81 supra-THz beams generated by a Fourier grating and a quantum cascade laser

Large heterodyne receiver arrays (~100 pixel) allow astronomical instrumentations to map more area within limited space mission lifetime. One challenge is to generate multiple local oscillator (LO) beams. Here, we succeeded in generating 81 beams at 3.86 THz by combining a reflective, metallic Fourier grating with an unidirectional antenna coupled 3rdorder distributed feedback (DFB) quantum cascade laser (QCL). We have measured the diffracted 81 beams by scanning a single pyroelectric detector at a plane, which is in the far field for the diffraction beams. The measured output beam pattern agrees well with a simulated result from COMSOL Multiphysics, with respect to the angular distribution and power distribution among the 81 beams. We also derived the diffraction efficiency to be 94 ± 3%, which is very close to what was simulated for a manufactured Fourier grating (97%). For an array of equal superconducting hot electron bolometer mixers, 64 out of 81 beams can pump the HEB mixers with similar power, resulting in receiver sensitivities within 10%. Such a combination of a Fourier grating and a QCL can create an LO with 100 beams or more, enabling a new generation of large heterodyne arrays for astronomical instrumentation. © 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

In the applications for homeland security, heterodyne techniques can be useful for both active imaging and passive imaging system [5]. A heterodyne imaging system is attractive because such a system detects not only the intensity, but also the phase, making the detection more decisive than the direct detection [6]. Both astronomical and homeland security applications will benefit from a heterodyne array to increase the imaging speed or observation efficiency [7]. Currently, due to the high power consumption in the backend of a heterodyne receiver, including low noise amplifiers and spectrometers, and the needs of the local oscillator (LO) array [8], the practical heterodyne receivers operating at supra-THz (> 1 THz) are limited to an order of 10 pixels. With recent advances in backend technology, involving low-noise-amplifiers [9] and CMOS digital spectrometers [10], both of which have demonstrated impressively low power consumptions, one can envision large heterodyne arrays, for example, 100 pixels or more, for future astronomical instrumentations in space. This allows for mapping more regions of astronomical interest within a space mission lifetime compared to current heterodyne arrays. Similar statements also apply to ground-based telescopes especially at frequencies where suitable weather conditions are rare. The challenge to develop large arrays now is to generate a large LO array with sufficient power and good uniformity among individual beams [11]. In comparison with LO sources based on multipliers [12], quantum cascade lasers (QCLs) are a more promising LO source in the supra-THz region due to its high output power. Current QCLs working at supra-THz frequencies have been demonstrated to cover the range from 1.2 THz to 4.9 THz, and their output powers typically are milliwatts or more [13], which is sufficient as an LO for a heterodyne array with more than 100 mixers. Besides, the distributed-feedback (DFB) laser structures based on double metal waveguide designs allow single-mode emission and hightemperature operation [14]. Therefore, multiplexing a single beam from a QCL by a diffraction grating to generate an LO array is an attractive approach for large heterodyne arrays.
A phase grating has played a crucial role in optics-related science and applications, including orbital angular momentum beams for optical communications, chromatic broadband nulling, UV and terahertz beam multiplexing, and complex beam shaping [15][16][17][18][19]. A phase grating is a periodic structure converting a single coherent radiation beam into multi-beams in different directions by manipulating the phase of the input beam. In [20], Dammann and Görtler proposed gratings with binary groove shapes, called Dammann gratings, and also structures with continuous phase-only groove shapes. Compared to a Dammann grating, a grating with a continuous phase-only groove shape achieves a higher diffraction efficiency, which is defined as the ratio between the power diffracted into the desired orders and the power of the input beam. The use of a Fourier synthesis technique to achieve the continuous phase-only groove shapes was proposed by Schmahl [21], and gratings having a fundamental and a finite number of harmonics are called Fourier gratings [22]. Better milling tool permits fabrication of Fourier gratings with finer features, thus higher diffraction efficiency. In essence, such a structure can also be viewed as a metasurface, manipulating the phase of the incoming beam [23]. Lanigan et al. fabricated a 3-pixel Fourier grating at 0.1 THz [24]. Graf and Heyminck [25] demonstrated an 8 beams grating operated at 490 GHz with a measured diffraction efficiency ~84%. A 7 pixel grating operated at 1.1 THz with diffraction efficiency ~80% has also been realized in [26]. Recently, Mirzaei et al in [27] have pushed the frequency further to 1.4 THz, and even to the 4.7 THz with a 2 × 4 Fourier grating with a diffraction efficiency of 74% in [18]. A 4.7 THz grating of 2 × 4 will be used as an LO multiplexer for mapping [OI] line emission in Galactic/extragalactic Ultra long duration balloon Spectroscopic Stratospheric THz Observatory (GUSTO), which will be launched in 2021 [28]. Fourier gratings have good performance in terms of diffraction efficiency and uniformity among the image beams when the number of output beams is less than 10. In this case, about 20 -30% of the incoming beam power is diffracted into unwanted higher diffraction order modes. Interestingly, a Fourier grating aiming for more diffraction orders (or a large number of diffracted beams) can offer a higher diffraction efficiency according to simulation shown below.
From the manufacturing point of view, accurately duplicating the surface profile of a Fourier grating is crucial [29], especially for large-pixel gratings working at supra-THz regions, since the performance of the grating is dominated by the surface profile. A Fourier grating operated at a higher frequency is more difficult to fabricate since the accuracy is more demanding. In this paper, we present a Fourier grating that generates 81 beams at 3.86 THz, where the incoming beam is provided by an unidirectional antenna coupled 3rd-order distributed feedback (DFB) QCL [30].

Diffraction theory
Based on the diffraction theory [31], the diffracted far field beam distribution from a phase grating is mathematically represented by where the term 2 2 ( , ) exp(( / 2 )( )) U j k z ξ η ξ η + describes the field distribution in the grating plane. Based on Eq. (1), the far field distribution is the Fourier transform of the field distribution in the grating plane. Since a phase grating manipulates the phase component of the incoming coherent beam, the field distribution in the phase grating plane is exp( is called a phase modulation function and is expanded in a Fourier series with a set of Fourier coefficients a n . Using the Fast Fourier Transform algorithm in Matlab, we define the far field multi-beams distribution. Then a set of a n is found for the desired number of diffraction orders with high diffraction efficiency and good uniformity using the Standard multidimensional minimization algorithm in Matlab. According to the relation between phase difference and groove depth of the surface structure on the grating / 4 cos i d λ φ π θ = Δ , where λ is the wavelength of the input radiation, ∆φ the phase difference, and θ i the incident angle with respect to the normal direction of the grating, we can define the groove profile of the grating based on the phase modulation function.

Grating design and fabrication
We generate a two-dimensional (2D) grating by superposition of two 1D gratings orthogonally. Thus its beam pattern is orthogonally distributed. In order to maintain the symmetry, gratings with an even number of employed diffraction-orders (pixels) are added with a half-wave phase step in one half of the unit cell to suppress all the even orders [25], making their surface structure finer than odd-pixel gratings. Besides, even-pixel gratings have lower diffraction efficiency than odd-pixel gratings since the intensities in even diffractionorders cannot be fully suppressed in most cases. Table 1 shows the diffraction efficiency of 1D gratings with different number of beams and their minimum radius of curvature (MRC) by assuming that the input beam is normally incident to the grating and by assuming a 1.2 mm unit cell. MRC is a crucial parameter for the manufacturer because the designed fine structure of the grating surface can only be reproduced when MRC of the grating is larger than the radius of the ball end mill used in a micro-milling machine. Our initial goal was to demonstrate a 100 beam grating. However, we chose to start with a grating for 81 beams because our simulation shows that a 9 × 9 pixel grating can achieve 98% diffraction efficiency and also a relatively uniform power distribution among the diffracted beams. Generally speaking, odd-pixel gratings have higher diffraction efficiency and larger MRC than what those from even-pixel gratings because even-pixel gratings are added a half-wave phase step in one half of the unit cell. Based on the equation for a 1D reflection grating, (sin sin ) where D is the periodicity, θ m is the angle of the m'th diffraction order, and θ i is the incident angle with respect to the normal of the grating, the angular separation of the image beams is calculated to be ~3.8° (in our case, D = 1.2 mm, θ i = 12°, and λ = 78 μm). The grating is machined on an aluminum plate (Alcoa QC-10 Mold Alloy, of 5 mm thick) using an KERN EVO micromilling machine. The MRC for this design is 357 μm, which is about 4 times the radius (90 μm) of the smallest ball end mill currently available for the micro-milling machine used. In this case, this machine is capable of patterning all the fine surface features. A photo of the manufactured grating and its 3D optical microscope image are shown in Fig. 1(a) and (b), respectively. The entire grating contains 16 × 16 unit cells and is 2 cm × 2 cm in size. We also measured the surface profile of a unit cell in two orthogonal directions using a Dektak XT30 stylus profiler. We found good agreement between the measured profile and the designed one with a deviation less than 1 μm in height (1/78 of the working wavelength). The effect of such a difference on the grating performance will be discussed in the results section.

Measurement setup
We apply an unidirectional antenna-coupled 3rd order distributed feedback (DFB) quantum cascade laser (QCL) at 3.86 THz as the input source for the grating. This laser can provide more than 10 mW output power. The unidirectionality, resulting from the reflector shown in Fig. 2(a), enhances the power level of the forward beam by a factor of ~2. In our measurement, we used one of the lasers out of an array of 20 DFB lasers as the input source. Figure 2(b) shows the laser array chip mounted on a carrier. Figure 3 shows our measurement setup schematically. The QCL was mounted in a pulse tube cooler working at ~10 K. We measured the far field beam pattern of the laser at a distance of about 3.5 cm from the laser, which is shown in Fig. 2(c). The beam has a size ~25 mm in the vertical direction and ~15 mm in the horizontal direction, implying the full width at half-maximum (FWHM) divergence angles to be 23° and 14° in the vertical and horizontal directions, respectively. Obviously the beam has considerable deviation from a fundamental Gaussian beam profile. Since the Fourier grating duplicates the beam pattern of the input beam in its output, directly applying the beam in Fig. 2(c) to the grating requires a large distance to image the output beams with sufficient angular separation to be spatially resolved. However, taking into account the absorption by water vapor in the air at this frequency, we have to choose a short optical path that is within 30 cm in practice. To overcome this issue, we have to make a smaller and collimated incident beam to the grating. We introduced a high-density polyethylene plano-convex lens with a 20 mm focal length to collimate the QCL beam first and then we filtered it with a 5 mm iris aperture. The resulting beam is symmetric and Gaussian-like, shown in Fig. 2(d). This beam is nearly collimated with its FWHM divergence angle to be < 1°. It is also important to notice that the incoming beam at the position of the grating has a diameter of 7 mm and covering more than 5 unit cells on the grating. Although the power of the modified beam is lower than the output power of the QCL (5-10% powers), there is still sufficient power to map output beams of the gratings with a good signal-to-noise ratio.
The incident angles of the incoming beam are 12 ± 2° and ± 2° with respect to the normal of the grating plane in the horizontal and vertical directions, respectively (see Fig. 3). A pyroelectric room temperature detector mounted on a 2D scanner is used to map the image beam pattern at a distance of 12 cm from the grating surface. The detector has a 2-mm diameter aperture and is mounted inside an aluminum housing. The latter together with the planar mapping makes the measured power direction dependent with the maximum if the radiation is incident normally. The detector is read out by a lock-in amplifier. An arbitrary waveform generator is used to modulate the power supplier of the QCL and provides a reference to the lock-in amplifier with a 70 Hz sinusoidal function. A PC is used to control the 2D scanner and record the data from the lock-in amplifier.
To compare the total power of the image beams of the desired modes with that of the input beam, we measured the input beam power in the following way, where the grating is replaced by a gold coated mirror, which reflects nearly 100% power. The reflected single beam is also mapped by the pyro-electric detector at the same distance as measuring the grating imaging beam (12 cm). In this way, we can compare the relative power values accurately since the effect of water absorption due to the optical path is canceled out even though we do not know the absolute power. Caution has been taken to reduce stray light by introducing THz blackbody absorbers to cover the outside area of the circular aperture and the unilluminated area of the grating.    For the non-uniformity in the measured beams, there are a few practical factors in the measurement that can affect the uniformity in the beam distribution: (1) There were humidity changes in the laboratory during the beam mapping, which took roughly 20 hours. During this period, the relative humidity varies between 25% and 45%, which in turn changes the absorption and influences the image beam intensity distribution. (2) Power coupling between the pyro-electric detector and the image beams. During the mapping of the image beams, the measured power is angle-dependent, with more power for the beams in the center, but less power for the outer beams. (3) There are possible errors to determine the exact incident angle. The grating is designed for an incident angle of 12° in one direction and 0° in the orthogonal direction. However, in our experiment, it is problematic to determine the incident angle since THz radiation is invisible. We estimate an error of ± 2° on the incident angles, which can also influence the intensity distribution of the image beams. (4) Different distances between the grating and individual image beams. Since we scan the detector along a plane, the distance between the grating and scanning position for each beam varies and it is larger for the outer beams. A longer distance means more loss due to the air absorption. All these experimental errors can influence the measurement of the grating with regard to the diffraction efficiency and uniformity.
To correct the error due to the angle-dependent power measured by the pyro-electric detector, we calibrated the dependence by changing the facing angle of the detector to the beams in both vertical and horizontal directions. We then adjusted the measured intensity of the beams. We also measured the power loss due to different distances between the grating and individual image beams, and found a maximum loss of 0.8%, which is negligible. Other errors are difficult to correct so the only correction is the angular dependence of the detector power in our data analysis. However, we removed the noise floor caused by the pyro-electric detector itself in the image beam data and in the input beam data. Figure 4(c) shows the corrected measured beam pattern, which agrees better with the simulated one in Fig. 4(d). We then calculated the diffraction efficiency based on the corrected data set. We found that the diffraction efficiency is 94 ± 3%. This efficiency suggests that 94% of power has been transferred from the input beam to the diffracted 81 beams by the grating. Clearly, it is a highly efficient Fourier grating. The error bar accounts mainly for the following two sources: the humidity changes in the laboratory and the ± 2° error in the incident angle.
In running COMSOL Multiphysics, we used the periodic port with periodic boundary conditions in the RF (radio frequency) module to get the simulated diffraction efficiency. The simulated diffraction efficiency of the designed grating is 98%, but as stated in section 3, there is a small mismatch (~1 μm) between the manufactured and designed surface profile. So we also imported the manufactured surface profile into COMSOL and found a diffraction efficiency of 97%. It reduces the efficiency by 1%. This 97% value agrees with the measured value of 94 ± 3% within the margin of error.   Fig. 4, hat in the ons, being g with a non-zero asurement influence using two

Conclusion
We report a 81-beams supra-THz LO array generated by a Fourier grating with a unidirectional 3rd-order DFB QCL emitting single mode radiation at 3.86 THz. We succeeded in measuring 81 diffraction beams and, due to a high output power of the QCL, we have achieved a good signal-to-noise ratio allowing us to determine the diffraction efficiency of the grating, which is 94 ± 3%, and evaluate the power uniformity of the image beams. The measured diffraction efficiency agrees well with the simulated result using the profile of the manufactured Fourier grating. The latter gives 97% efficiency. Based on the measurements and uniformity requirement of the LO power for superconducting HEB mixers, we find 64 beams out of 81 have their power varying within ± 20% around the nominal value and that the 64 beams can be used to pump a 64 uniform array, maintaining the sensitivity degraded less than 10%. Our results open a new route towards a large heterodyne array of order of 100 pixels for future space instruments. Our Fourier grating approach can improve the functionalities of the phase gratings, such as the diffraction efficiency, for orbital angular momentum beams for optical communications, UV beam splitting, and complex beam shaping.