Melles Griot Optics Guide - Transformation and Magnification by Simple Lenses

Gaussian Beam Propagation

Transformation and Magnification by Simple Lenses

It is clear from the previous discussion that Gaussian beams transform in an unorthodox manner. Siegman uses matrix transformations to treat the general problem of Gaussian beam propagation with lenses and mirrors. A less rigorous, but in many ways more insightful, approach to this problem was developed by Self (S. A. Self, "Focusing of Spherical Gaussian Beams,"). Self shows a method to model transformations of a laser beam through simple optics, under paraxial conditions, by calculating the Rayleigh range and beam waist location following each individual optical element. These parameters are calculated using a formula analogous to the well-known standard lens formula.

The standard lens equation is written as

where s is the object distance, s " is the image distance, and f is the focal length of the lens. For Gaussian beams, Self has derived an analogous formula by assuming that the waist of the input beam represents the object, and the waist of the output beam represents the image. The formula is expressed in terms of the Rayleigh range of the input beam.

In the regular form,

In the normalized form,

In the far-field limit as z_R approaches 0 this reduces to the geometric optics equation. A plot of s / f versus s" / f for various values of z_R / f is shown in the figure below. For a positive thin lens, the three distinct regions of interest correspond to real object and real image, real object and virtual image, and virtual object and real image.

The main differences between Gaussian beam optics and geometric optics, highlighted in such a plot, can be summarized as follows:

There is a maximum and a minimum image distance for Gaussian beams.
The maximum image distance occurs at s = f + z_R, rather than at s = f.
There is a common point in the Gaussian beam expression at s / f = s"/ f = 1. For a simple positive lens, this is the point at which the incident beam has a waist at the front focus and the emerging beam has a waist at the rear focus.
A lens appears to have a shorter focal length as z_R / f increases from zero (i.e., there is a Gaussian focal shift).

Plot of lens formula for Gaussian beams with normalized Rayleigh range of the input beam as the parameter

Self recommends calculating z_R, w₀, and the position of w₀ for each optical element in the system in turn so that the overall transformation of the beam can be calculated. To carry this out, it is also necessary to consider magnification: w ₀"/w₀. The magnification is given by

The Rayleigh range of the output beam is then given by

All the above formulas are written in terms of the Rayleigh range of the input beam. Unlike the geometric case, the formulas are not symmetric with respect to input and output beam parameters. For back tracing beams, it is useful to know the Gaussian beam formula in terms of the Rayleigh range of the output beam:

Beam Concentration

The spot size and focal position of a Gaussian beam can be determined from the previous equations. Two cases of particular interest occur when s = 0 (the input waist is at the first principal surface of the lens system) and s = f (the input waist is at the front focal point of the optical system). For s = 0, we get

and

For the case of s = f, the equations for image distance and waist size reduce to the following:

and

Substituting typical values into these equations yields nearly identical results, and for most applications, the simpler, second set of equations can be used.

In many applications, a primary aim is to focus the laser to a very small spot, as shown in the figure below, by using either a single lens or a combination of several lenses. Melles Griot has designed a series of single lenses optimized for this specific purpose.

Concentration of a laser beam by a laser-line focusing singlet

For example, by using a 25 LHR 151 helium neon laser and a focusing singlet, 01 LFS 033, the formula should be modified as follows:

The factor 4/3 arises because of the careful balance of spherical aberration and diffraction designed into the singlet. The ratio f/w is proportional to lens f-number, but is not equal to it.

If a particularly small spot is desired, there is an advantage to using a well-corrected high-numerical-aperture microscope objective to concentrate the laser beam. The principal advantage of the microscope objective over a simple lens is the diminished level of spherical aberration. Although microscope objectives are often used for this purpose, they are never designed for use at the infinite conjugate ratio. Suitably optimized lens systems, which Melles Griot can design and build on special request, are more effective in beam-concentration tasks.

Depth Of Focus

Depth of focus (�Dz ), that is, the range in image space over which the focused spot diameter remains below an arbitrary limit, can be derived from the formula

The first step in performing a depth-of-focus calculation is to set the allowable degree of spot size variation. If we choose a typical value of 5%, or w₀, = 1.05w ₀, and solve for z = Dz, the result is

By applying this result to the combination of the 05 LHR 151 laser and laser-line focusing singlet 01 LFS 033, we find

Since the depth of focus is proportional to the square of focal spot size, and focal spot size is directly related to f-number, the depth of focus is proportional to the square of the f-number of the focusing system.

Truncation

In a diffraction-limited lens, the diameter of the image spot is

where K is a constant dependent on truncation ratio and pupil illumination, l is the wavelength of light, and f/# is the speed of the lens at truncation. The intensity profile of the spot is strongly dependent on the intensity profile of the radiation filling the entrance pupil of the lens. For uniform pupil illumination, the image spot takes on the Airy disc intensity profile illustrated below.

Airy disc intensity distribution at the image plane

If the pupil illumination is Gaussian in profile, the result is an image spot of Gaussian profile, as shown below.

Gaussian intensity distribution at the image plane

When the pupil illumination is between these two extremes, a hybrid intensity profile results.

In the case of the Airy disc, the intensity falls to zero at the point d_zero = 2.44 x l x f/#, defining the diameter of the spot. When the pupil illumination is not uniform, the image spot intensity never falls to zero making it necessary to define the diameter at some other point. This is commonly done for two points:

d_FWHM = 50% intensity point

and

d_1/
e² = 13.5% intensity point.

It is helpful to introduce the truncation ratio

where D_b is the Gaussian beam diameter measured at the 1/e² intensity point, and D_t is the limiting aperture diameter of the lens. If T = 2, which approximates uniform illumination, the image spot intensity profile approaches that of the classic Airy disc. When T = 1, the Gaussian profile is truncated at the 1/e² diameter, and the spot profile is clearly a hybrid between an Airy pattern and a Gaussian distribution. When T = 0.5, which approximates the case for an untruncated Gaussian input beam, the spot intensity profile approaches a Gaussian distribution.

Calculation of spot diameter for these or other truncation ratios requires that K be evaluated. This is done by using the formulas

and

The K function permits calculation of on-axis spot diameter for any beam truncation ratio. The chart below plots the K factor vs T(D _b/D_t).

K factors as a function of truncation ratio

The optimal choice for truncation ratio depends on the relative importance of spot size, peak spot intensity, and total power in the spot as demonstrated in the table below. The total power loss in the spot can be calculated by using

for a truncated Gaussian beam. A good compromise between power loss and spot size is often a truncation ratio of T = 1. When T = 2 (approximately uniform illumination), fractional power loss is 60%. When T = 1, d _1/e² is just 8.0% larger than when T = 2, while fractional power loss is down to 13.5%. Because of this large savings in power with relatively little growth in the spot diameter, truncation ratios of 0.7 to 1.0 are typically used. Ratios as low as 0.5 might be employed when laser power must be conserved. However, this low value often wastes too much of the available clear aperture of the lens.


Spot Diameters and Fractional Power Loss for Three Values of Truncation
Truncation Ratio	d_F_W_H_M	d₁_/_e²	d_z_e_r_o	P_L(%)
Infinity	1.03	1.64	2.44	100
2.0	1.05	1.69	-	60
1.0	1.13	1.83	-	13.5
0.5	1.54	2.51	-	0.03

Spatial Filtering

Laser light scattered from dust particles residing on optical surfaces may produce interference patterns resembling holographic zone planes. Such patterns can cause difficulties in interferometric and holographic applications where they form a highly detailed, contrasting, and confusing background that interferes with desired information. Spatial filtering is a simple way of suppressing this interference and maintaining a very smooth beam irradiance distribution. The scattered light propagates in different directions from the laser light and hence is spatially separated at a lens focal plane. By centering a small aperture around the focal spot of the direct beam, as shown in the illustration below, it is possible to block scattered light while allowing the direct beam to pass unscathed. The result is a cone of light that has a very smooth irradiance distribution and can be refocused to form a collimated beam that is almost equally smooth.

Spatial filtering smooths the irradiance distribution

As a compromise between ease of alignment and complete spatial filtering, it is best that the aperture diameter be about two times the 1/ e² beam contour at the focus, or about 1.33 times the 99%

throughput contour diameter.

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