Problem Statement:

An optical system requires a beam expander to increase the diameter of a 1550 nm Gaussian laser beam by a factor of five. The beam expander must fit within a maximum optical length of 75 mm.

Objective:

Our objective is to design a 2-element Galilean (has a shorter track lenght and with a lasr beam, it is better as it does not have a internal focal point) Beam Expander which expands the 3mm wide gaussian beam to 5 times while maintaing the wavefront quality (quantified by wavefront aberrations and rms spot diagram). We assume that the input beam is at its waist (i.e., collimated with R-> inf) when enterign the first lens.

Solution:

Starting with first order solution:

A Galilean Beam Expander has a negative lens (lens 1 ) and a positive lens (len 2) and the distance between the two lenses is f1 + f2 (here f1 is negative). Thus, f1 + f2 < 75 mm and f2/f1 = -M. Using the two equations, we come to the following constraints:

f1 < 75 mm / (1-M) or |f1| < 75mm / (M -1) => |f1| < 75/4 => |f1| < 18.75 mm

and thus, f2 < M*|f1| mm

If we have a list of available lenses to compare from, the above focal lenghts can be chosen from there. Otherwise, here we start from f1 = -15 mm and f2 = 74 mm.

We start with plano-convex/concave lenses since they are bound to give the minimum spherical aberrations (we will face the plano side towards the input beam). Using the first order thin lens formula, we can predict the radii of the two lenses. The thickness is taken to be 10% of the radii.

For the diameter of the two lenses, we take the first lens to be 6 mm (so that the entire 3 mm beam can go through) and the second lens to be 30 mm (for the same reason but for the output beam). Now, the system is designed in our custom sequential ray tracer. Once the initial system is designed, we trace the rays in the beam and plot the spot diagram and wavefront error. Since this is afocal configuration, the spot diagram is not spots, instead they are the angles of each ray, and rms deviation would be angular rms deviation. Also, for the same reason, the wavefront error is calculated wrt a planar reference.

The PV wavefront error is ~9 waves, very large. We need to optmize the lens design.

Optimization:

To optimize, we decide the following degree of freedoms that can be moved by the optmizer:

1. Back radius of the first concave lens

2. Back radius of the second concave lens

3. distance betwen the two lens

The operands we use - conditions/ constraints we need to achieve/follow are magnification (here we have calculated the magnification from system's ABCD matrix) of 5.0, divergence to be minimized (afocal behavior). Here divergence is RMSangular spread of the output rays, and wavefront error to be minimized. We also keep the distance betwen the two lens to be bounded by a range (to make sure it is less than the maximum track length given to us as design criteria). Once optimization is done, a trace is run again. Folliwing are the results for that.

After optimization, the peak to valley wavefront error has not become 0.0922 waves - less than 0.25 waves (lambda/4). This is now a diffraction limited design. The rms angular divergence is also 0.06 mrad, smaller than the airy radius in terms of angle. The maximum beam diameter (using the maximum distance a ray traces to on the last surface) for output beam and input beam gives a ratio of 5.10X.

Note the optimization here might change if the focal lenghts of the lens is from available lenses - they would no longer be design variables.

The layout and the resultant lens design is shown below.

Optimized beam expander final design:

Sensitivity and Monte Carlo Tolerance Analysis

Let's finish this study with a sensitivity and tolerance analysis to see which alignment and fabrication degree of freedom has the most impact on the performance (wavefront error and divergence) of this beam expander.

In sensitivity analysis each degree of freedon is changed within a range to see the effect on the two performance criteria. This analysis identifies the design parameters to which optical performance is most sensitive, helping determine which parameters require tighter manufacturing and alignment tolerances.

The sensitivty analysis is shown below. It shows that all three - wavefront error, rms spot (angular) and divergence are most sensitive to the two curved radii and the refractive index of the second lens followed by the gap between the lens.

In Monte Carlo tolernace analysis, all the alignment and fabrication degree of freedoms are randomly chosen from a normal distrbution, thus all the errors might be present (closer to real scenario), and then each of those random samples is judged to eitehr fail or pass on the basis of perfomance criteria. Here we have chosen rms wavefront error to be less than 0.07 waves (marechal condition) and divergence to be less than 0.2 mrad (calculated using divergence of gaussian beam with waist equal to the output beam diameter). The tolerance analysis without compensation mechanism shows that only 34% of the total samples would pass the performance criteria we have set.

Under compensation mechanism, however, most of that yield loss can be recovered. Here the compensation mechanism we have chosen is the lens spacing. The lens spacing when compensating had a maximum value of 0.4 mm with 95% samples being under 0.287 mm.

Concluding Remarks:

A 5× Galilean beam expander was designed and optimized for a 1550 nm laser using a custom sequential ray tracer. The optical system consists of a negative plano-concave lens followed by a positive plano-convex lens, both made from N-BK7 glass.

First-order calculations were used to calculate the initial lens focal lengths, surface curvatures, and the distance between the lens. These parameters were then refined using a damped least-squares optimization algorithm, where the lens curvatures and lens spacing were used as design variables. The optimization target was to maintain the magnification 5X while minimizing output ray divergence and wavefront aberrations. Sensitvity Analysis showed the system performance was most sensitive to the lens' curved radii and the interspacing. Monte Carlo Analysis showed that under a compensation mechanism, all the wavefront error casued due to alignment and fabrication errors can be recovered, and the yield can increase from 34% to 99%.

The optimized system achieved:

5.00× paraxial beam expansion, with a traced output-to-input beam diameter ratio of approximately 5.10×

65 mm total optical track, satisfying the 75 mm packaging constraint

0.0611 mrad RMS angular spot radius, reduced from 2.6327 mrad.

0.0269 waves RMS wavefront error, reduced from 2.6908 waves.

Furthermore the monte carlo tolerance and sensitivity analysis showed that the sytem was most sensitive to fabrication tolerances and interlens spacing, and that under compensation mechanism, the performance yield can increase from 34% to 99%.

The final RMS wavefront error is << λ/4, indicating good geometrical wavefront quality.

This project demonstrates the integration of first-order optical design, exact ray tracing, numerical optimization, sensitivity/tolerance analysis and wavefront analysis for an afocal laser optical system.