1. Introduction

Phase gratings are popular beam multiplexers for submillimeter and THz instruments, usually in the form of Fourier gratings (FG) [1] or collimating Fourier gratings (CFG) [2]. All currently operational astronomical heterodyne array receivers at frequencies beyond 400 GHz [3–6] successfully use CFGs in their local oscillator (LO) beam distribution. Former and future instruments in this frequency range employ CFGs [7] or FGs [8,9].

Since LO power is often scarce, it is important to optimize the efficiency of the beam multiplexer. In the case of a phase grating multiplexer this requires to optimize the phase structure to maximize the power diffracted into the desired orders and to suppress as much as possible all other (parasitic) diffraction orders. An added benefit of an efficient grating is the reduction of stray light in the system. Nonetheless, several of the instruments mentioned above [3,7–9] use grating designs that fall short of the maximum achievable efficiency by a significant margin.

The purpose of this paper is to propose a method to obtain higher efficiency gratings for beam arrangements that we define to call pseudo-two-dimensional.

2. Pseudo-two-dimensional gratings

Two-dimensional (2D) beam arrangements that can be obtained by superimposing two orthogonal one-dimensional (1D) arrangements are considered to be pseudo-two-dimensional. This in particular includes commonly used patterns on a rectangular grid like a 2×2 [7] or 2×4 [3,8,9] beam arrangement. In contrast, the 2-4-2 beam arrangement studied in [1] or the hexagonal structures used in CHAMP⁺ [4] and upGREAT [5,6] inherently require true 2D structures.

Modeling the phase grating as a pseudo-two-dimensional structure implicitly imposes design constraints that may unnecessarily limit the grating efficiency. For instance, in a 1D grating producing two symmetric beams, the two beams are in phase, the same is true for the superposition of two such gratings. In other words, all four beams in a pseudo-2D 2×2 grating share the same phase. Many applications do not require a specific phase relationship between beams. Therefore, imposing a common phase to the four beams is often a dispensable constraint. Allowing for phase differences in the individual beams releases additional degrees of freedom to the grating design, potentially resulting in a higher grating efficiency.

Obviously, the full two-dimensional grating optimization is more complex than designing two one-dimensional grating structures, but computationally it is not really challenging for a standard computer.

3. Application

In the following we discuss two examples of pseudo-2D gratings and show how a true 2D approach can enhance the grating diffraction efficiency.

3.1. Design method

The unit cell structures are optimized in a similar way as sketched in Section IV of [1]: in a first step we reverse the problem and aim to model the grating as a beam combiner that combines N planar waves into an electric field with a constant amplitude in the grating plane. The amplitudes of those incident waves, which represent the wanted diffraction orders are set to the desired values. All other diffraction orders are set to zero amplitude. This step reduces the number of free parameters in the problem to the N phases of the incident waves. The algorithm first performs an unbiased search of the N-dimensional parameter space to find the combination of phases that minimizes the amplitude variations of the combined field in the grating plane. The set of incident phases is further improved by standard χ² minimization.

The resulting grating field distribution is used as the starting field for optimization by a Gerchberg-Saxton algorithm (GSA) [10]. Two-dimensional FFT is used to convert back and forth between grating field and diffraction pattern.

The initial step of optimizing the phases of the diffraction beams may impose the phase conditions mentioned in Section 2. In the most general case, no phase constraints will be applied, if only the amplitudes of the beams are of interest. If, however, a phase relation between beams is imposed, the number of free parameters in the optimization is reduced and in general the achievable amplitude flatness of the combined field in the grating plane will be reduced. This is equivalent to a loss in grating efficiency, because more power in unwanted diffraction orders is needed to flatten the grating field.

To facilitate the initial phase search, we impose an inversion symmetry constraint: the phase of diffraction order (x, y) is kept equal to the phase of order (−x, −y). Although the GSA does not maintain our phase constraints, the resulting diffraction patterns still show the inversion symmetry of the phases within small numerical errors. Running the same algorithm without the inversion symmetry constraint did not improve the results.

All calculations are performed with a specially developed code on a 256×256 pixel grid. The grating efficiency, as usual, is defined as the power diffracted into the desired orders divided by the power diffracted into all orders. The finite grid size implies that the grating efficiency estimates neglect the very small amount of power diffracted beyond ±127^th diffraction order. Our method results in very high diffraction efficiencies, but, since the search pattern may still be incomplete, we can not prove that the maximum achievable grating efficiency is found.

The optimized grating structures contain sharp phase steps, which may be difficult to manufacture for practical applications. However, approximative methods like for instance the Fourier grating approach usually yield efficiencies not far below the theoretical maximum [1].

As an independent check of the optimization results, we simulate the grating using scalar diffraction theory. The diffraction patterns shown in the following were obtained by numerical integration of the Kirchhoff integral for an illuminating field corresponding to a Gaussian beam waist in the grating plane with the phase modulation from the GSA result. The waist size was chosen to be 0.56 times the size of the unit cell, creating diffracted beam FWHM sizes of 2/3 of the order spacing. Since we are designing for an even number of beams in each direction, the grating has to suppress all even diffraction orders. Thus the effective beam size is 1/3 of the spacing of non-suppressed orders.

3.2. 2×4 beam grating

The pseudo-2D 2×4 beam grating is composed of a 1D two-beam grating and a 1D four-beam grating. The latter has been discussed in [1] and reaches an efficiency of 92%. The two-beam grating has an efficiency of 81%. Its unit cell consists of two equal halves with a phase step of π between them. Orthogonal superposition of the two 1D gratings yields a 2×4 beam grating with an efficiency of 74%. Figure 1(a) shows the unit cell phase structure and Fig. 2(a) gives the diffraction pattern of this grating.

 

Fig. 1 Unit cell phase structure of the pseudo-2D 2×4 beam grating (a) the true 2D 2×4 (b) and the true 2D 2×2 (c) beam gratings. The axis labels are relative linear dimensions of the unit cell size, The corresponding diffraction patterns are shown in Fig. 2.

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Fig. 2 Simulated diffraction patterns of the unit cell structures of Fig. 1. The color code is on a logarithmic scale. The power contours range from 0.05 to 0.95 in steps of 0.45 of the peak power in linear units. The axis labels denote diffraction orders. The efficiency improvement by the true 2D structures (b) and (c) is evident from the reduction of power lost into unwanted diffraction orders.

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Modeling the 2×4 beam grating as a true 2D structure creates a grating with a diffraction efficiency of 92%. Its unit cell and diffraction pattern are shown in Figs. 1(b) and 2(b). The reduction of power lost into parasitic diffraction orders compared to the pseudo-1D case is obvious.

3.3. 2×2 beam grating

The pseudo 2×2 beam grating uses the 1D two-beam structure with 81% efficiency in both spatial dimensions. The resulting diffraction efficiency is 66%. Its grating structure corresponds to a chess board pattern, where the black squares have a phase offset of π relative to the white squares. The diffraction pattern in both directions is equal to the horizontal pattern in Fig. 2(a).

When modeled as a true 2D grating, the efficiency of the 2×2 beam grating increases to 92% [Figs. 1(c) and 2(c)], which represents a 39% improvement over the pseudo-2D grating.

It is interesting to note that our 2×2 beam unit cell is very similar to the phase profile obtained with an non-periodic surface design method [11].

4. Conclusion

If grating diffraction efficiency is of importance, a 2D grating - even if it can be modeled as a superposition of two 1D gratings - should always be modeled as a true 2D grating. Doing so can significantly enhance the efficiency. For two commonly used beam arrangements we found improvements of up to 40%.