Gauss

meshgrid函数用于根据给定的横纵坐标点生成坐标网格，以便计算二元函数的取值。

设二维高斯函数表达式为：

程序如下：

u=[-10:0.1:10];
v=[-10:0.1:10];
[U,V]=meshgrid(u,v);
H=exp(-(U.^2+V.^2)./2/3^2);
mesh(u,v,H); %绘制三维曲面的函数
title('高斯函数曲面');

运行结果为：

Transforming Gaussian Beams into Uniform, Rectangular Intensity Distributions

Sunday, January 01 2012

Page 1 of 3

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The majority of laser types in current use produce output beams with circular or elliptical crosssections, with either Gaussian or near- Gaussian intensity profiles. This Gaussian intensity distribution is acceptable, and often beneficial for many applications in which the laser beam is being focused to a small spot. However, there are also many different uses for which a uniform intensity distribution (often referred to as a “flattop”) would be more optimal. For example, in materials processing tasks, a uniform intensity distribution ensures that the entire laser illuminated area is processed evenly. It is also valuable in situations where the laser light is used essentially for illumination. This is because uniform illumination makes identical features that all appear to have the same brightness, regardless of where they are located in the illuminated field, simplifying the image processing task and increasing contrast and resolution. These same benefits apply over a wide range of other applications that can be broadly classed as “illumination,” from machine vision, through flow cytometry, inspection, and even some medical uses.

Figure 1. The simplest way to transform a Gaussian beam to a uniform intensity profile is to truncate it, but this approach yields the poorest results.

There are several ways to convert a Gaussian beam into a uniform intensity distribution (in both one and two dimensions).

Achieving Uniform Illumination

The most simple and direct way to transform a Gaussian beam into a uniform intensity distribution is to pass the beam through an aperture which blocks all but the central, and most uniform portion of the beam (Figure 1). There are two disadvantages to this approach. First, a very large fraction of the laser power is discarded, as much as 75%. Second, the resulting beam still has a substantial falloff in intensity from the center to the edge. Additionally, other optics are often needed to clean up the beam by removing stray lobes produced by diffraction from the aperture edge.

Transforming a Gaussian beam to flattop without substantial light loss, therefore, requires some alternative technique which can redirect energy from the center to the edges of the distribution without simply blocking it. This can be accomplished with either diffractive or refractive techniques.

Figure 2. Two configurations of cylindrical lens arrays.

Diffractive optics offer a very powerful means for reshaping the Gaussian intensity distribution. Specifically, they can be used to produce virtually any arbitrary intensity profile, including nearly flattop, as well as a wide variety of patterns. The latter can include arrays of spots and lines, crosshairs, circles, concentric circles, squares, and so on.

Diffractive optics operate by creating interference between various diffracted orders to redistribute the incident intensity distribution. Of course, diffraction effects are by their very nature highly wavelength dependent, so a given component will only work over a narrow wavelength range. This wavelength sensitivity becomes particularly problematic when pairing diffractives with diode lasers because these have a relatively large wavelength bandwidth as compared to other laser types. Also, there are large unit-to-unit variations in the nominal output wavelength of laser diodes.

Diffractive optics also always put at least some light into unwanted diffraction orders. The simplest and lowest cost of diffractive optics for beam shaping is binary, etched gratings. Un fortunately, manufacturing tolerances in the type of optic usually result in a substantial decrease in efficiency due to this phenomenon; an overall efficiency of 70% would be considered excellent for a diffractive beam shaper. Similarly, the large and small scale (ripple) uniformity of the patterns produced with diffractive optics are limited by grating manufacturing tolerances. Finally, diffractive optics for creating two dimensional uniform distributions typically have a relatively limited working distance outside of which the desired intensity pattern will not be produced.

Figure 3. A Powell lens is a type of aspheric cylindrical lens.

Another quite different approach is to use cylindrical lens (Figure 2) arrays to construct a purely refractive beam shaping system. The incoming beam covers several of the lenslets, and the pattern from each overlaps in the far field, creating the desired uniform intensity distribution.

Cylindrical lens arrays are most frequently employed for homogenizing excimer lasers, which have a rectangular output beam that is well-matched to this approach. These types of arrays can also be used with round, Gaussian beams but in this case, they tend to produce patterns which are not highly uniform and usually have a substantial amount of high frequency ripple. The optical systems utilized with lenslet arrays usually have a limited working distance as well.

Flat-top Technology

Figure 4. The Powell lens efficiently transforms a Gaussian input into a flattop in one dimension.

To avoid such limitations, Coherent’s approach to transforming Gaussian beams into uniform, rectangular distributions is based on Powell lenses (Figure 3). The Powell lens is an aspheric cylindrical lens that purposefully aberrates a collimated Gaussian input beam so that the energy is efficiently redistributed from the beam center to the edges in the far field (which usually begins at 100mm from the last lens surface). This can be seen clearly in the ray trace diagram (Figure 4). Because a Powell lens is a type of cylindrical lens, it only homogenizes the beam in one dimension. So, for applications that require uniform radially symmetric intensity distributions, we use a patented combination of Powell optics with their cylindrical axes oriented at right angles to each other to achieve a uniform, two dimensional distribution. 

Figure 5. Two different Powell lenses can be used to transform the asymmetric output from a diode laser into a square or a rectangle having virtually any aspect ratio.

This approach delivers superior results over diffractive optics in almost every aspect of performance, especially when utilized with diode lasers. In particular, Coherent’s flat-top technology yields very high efficiency (over 90%), and produces a steep edged pattern, with little light outside the desired region. This configuration is also fairly insensitive to input wavelength, meaning it is unaffected by unit-to-unit variations in diode laser wavelength, as well as the inherent bandwidth and wavelength temperature dependence of these sources. The result is that an overall intensity uniformity of ±5% over the entire pattern can be routinely achieved in production beam homogenizers without having to wavelength select or bin diode lasers (Figure 5).

Diode laser homogenization and reshaping is also aided by the fact that a Powell lens only operates in a single dimension. Diode lasers typically exhibit very large divergence differences in orthogonal axes. As a result, two Powell lenses of differing characteristics can be used in the high divergence (fast) and low divergence (slow) axes to simultaneously achieve both optimum homogenization performance, and the desired beam dimensions in each.

Figure 6. Definitions of various flat-top system parameters.

Coherent’s flat-top technology is also flexible from a design standpoint, and can be readily adapted to meet specific requirements. For example, designs can be produced that deliver diverging (fan angle from 1° to 120°) beams, or incorporate additional lenses to produce focused or collimated output beams with patterns over a very wide range of aspect ratios. Systems can also be designed to work with varying input beam shapes and sizes, so as to mate with a user’s existing optical system.

Design Considerations

Virtually all Coherent flat-top optical systems are custom made to meet customers’ exact requirements. It’s useful to understand some of the basic design parameters for these systems so you may present engineers with all of the information necessary for them to create a flat-top optical system that will work perfectly in your application.

φi is the input beam diameter (at the 1/e² points)
FA is the fan angle output from the Powell Lens
E is the beam expansion power, i.e., the expander’s output beam diameter divided by its input beam diameter
f is the focal length of the final focusing lens
WD is the distance from the last optic to the image plane

The most important flat-top performance parameters determined by these variables are:

Depth of (intensity) uniformity specifically refers to the maximum intensity variation over the width of the focused, flat-top pattern.

Conclusion

Coherent’s flat-top technology provides a powerful means for creating highly uniform, rectangular focused patterns from round or elliptical Gaussian input beams. These optical systems are particularly useful with diode lasers because they deliver superior performance even if there are large unit-to-unit variations in source wavelength or changes in wavelength during device operation.

This article was written by Wallace Latimer, LASIRIS Product Manager, Coherent, Inc. (Santa Clara, CA). For more information, contact Mr. Latimer atWallace.latimer@coherent.comThis e-mail address is being protected from spambots. You need JavaScript enabled to view it, or visit http://info.hotims.com/40429-200.