## Abstract

A novel design method is presented for a simple laser beam shaper. Unlike earlier reports and designs based on the 2-element model, we prove it is possible to convert a laser beam from a non-uniform profile to a uniform flat-top distribution with one single aspherical lens.

© 2003 Optical Society of America

## 1. Introduction

Uniform spatial profiles across laser beams have always been desirable in laser applications such as holography, material processing, lithography, etc. where uniform illumination on targets is crucial. However, most laser devices typically produce Gaussian or similar beam profiles. To solve this problem, a number of proposals were put forward and some of them proved to be very successful [1–3]. A flat-top beam can be obtained by refractive, diffractive and even absorptive elements. Among these methods, refractive system has some advantages such as high efficiency, simple structure and less wavelength-dependence, which are essential for high power lasers. A widely-used refractive laser beam shaper consists of two separate aspherical lenses as shown in Fig. 1. Although this optical system is neither difficult to align or complex, a single lens system, if it exists and does the same job, will be much preferred, especially if it is mass produced. We will prove in this paper that a simpler, integrated system can be obtained by taking the alternate solution to the same 2-element model. In addition, the overall thickness of this kind of shaper can be minimized, which is highly desired for ultrashort pulse applications.

## 2. Basic concept and analysis

Figure 1 is the conventional design proposed by Shealy [3] and Frieden [4]. Two aspherical lenses are coaxially placed at a certain distance apart. The collimated input rays are refracted on the first lens and then recollimated by the second lens. Since the rays near the axis experience larger radial magnification than those near the edge, the irradiance across the beam is non-linearly redistributed and a uniform flat-top profile is produced.

In order to give a clear description about our idea, it is necessary to briefly go through the basic equations. Mathematical details can be found in Refs. [1, 3]. Within the geometrical model described here, in order to have a uniform collimated output spatial profile and maintain the original wavefront, the following conditions have to be met. The energy density at the output must be a constant and all rays must maintain the same optical path length (OPL).

Assume the whole system is rotationally symmetric, a ray, starting from point A through the first lens with refraction index *n*
_{1}, as shown in Fig. 2, propagates through a medium (refraction index *n*
_{0}) and is refracted again at the curved surface of the second lens (refractive index *n*
_{2}). If the input energy density is *E*_{i}
(*r*) and the maximum beam radius is *r*
_{0}, then it follows from the energy conservation,

where *E*_{o}
is the output energy density which is a constant and will be decided by the system magnification factor M=R/r. Applying Snell’s Law at both refractive surfaces and performing ray transform yields the following differential equation,

$$\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}-{\left(z\text{'}\right)}^{2}\left(1-{\gamma}_{1}^{2}\right)\left[{\left(R-r\right)}^{2}+{\left(Z-z\right)}^{2}\right]$$

$$\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}-z\text{'}\left[2\left(R-r\right)\left(Z-z\right)\right]-{\left(R-r\right)}^{2}=0$$

where *z*'=*dz*/*dr*, *γ*
_{1}=*n*
_{1}/*n*
_{0}. The constant optical path requires the OPL of any arbitrary ray equals to the OPL of the central ray, which leads to

$$={n}_{1}z+{n}_{0}{\left[{\left(R-r\right)}^{2}+{\left(Z-z\right)}^{2}\right]}^{\frac{1}{2}}+{n}_{2}\left({Z}_{0}+{t}_{2}-Z\right)$$

It can be solved to give

$$\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}+\left({n}_{2}^{2}-{n}_{0}^{2}\right){\left(R-r\right)}^{2}{]}^{\frac{1}{2}}\}$$

Similar calculation on the curved surface of the second element at point C(R, Z) results in

$$-{\gamma}_{2}{\left[1+{\left(z\text{'}\right)}^{2}\left(1-{\gamma}_{1}^{2}\right)\right]}^{\frac{1}{2}}\}$$

where *Z*'=*dZ*/*dr*, *γ*
_{2}=*n*
_{0}/*n*
_{2}. The solutions to Eq. (1) through Eq. (5) will give the exact profiles for the 2 surfaces on the first and the second lens. Equation (2) can be further simplified to a quadratic equation which has following roots,

$$\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}/\left[\left({\gamma}_{1}^{2}-1\right){\left(Z-z\right)}^{2}+{\gamma}_{1}^{2}{\left(R-r\right)}^{2}\right]$$

Eq. (6) is a differential equation and can be solved numerically.

The plus and minus signs in Eq. (6) represent the different configurations. Apparently the rays are expected to be divergent after the first lens. Assuming the first refractive surface is convex and the second one is a concave to the incoming rays, then the positive root has to be taken in Eq. (6) and also the plus sign for Eq. (4) under the condition that *n*
_{1}>*n*
_{0}<*n*
_{2} holds. This means that rays get defocused at the first surface and recollimated because of the focusing effect at the second refractive surface. This scenario was taken for granted and has been, to the best of our knowledge, the basis for this kind of beam reshaping systems ever since it was proposed.

The negative root in Eq. (6) indeed does not make sense in a configuration mentioned above where *n*
_{1}>*n*
_{0}<*n*
_{2} has been assumed. But it does represent the right solution in the case of *n*
_{1}<*n*
_{0}>*n*
_{2}. If *n*
_{1}=*n*
_{2}=1,*n*
_{0}>1, then the whole picture will be exactly the same as rays coming from air, entering and refracted by a lens (with refractive index *n*
_{0}) before exiting into air again. In this case the minus sign in Eq. (4) has to be taken, corresponding to a convex surface which is illustrated in Fig. 2 with dotted lines. Fig. 3 is a clearer picture for the single-lens system that converts the non-uniform input spatial distribution into a uniform flat-top one.

## 3. Calculations and evaluation

Numerical calculations have been performed to verify this idea. In principle the input beam spatial profile can be arbitrarily chosen. More specifically, we take the initial input to be a Gaussian beam shape expressed by *E*_{i}
(*r*)=exp(-2*r*
^{2}/${r}_{0}^{2}$), and *r*
_{0} is the beam radius where the irradiance falls to *e*
^{-2} of its maximum. Fused silica is taken as the lens substrate and all other parameters have been listed in the graphs. The calculated lens surface profiles along with the ray trace are shown in Fig. 4. The irradiance distribution before and after the shaping system can be seen in Fig. 5. A non-uniform input beam profile has been converted to a uniform flat-top profile at reduced irradiance which is determined by the beam magnification. The wavelength for all calculations is 527nm.

It is expected the performance of the single-lens system will resemble the 2-lens system. To look further into the issue of performance, ray tracing was done to examine the profile variation caused by actual deviations from the ideal input beam. Results can be seen in Fig. 6. Shown are three Gaussian input profiles and one described by sec *h*
^{2}(*a*_{s}*r*/*r*_{s}
) (*r*_{s}
is the beam radius at *e*
^{-2} of maximum, *a*_{s}
=1.657). When the two input profiles are close to each other (as denoted by GSN2 and Sech2 in Fig. 6), the deviation at the output is very small across the whole aperture. The inputs with large deviation from the desired input profile results in obvious non-uniformity at the output. But even with large variation, the output uniformity within a certain radius is still superior to the beam resulting from a Gaussian profile cut by a hard aperture.

One concern over the single-element system is that the lens thickness may be considerably larger than that for a 2-element system. This is undesirable for ultrashort pulse lasers although it is not critical for picosecond or longer laser pulses. Our calculations show that the overall material length can be reduced to a minimum that may not be realized with a 2-element configuration. Figure 7 shows a lens design with a thickness of only about 2mm over a 8mm diameter aperture. Given the same input beam, although the thickness of each lens in a 2-element system can also be minimized within 2mm at the cost of increasing the lens separation, the overall thickness of two lenses always tends to be greater than or about twice as much as that of a single lens due to the minimum thickness of each lens required during optical fabrication. This makes big difference in the case of very short pulses.

Further calculations in regard to other issues such as optical and mechanical tolerances and diffraction effects induced by the lens edge will be helpful, but beyond the scope of this paper. A well-designed beam shaper will be able to work effectively with fairly good tolerance. The past decade has seen dramatic advances in the fabrication of precision optics. The current manufacturing capability allows asphere profile tolerance to be within ±1micron. This will ensure the aspherical shapers work as designed with minimal distortions to the beams.

## 4. Summary

We have proven, through theoretical analysis and numerical calculations, that a single-element beam shaper can be obtained by taking the alternate solution to the well-established 2-element model. This has obviously extended the capability of the current method and made it possible to incorporate the design of both 2-element and single-element shapers into one simple and fast calculation program with high precision, which constitute the primary advantages over other methods [5–6] for such beam shapers.

In addition, we presented analysis on profile variations and thin lens design which are important to real applications. Such a beam shaper simplifies the overall structure in comparison with the commonly-used 2-element systems and therefore provides great ease in alignment and reduction of cost. A very thin shaper is possible through careful design and can be used in case where ultrashort laser pulses are needed. Like the aspherical lenses widely used for fiber optics, this kind of beam shaper has the potential to be mass produced.

## Acknowledgement

Discussions with S. Benson and F. Dickey are gratefully acknowledged. This work is supported by the Office of Naval Research, the Joint Technology Office, the Commonwealth of Virginia, the Air Force Research Laboratory, and by DOE Contract DE-AC05-84ER40150.

## References

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**3. **W. Jiang and D. L. Shealy, “Development and testing of a laser beam shaping system,” in Laser Beam Shaping, F. M. Dickey and S. C. Holswade, eds., Proc. SPIE **4095**, 165–175 (2000). [CrossRef]

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**6. **S. R. Jahan and M. A. Karim, “Refracting systems for Gaussian-to-uniform beam transformations,” Opt. Laser Technol. **21**, 27–30 (1987). [CrossRef]