Fourier gratings
DevelopmentsWorkgroup for SubMillimeter and Terahertz Receivers 
Fourier gratingsContents 

IntroductionOwing to their high diffraction efficiency, phase gratings are widely used in optics to shape and split monochromatic beams. With the advent of multibeam heterodyne receivers in submillimeter astronomy, they have also become popular in this part of the spectrum for localoscillator beam multiplexing. Splitting the beam from one oscillator into a given number of equal beams by use of a phase grating is a very elegant way to distribute the localoscillator power efficiently to an array of mixers. 
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Stepped grating Structures
These grating structures have been the starting point in our development of a new type of grating for the use as LOmultiplexer.
Fig. 1: Stepped grating structure with the resulting beampattern (measurement of the far field intensity distribution). 2 Layer grating (top row) and an optimized 4 layer structure (second row). 
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Theory of Fourier gratings (FG)A phase grating has a transmission function like using the phase modulation at the position (x,y). The structure of a one dimensional symmetric Fourier grating is given by a finite Fourier series expansion : where the unit cell extends from Delta x/2 to +Delta x/2. The transmission function is accordingly written as The diffraction pattern of the general Fourier grating is given by the Fourier transform of t(x). This results in a multiple convolution of the diffraction fields of the individual Fourier components: where J_{q}(a) denotes the Besselfunction of the first kind of order q. Although this is a simple equation we can not analytically calculate the coefficients a_{n}. We thus developed a numerical optimization method to calculate them. A direct numerical FFT is usually faster than evaluating T(k), and is therefore used in our calculation process.
Fig. 2: Structures of 13 coefficients Fourier gratings producing 2, 5 and 8 beams (from left to right). The theoretical efficiencies are 81%, 92% and 96%. 

A comparison of the intensity distribution in the diffraction pattern using a standard Dammann grating (from left to right in Fig. 3), a multilevel grating, a 13 coefficient Fourier grating and and the optimum Fourier grating (right hand panel) shows the increase of the efficiency by using our type of grating. 

The efficiency of Fourier gratings producing a given number of equal beams (Fig. 4). The dotted line marks the maximum efficiency achievable with N=5 Fourier coefficients, the solid line gives the efficiency for N>infinity. 

Number of Fourier coefficients is calculated from the tradeoff between:
==>results in approximatly 13 Fourier coefficients (Fig. 5). 

Bandwidth of the gratings:Detuning the frequency results either in position errors of the beams and in a power inbalance between them. Our SISsystem can tolerate a frequency shift of approximatley +5%. For higher bandwidth we have to exchange the grating. Because the reference planes are machined together with the grating structure, exchanging the gratings is simple and reproduceble.

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1dimensional Fourier gratingsFor the calculation of onedimensional structures we use the following 2step method:
The grating structures can be superimposed in different orthogonal directions to get a 2dimensional beampattern. Of course, this method only produces rectangular arrengements.
Fig. 6: Measured intensity distribution of a 490GHz beam, split by an 8 beam grating (left hand side). Contour levels are from 5% to 95% of the maximum peak intensity in steps of 10%. The grating consists of a 2 beam grating, superimposed with a 4 beam grating. The resulting surface topology of the unit cell using 13 Fourier coefficients for each direction is shown on the right hand side. Contour levels are in steps of .5 radians. The 1dimensonal structures are plotted along the axis of the graph. 
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2dimensional Fourier gratings
Calculation of twodimensional structures:Because of the higher number of coefficients N^{2} a random search is inefficient. We thus established the following method:
Fig. 7: Measured intensity distribution of a 490GHz beam, split by the grating for STAR (left hand side). Contour levels are from 5% to 95% of the maximum peak intensity in steps of 10%. The theoretical (in parenthesis) and the measured efficiency is: (83.0%) 84.0%. And the surface topology of the unit cell using 13 Fourier coefficients for each direction (right hand side). Contour levels are from 1.5 to 1.5 radians in steps of .25 radians.


To show the power of the design process, we calculated some letters as Fourier gratings using also 13 coefficients per lateral dimension. Fig. 8 shows the unit cells of the letters (top row) and their calculated diffraction pattern. These grating structures would be very hard to manufacture as stepped gratings, but easily as Fourier gratings. 

Collimating Fourier gratings (CFG)Using planar gratings, the LO source had to be reimaged to form a plane wave at the location of the grating. The new collimating Fourier grating is matched to the wave front curvature of the divergent beam and does not require an additional collimating element. Because of the smooth grating structure it is possible to produce the collimating Fourier gratings on a commercial numerically controlled milling machine.
Fig. 9: Comparison of a LOunit using a standard flat Fourier grating with a collimating mirror and (left hand side) with LOunit using the collimating Fourier grating.
Fig. 10 : Measured intensity distribution of a 350GHz beam, split by the grating for the DesertStar project of SORAL (left hand side). Contour levels are from 5% to 95% of the maximum peak intensity in steps of 10%. The surface topology of the unit cell using 8 complex Fourier coefficients for each direction is shown at the right hand side. Contour levels here are from 0 to 6 radians in steps of .5 radians. The unit cell ist streched in one direction by sqrt(3) to match the hexagonal arrangement. 

The collimating Fourier gratings can as well be manufactured easily on our in house CNC milling machine. Fig. 11 shows a photograph of the grating for the DesertStar project. The 4 corners are manufactured as referenc planes for easy repositioning of the grating. 

ContactFor Further informations contact: Dr. Urs Graf
