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We discuss thin optical structures that allow chromatic aberrations to be avoided in the THz domain. The paper contains the theoretical considerations, computer modeling and experimental evaluation of the high order kinoform diffractive elements in the THz range. According to the obtained results application of the high order kinoforms enables broadband operation in the THz range.
©2014 Optical Society of America
1. Motivation
Active and passive systems working in the terahertz (THz) radiation range require optical elements, like lenses or mirrors, which efficiently collect energy and concentrate properly the incoming radiation. This is compulsory as most of the THz sources and detectors exhibit a rather low efficiency [1–4]. Thus a sufficient THz energy, exceeding the noise level, must reach a single element detector or each pixel of a matrix detector. Therefore, the optical system transmission must be optimized. Simultaneously, the geometrical aberrations of the optical system must be low in order to both concentrate the energy at the focal plane and produce sharp images. Typically, the size of pixels in available THz matrix detectors is of the order of the wavelength, or even sub-wavelength in sensors derived from the infrared technology (in that case, the delivered signal corresponds to an average over several pixels, which allows noise processing) [5–8]. Therefore, the optics must operate at the limit of diffraction. In practice, the aperture of imaging or focusing optical component is often large and its diameter usually spreads from 50 mm up to 300 mm or even more. To reach the diffractive resolution limit, the focal length of such components is often equal to their diameter. Moreover, chromatic aberrations must also be reduced in many practical cases. For example, a chemical identification of the observed items, in view of security applications, may be performed by recording the images at several different wavelengths, for which the items show a differentiated THz absorption [9, 10]. Another relevant example is given by pulsed THz systems, like time-domain spectroscopy set-ups, which deliver broadband spectra spreading over several frequency decades [11–14].
Common THz optical components include refractive dielectric devices and metallic mirrors. Metallic mirrors are known to be achromatic and to show a very high efficiency, as actual metals in the THz range behave almost as ideal conductors [15–18]. However, reflecting and focusing mirrors are very sensitive to any misalignment and they do not work properly off axis. Thus systems based on focusing mirrors show high geometrical aberrations, larger than systems based on refractive lenses. Moreover, large numerical aperture mirrors are heavy and have significant sizes. On the other hand, dielectric lenses are easy and cheap to manufacture, especially when made from common organic plastics like high density polyethylene (HDPE) or polytetrafluoroethylene (PTFE). They are approximately achromatic because the refractive index n of these materials is almost constant over the whole THz range. Unfortunately, devices with the requested f-number (N ≈1) and focal lengths (100~300 mm) are tremendously thick. For example, a plano-convex lens with 300 mm diameter and N = 1 made from HDPE (n ≈1.54) is 100 mm thick! Not only such lenses are heavy, but they do not well transmit high frequency signals, as the absorption of polymers is rather substantial over 1 THz. In addition, we have already demonstrated that strong absorption of the medium boosts aberrations of bulky lenses [19]. In the THz range of radiation the first order kinoform can be used [20, 21], but they suffer from chromatic aberration. A good improvement of the bulky design is given by the so-called high order kinoform (HOK) scheme [22, 23]. HOK lenses consist of zones with a thicknesses being of the order of the THz wavelength, while classical Fresnel lenses have much thicker zones (in comparison with the wavelength) and can be described by a ray-tracing approach. By a precise control of the phase shift introduced by each zone, a constructive interference of all the waves diffracted by the HOK structure occurs only at the focal spot, while destructive interferences make the signal dropping to zero elsewhere in the focal plane. Thus, as compared to Fresnel lenses, HOK show much better optical performance.
We already proposed HOK [19, 24] as a relatively thin diffractive optical element for THz beams. Such an element is lightweight and offers lower attenuation due to its smaller thickness. The goal of the present paper is to give a theoretical outline of the HOK response with significantly suppressed chromatic aberration. Numerical simulations and experiments illustrate focusing in the broadband 300-600 GHz. This whole study demonstrates that thin HOK refractive elements help at nicely manipulating THz beams and moreover exhibit fairly good achromatic properties over the whole THz spectrum.
2. Theoretical background
First, we describe the difference between few lenses to facilitate the understanding of the optical response of the HOK element, also discussed in [25]. A classical refractive lens (here a plano-convex) is a thick element [Fig. 1(a)]. Thanks to the original idea by Fresnel, its thickness can be strongly reduced by removing circular layers whose thicknesses correspond to phase differences equal to multiples of 2π. In Fresnel lenses, the radial width of such zones is larger than 1000 times the wavelength. Kinoforms [26] [Fig. 1(b)] are improved devices based on a similar scheme. The radial width of each zone if of the order of the wavelength and the phase shifts introduced by all the zones are precisely controlled in order to achieve constructive interference at the focal spot. In a classical kinoform (1st order kinoform), the phase jump between two adjacent zones is 2π, resulting in a very thin and thus highly transparent structure. However, because of the single 2π phase jump, 1st order kinoform reveals very strong wavelength dependence. This inconvenience is minimized by designing HOK [22, 23] [Fig. 1(c)], whose phase jump is 2pπ, with the order p >> 1. The HOK element remains still a thin component, leading to a low attenuation of the illuminating beam as required for THz applications.
Fig. 1 Different schemes for a converging lens: a) classical plano-convex lens, b) 1st order kinoform, c) HOK (p>>1).
Let us briefly discuss the chromatic properties of the HOK in a case of a broadband illuminating radiation. We consider a HOK that is designed for the pth order at wavelength λ and we study its chromatic aberrations, i.e. its focusing properties for a smaller wavelength λ’ of the spectrum. Generally, the phase φ of a kinoform can be arbitrary chosen to design the desired optical properties. Here, for a better clarity, we concentrate on a HOK focusing lens [23, 25], with the well-known phase function of a thin converging lens in the paraxial approximation:
where r denotes the radial coordinate and f is the focal length of the lens. In the THz range, it is reasonable to assume a constant refractive index n of the kinoform medium. The local phase shift induced by the propagation through the lens material of thickness h is equal to:Then, in a case of the illumination by the smaller wavelength λ’ (λ’<λ), the phase shift is . The transmitted wave of wavelength λ is multiplied by the phase transmittance of the lens . Thus, for a different wavelength λ’, the transmittance converts into the form being a function of φ . We design the HOK in such a way that phase returns to zero each time when φ exceeds . Then the function T1(φ) defines the HOK transmittance within the period and we can expand this transmittance for a wavelength λ’ into the following Fourier series:where m denotes the diffractive order with the amplitude given as follows:Therefore, the diffraction efficiency of the m-th order, i.e. the square of the modulus of the Fourier coefficient, is:We can choose a natural number m fulfilling the following condition:where and according to the assumption <λ. This leads to:As , weaker orders contribute less to the output wave field. The phase at wavelength λ’ of the m-th order is deduced from (3):Then we substitute in φm the value of φ given by (1) for the wavelength λ. The focal length for the wavelength obeys the same expression (1) but the phase is defined by φm. Finally, we obtain:Using Eq. (6), we easily get:Taking into account that and , . Therefore, the chromatic aberration is substantially suppressed and the HOK has roughly the same focal length for different wavelengths λ and λ’.Although we have only discussed the case of a converging lens, the same conclusion can be drawn for an arbitrary phase function φ that is inversely proportional to the wavelength λ. The above method can be applied to design a large variety of HOK components as for example spherical, cylindrical, elliptical, hyperbolical, conical lenses, or axicons, or prisms. Relation (8) indicates that the geometry of the reconstructed wave front is almost independent of the wavelength for a sufficient large p. Hence chromatic properties of such HOKs can be substantially avoided.
3. High order kinoform lens for the THz range – numerical modeling
The numerical modeling is performed using a non-paraxial version of the modified convolution method [27, 28] combined with a non-paraxial modification of the beam propagation algorithm [29]. The volume element is divided into slices, each of them is 50 µm wide. The studied HOK structure exhibits a diameter d = 300 mm and the desired THz focal distance is f = 300 mm. The HOK is supposed to be made from HDPE, whose refractive index is n = 1.54 and the attenuation coefficient is α = 0.23 cm−1 at 0.3 THz (λ = 1 mm). The HOK is designed to be optimized for the p = 8 order at λ = 1 mm. We perform the modeling of the HOK focusing response in the wavelength range from 0.5 mm to 1 mm.
Generally Eq. (6) can be rewritten in the following way: λ’ = λp/m’, where m’ is a number equal to m ± ξ/2. According to Eq. (5), the designed HOK lens shows one main order of diffraction for the wavelengths corresponding to the integer m’ (ξ = 0). Table 1 gives the λ’s in the range 0.5-1 mm, i.e. for m’ varying from 8 to 16. For these wavelengths, the focal spots are diffraction limited (the spot diameter is almost equal to the wavelength) and, as the lens is achromatic, they lie in the same focal plane. Figure 2 shows the computed images of the beam in the focal plane for different λ’, i.e. when m’ varies from 8 to 16. For integer values of m’, the light beam is properly focused down to the limit of diffraction. The focal spot size increases with the wavelength as expected from the diffraction law. According to the description given in the former section, a non-integer number m’ defining λ’ by the equation λ’ = λp/m’ should lead to a focal shift. This effect is well observed in Fig. 2 where spots corresponding to non-integer m’ are defocused. On the other hand, for practical imagery applications, the chosen order p = 8 is sufficiently high to maintain the concentrated beam within an assumed detector area, which is a 5 × 5 mm2 square. Detectors with this dimensions are available and were used in our experiments.
Table 1. Wavelengths with Exact Focusing for p = 8 Order for λ = 1.0 mm
Fig. 2 Output radiation (PSF – Point Spread Function spots) in the designed focal plane. Squares indicate the area of the assumed detector which is 5 × 5 mm2. The number m’ defines the wavelength λ’ according to the equation λ’ = 8λ /m’.
4. High order kinoform lens for the THz range – experiment
We characterized HDPE HOK lenses that have the same geometrical parameters as the ones used in the modeling as described above. The experimental setup is presented in Fig. 3. A sub-THz beam is delivered by a Virginia Diodes Inc. (VDI) all-electronic system. The frequency is tunable in the range 0.14-1 THz. The output of the VDI source is a diagonal horn. A f = 75 mm HDPE lens (diameter 5.08 cm) focuses the beam on a 2.5 mm diameter pinhole. The pinhole was imaged in the 2f-2f configuration by the HOK lens, thus leading to the point spread function (PSF) of the HOK lens in the focal plane. The image plane was scanned with a movable detector (Zero Bias Detector – VDI Schottky Diode) attached to a 5 × 5 mm2 horn. The image was recorded within a 25 × 25 mm2 paraxial region at the focal plane by steps of 5 mm. Due to the relatively large geometrical size of the detector, a translation of the lens or the detector by 10 mm along the optical axis did not change significantly the readout
The recorded images are presented in Fig. 4. Each image was recorded for a given wavelength λ’, which corresponds to the m’ parameter ranging between 8 and 10, and is normalized due to the different efficiency of the source and detector for different frequencies. The average transmission of the volume of the HOK (0.3 THz) is 83% of the illumination energy (taking into account only the attenuation of the lens material). Additionally, this percentage will be smaller due to the Fresnel losses. Moreover, the attenuation of HDPE changes with frequency. The spatial resolution of the set-up is enough to allow the recording of the PSF even when the THz beam is strongly focused. However, the diameter (3 mm) of the recorded central peak is here limited by the set up resolution, and its actual size is smaller, i.e. certainly close to the diffraction limit (~1 mm). The recorded images are in a very good agreement with the computed ones, as it may be seen in Fig. 2. It should be noticed that experimental results have the sampling (pixel size) 5x5 mm2 and numerical simulations 0,1x0,1 mm2. That is why in the experiment we cannot see big differences between the consecutive images. The HOK lens was designed to work properly for the detector having the diameter 5x5 mm2, which means to focus well on this area for all frequencies. Even when in modelling we can notice some focal shifts for non-integer numbers of m’, in the experiment they are slightly visible due to the averaging of the detector. The differences between all the recorded PSFs are not significant even if, for the fractional values of m’, they exhibit additional blur (see for example the cases m’ = 8.50, m’ = 8.75, m’ = 9.5). This proves that chromatic aberrations are strongly reduced for this HOK lens, and thus this component can be usefully employed in THz broadband imaging systems.
Fig. 4 Output radiation PSF spots in the focal plane. Each image size is 25 x 25 mm. The number m’ is related to the experimental wavelength λ according to an equation λ’ = 8 λ /m’. The black squares indicate the area of the assumed detector which is 5 × 5 mm2.
5. Conclusion
The paper analyzes the usefulness of the transmissive focusing HOK lenses for broadband THz radiation. In agreement with the above theoretical outline, these HOK lenses exhibit negligible chromatic aberrations and good focusing properties in the THz range. Although theory specifies that ideal achromatism is achieved for HOKs running in the large p order regime, calculation and experiment show that acceptable effects are obtained even with a relatively small value of p (here p = 8). This small value permits the fabrication of thin HOK lenses, and thus to reduce the absorption losses in the lens material. In conclusion, such HOK components constitute an interesting alternative to more classical THz optics, with many advantages like improved transparency, low chromatic aberrations and reduced weight.
Acknowledgments
This work was supported by the Polish Ministry of Science and Higher Education under the Project O N515 020140 and by the Polish National Science Center under grant N N515 498840, with a complementary support from the European Union in the framework of European Social Fund through the Warsaw University of Technology Development Program.
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