ABSTRACT
Beam splitting gratings are used to split a laser beam into multiple beams for industrial and scientific applications. They are used in a range of applications, including parallel processing in laser machining and material processing, sensor systems, interferometry, communication systems, and image processing and gathering systems. An arbitrary periodic grating will split an incoming beam into a large number of outgoing beams (or orders). For many applications, it is desirable to put as much energy as possible into certain orders, while keeping the energy in all of these
3.1 Introduction .................................................................................................. 103 3.2 Fourier Optics ............................................................................................... 104 3.3 The Constrained Optimization Problem for Line Gratings .......................... 105 3.4 Some Alternative Optimization Problems .................................................... 107 3.5 Results for Line Gratings .............................................................................. 108 3.6 Experimental Results .................................................................................... 112 3.7 Square Gratings ............................................................................................ 112
3.7.1 A Two-Dimensional Beam Splitting Problem .................................. 112 3.7.2 Separable and Nonseparable Solutions ............................................. 115
3.8 The Symmetry of Beam Splitting Problems ................................................ 116 3.8.1 Symmetry Considerations for One-Dimensional Gratings .............. 116 3.8.2 Symmetry and Two-Dimensional Gratings ...................................... 117
3.9 Beam Shaping on Hexagonal Grids .............................................................. 119 3.10 Summary ...................................................................................................... 120 References .............................................................................................................. 120 Appendix A ............................................................................................................ 122 A.1 Four Beam Splitting ...................................................................................... 122 A.2 Five Beam Splitting ...................................................................................... 123 A.3 Nine Beam Splitting ..................................................................................... 123 Appendix B ............................................................................................................124 B.1 Six Beam Splitting ........................................................................................124 B.2 Seven Beam Splitting .................................................................................... 125
orders equal to each other (or more generally in some fixed proportion). If the grating does not absorb any light, it is referred to as a phase grating.
