to \infty $. With these assumptions, Eq. (11) can be transformed to

Inserting Eq. (13) into Eq. (12) yields

Thus, for any given focal length *f,* Eqs. (13) and (14) fully describe the Brewster lens geometry.

Please note, that in case of THz Brewster lenses the ratio between the diameter of the incoming beam and the lens diameter is rather large. As the Coddington equations, which form the basis of the derivation given above, assume that only a small bundle of light rays is incident, the radii *r _{c}* and

*r*given by Eqs. (13) and (14) might become inaccurate if short focal lengths are considered. In that case, an additional numerical optimization using a ray-tracer becomes advisable to achieve a high imaging performance. Yet, the radii

_{s}*r*and

_{c}*r*given by Eqs. (13) and (14) serve as a good starting point and only small corrections will be necessary. Furthermore, it should be noted that due to the tilting of the lens the effective field of view will be decreased according to $A={A}_{0}\mathrm{cos}\left(\beta \right)$, where ${A}_{0}$is the field of view for a tilting angle $\beta =0\xb0$. This will be essential, especially in the case of high refracting materials. A larger lens diameter then has to compensate for this. The tilting will furthermore lead to a displacement

_{s}*x*of the beam which can roughly be estimated by $x=d\mathrm{sin}\left({\beta}_{1}^{(1)}-{\beta}_{1}^{(2)}\right)/\mathrm{cos}\left({\beta}_{1}^{(2)}\right)$, where

*d*is the lens thickness and ${\beta}_{1}^{(1)},{\beta}_{1}^{(2)}$are the propagation angles outside and inside the lens, respectively.

For the work presented in this paper, we chose HDPE as lens material. HDPE is highly transparent for THz waves and is a popular choice as base material for THz optics [18–21] due to its high mechanical and chemical stability, its excellent machinability and low cost. The refractive index of HDPE approximately is 1.54 [22], which corresponds to a Brewster angle of 57°. For a Brewster lens with a focal length of f = 120 mm, Eqs. (12) and (13) yield a spherical and a cylindrical curvature radius of *r _{s}* = 302.22 mm and

*r*= 127.43 mm, respectively.

_{c}## 3. Numerical simulations of the antireflection characteristics

While the analytical calculations given above are well suited for determining the correct lens geometry, they assume a small illuminated area. To account for the large illumination area and realistically predict the reflection, transmission and imaging properties of the investigated Brewster lens, we employ the commercially available ray-tracing and physical optics software ZEMAX for all following simulations.

To evaluate the antireflection properties of the Brewster lenses, Fig. 2
shows the frequency dependent intensity reflection coefficient of two *f* = 120 mm Brewster lenses (solid lines), one made from HDPE (blue line) and the other made from high resistivity silicon (HR-Si, red line). Simulations of the intensity reflection coefficient of two conventional lenses indicated by the dashed lines are provided for comparison.

While the frequency averaged reflection coefficient of the aspheric HDPE and HR-Si lenses is approximately 9% and 51%, respectively, the corresponding values for the Brewster lenses are only 0.56% and 0.39%. The smaller value for the Brewster HR-Si lens despite the materials high refractive index of approximately 3.418 is due to the lower curvatures of the two lens surfaces in comparison to the HDPE Brewster lens. These simulations clearly demonstrate the close to reflection-free performance of the Brewster lens and highlight the fact, that the reflection losses become even smaller when high refractive index materials are employed. This property could become very significant in the light of recent advances in the field of artificial dielectrics, such as polymeric compounds [23,24], which have recently been proposed as base material of THz optics [25]. Especially when techniques such as compression molding [26,27] or die casting are employed, inexpensive lenses with refractive indices well above 2 featuring very high numerical apertures could become widely available. Using these artificial materials, a Brewster lens design would significantly outperform conventional aspheric lenses in terms of transmitted power, as the simulations of the HR-Si lens clearly reveal. In the following section, the imaging properties of the lens will be further explored and compared to conventional lens designs.

## 4. Experimental results and discussion

All of the following measurements are performed using a fiber-coupled THz time domain spectrometer [28,29] as shown in Fig. 3 .

A train of femtosecond laser pulses with a center frequency of 1560 nm and a bandwidth of 40 nm, emitted from an Er^{+}-doped fiber laser, drive a photoconductive antenna, which emits a terahertz pulse containing frequency components from 100 GHz to approximately 1.5 THz. A second optically gated photoconductive antenna serves as coherent detector. To sample the terahertz time domain signal, a mechanical delay line is used to delay the gating pulse at the receiver antenna with respect to the emitted terahertz waveform. At each delay line position a data point is acquired by measuring the photocurrent at the receiver antenna until the full terahertz waveform is obtained. Both, emitter and detector are fiber coupled so that they can be freely positioned in space. The system has a peak dynamic range of 60 dB at 130 GHz.

The emitted THz radiation is collimated to a beam with a full width half maximum (FWHM) of approximately 45 mm and is guided to the Brewster lens which stands under an angle of 57° (the Brewster angle of HDPE) in the beam path. To characterize the imaging capabilities of the lens, a focal plane measurement is performed by raster scanning the receiver antenna in the x-y-plane using a motorized micro positioning stage. To obtain a point-like detector characteristic, a pinhole with a 1 mm diameter is placed directly in front of the receiver antenna’s substrate lens. While introducing the aperture drastically improves the spatial resolution, the overall transmitted power is reduced, so that the data evaluation range is limited to the spectral window between 100 GHz and 500 GHz. At each stage position, a full terahertz time domain waveform is acquired. Fourier transforming these waveforms yields frequency and spatially resolved intensity maps which reveal the focus spot characteristics.

Figures 4(a) and (b) show the measured intensity map in the focal plane and the corresponding cross-section plots along the x- and y-directions of an f = 120 mm Brewster lens at 150 GHz. The respective simulations are provided in Figs. 4(c) and (d). The qualitative agreement between measurements and simulations is very good. The false color images displayed in subfigures (a) and (c) reveal a slightly asymmetric shape instead of a purely Gaussian distribution. This asymmetry is expected due to the relatively large illumination area. The cross-section plots in the subfigures b) and d) allow for a quantitative evaluation: The measured FWHM in x-direction is determined to be 6.15 mm while in y-direction a value of 6.2 mm is found. The simulated FWHM focus spot sizes for the x- and y-direction are 6.85 mm and 7.49 mm, respectively. The deviations of the measured data points from a Gaussian fit are quite small. Interestingly, the measured focus sizes are even slightly smaller than the simulated ones but the deviations are negligible. To compare the imaging performance of the Brewster lens to a conventional aspheric lens, we conducted additional simulations. The FWHM in case of the conventional aspheric lens was found to be 6.61 mm, which is comparable to the focus spot dimensions of the Brewster lens. Thus, the Brewster lens exhibits a similar imaging performance while enabling significantly higher transmission.

To demonstrate the broadband performance of the Brewster lens, Fig. 5 shows measurements and simulations of the same f = 120 mm Brewster lens, now evaluated at 300 GHz, thus one octave higher than the 150 GHz discussed above. At 300 GHz, the overall focus size is considerably smaller as expected from the laws of diffraction. The measured FWHM in both x- and y-direction is approximately 4.9 mm. The corresponding simulated values are 3.1 mm and 3.5 mm in x- and y-direction, respectively.

In order to also experimentally investigate the polarization dependent antireflection properties of the Brewster lens, we rotate the fiber coupled antenna head by 45°, so that it emits half s- and half p-polarized terahertz waves. A wiregrid polarizer is introduced after the first collimation lens. Rotating the wiregrid polarizer allows for the convenient selection of either p- or s-polarized transmitted light. As shown in Fig. 6(a) , a terahertz waveform is acquired for each polarization state. The solid and dashed line shows the received signal in case of the p- and the s-polarized incoming wave, respectively. As expected from the theoretical predictions, the signal transmitted through the Brewster lens in case of p-polarized light is significantly stronger than in case of s-polarized light. To quantify this finding, the time integrated intensity ratio is calculated as

## 5. Conclusion

In summary, we proposed Brewster lenses for THz frequencies as inherently reflection free imaging components and experimentally demonstrated their excellent focusing capabilities by frequency and spatially resolved focus plane measurements using a fiber-coupled THz-TDS system. A very good agreement between simulations and measurement is achieved verifying the suitability of Brewster lenses for real world applications. In future, Brewster lenses could become a key-component in a variety of optical systems, also for other spectral ranges, when a linear polarized light source is used and reflection-free operation is a major concern.

## Acknowledgments

We are grateful for the support of the Federal Ministry of Economics and Technology (BMWi), grants 345 ZN and VP2376002AB0, adminis0tered by the German Federation of Industrial Research Associations (AiF). Furthermore, the Marburg group acknowledges financial support by the Federal Ministry of Food, Agriculture and Consumer Protection (BMELV) within the funds 2814504410, administered by the Federal Office for Agriculture and Food (BLE). Benedikt Scherger acknowledges financial support from the Friedrich Ebert Stiftung.

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