## Abstract

An anamorphic zoom lens is designed for altering the size and aspect ratio of a laser beam. The design is based on using a Galilean beam expander in which the optical elements consist of two rotating cylinders. The rotation of the cylinders allows the spherical and cylindrical powers of the rotating cylindrical lens to be altered in such a manner that the magnification is anamorphic. A simple anamorphic zoom lens based on rotating cylindrical lenses is designed and then implemented. Operation of the anamorphic lens demonstrates a 1 to 1 to 10 to 1 aspect ratio beam conversion.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Laser beam shaping has many applications in laser machining, lithography, welding and biomedical industries [1,2]. A wide variety of methods can be used for shaping laser beams, depending on the desired outcome. The simplest method of circularizing has been demonstrated using two cylindrical lenses to collimate the fast (high divergence angle) and slow (low divergence angle) axes of a laser with an elliptical beam [3]. Another simple method is the use of anamorphic prisms to stretch one axis of a collimated laser while compressing the other axis [4]. These methods change the geometric shape of the beam but do not change the power distribution of the laser beam.

In many applications it is desirable to redistribute the gaussian laser beam profile into a top hat or other distribution specific to the application. Methods of beam shaping include highly aspheric optical elements for power redistribution, use of lenslet also called fly’s eye integrators, and diffractive optical elements. While these methods are effective, they are often subject to difficulty in fabrication and alignment [1,2,4,5]. Another method for generating a specific laser beam profile is the use of specialized fiber optics [4–6]. In particular square core fibers have shown excellent beam uniformity for laser machining and plastic welding [7,8]. Using square fiber optics to generate a uniform profile eliminates the needs for the beam shaping methods previously mentioned.

What appears to be missing for the laser beam shaping industry is an anamorphic zoom lens for changing the aspect ratio and magnification for laser welding and machining [9]. A few researchers have demonstrated using anamorphic optics and microlens to generate various beam shapes and alter uniformity [5,10]. However, this system does not offer variable anamorphic magnification properties. To address this problem, the author explored the anamorphic zoom lenses. Most of the anamorphic zoom lenses are used in cinematography and are fairly expensive [11]. Based on the idea of a Jackson Crossed Cylinder, the author employed the concept of rotating crossed cylinders as a way of controlling the spherical and astigmatic (or cylindrical power), allowing for the anamorphic magnification to be altered while maintaining image focus [12,13]. Undergoing further research the author discovered another group of researchers were using rotating toroidal optics to achieve a zoom lens in a similar manner. Their work focused primarily on achieving a compact lens for imaging applications by placing two sets of rotating toroidal lens at a fixed separation. Variable magnification is adjusted by rotation of the toroidal lens. Their work focused primarily on achieving demonstrated a variable magnification lens. with good image quality, with minimal distortion and anamorphism. Their publication validated the idea of using rotational cylindrical optics to create a variable magnification lens [14]. In this paper the author uses the same principle of rotating cylindrical to demonstrate a design capable of generating anamorphic magnifications for laser beam shaping of a square core fiber, specifically generating lines with variable aspect ratios and magnification. Maintaining low anamorphic distortion and magnification is not required in this application. Allowing the spacing of rotating toroidal lens allows for a wider range of magnifications and anamorphic ratios. The mathematical treatment variable anamorphic optical magnification is simplified using power vector notion described in [12] rather than the complex derivation of the instantaneous surface power in the tangential and sagittal orientation shown in [14]. A design is simulated and then implemented. Several zoom configurations are implemented to demonstrate operation of the lens system.

## 2. Theory of operation

The anamorphic zoom lens relies on two simplistic lens forms discussed in texts on geometrical optics and optical design such as [15]. The first form is a two lens relay, Fig. 1, with the object at the back focus, *f*_{1}, allowing the rays to be collimated between lens 1 and lens 2. The image is formed at the front focal length, *f*_{2}.

The magnification, *M*, of this relay is given by

The second form is called a Galilean telescope or beam expander, Fig. 2, and consists of a positive and negative lens. This system is afocal when the distance between the lens is equal to the sum of the focal lengths of the individual lenses. The magnification of this lens system is given by Eq. (2).

If a Galilean beam expander is designed using cylindrical lens the beam will be expanded or magnified in only one axis. Placing the cylindrical beam expander in the collimated space of the first relay we can achieve anamorphic magnification. The combination of both systems allows for the construction of an anamorphic lens where the image and object conjugate points are in the same location for either axis, Fig. 3. The magnification of the system along the *x* and *y* axis is given by Eqs. (3a) and (3b). This simple anamorphic design form allows for the generation of a uniform line image of a uniform square source such as a fiber discussed by [8].

This form can be converted into a zoom or variable magnification lens by replacing the elements of the Galilean beam expander with a pair of rotating cylindrical lenses. To discuss how this yields a system of variable magnification, we need to discuss the application of power vectors, a convenient way to describe the cylindrical and spherical power of a toric lens. Power vector notation uses the spherical power, *S*, cylindrical power, *C*; θ is the rotation of the cylinder from the horizontal axis to mathematically describe the power of a lens. From these quantities we can calculate the following: *P*, spherical equivalent power; and *J*0 and *J*45, two crossed cylindrical lenses oriented at 0 and 45 degrees respectively [12,13].

The use of power vectors allows us to determine the power of the toric lens in terms of reduced spherical power *S _{R}*, reduced cylindrical power

*CR*and reduced axis θ

*.*

_{R}Given two thin cylinder lenses of equal power *C*, rotated in equal, but opposite directions $\theta $, the two lenses can be written in terms of crossed cylinders $J0$ and $J45$ and spherical equivalent power *P*, shown in Table 1.

Using the trig properties for negative angles and adding the columns for the crossed cylinders and spherical equivalent yield the results in Table 2.

Finally, we can use the formulas from the power vector formulas to determine the net prescription.

To verify and demonstrate this principle, two +10D cylindrical thin lenses rotating oppositely are modelled in Zemax (an optical ray tracing program). A 3mm pupil is used for the simulation. Spherical and cylindrical power are calculated by tracing paraxial rays along the *x-* and *y-*axis. The *S _{R}* and

*C*power terms can then be calculated by using the following equations

_{R}*u*is the margin ray angle in the

_{mx}*x*-axis and

*x*is the ray height at the last optical surface in the along the

*x*-axis.

*u*and

_{my}*y*terms are equivalent terms along the

*y-*axis.

The spherical and cylindrical power are computed as function of rotation angle and compared to Eqs. (10) and (11). Figure 5 shows excellent agreement between the results of Eqs. (10) and (11) and ray tracing.

To further demonstrate the principle of power vectors as it related to optical design, Fig. 6 shows the spot size as a function of lens orientation. The *S _{R}*,

*C*and θ are given for each orientation. It should be noted that in Fig. 6 θ refers to the rotation angle of the lenses and not the orientation of astigmatism denoted by θ

_{R}*. As the lenses rotate by θ, the amount of astigmatism generated by the cylindrical pair varies which is demonstrated in the spot diagram images. In the top configuration a line focus is generated showing only astigmatism,*

_{R}*C*= 20D, being generated with and no spherical power,

_{R }*S*= 0D. The middle shows a spot that consists of a combination of cylindrical and spherical power. The final configuration shows a completely rotationally symmetric spot that is defocused due to the elimination of astigmatism,

_{R}*C*= 0D, the reduced net spherical power

_{R}*S*= 10D. As demonstrated by the plot in Fig. 5 and the amount of astigmatism and spherical power is a function of θ which is easily visualized by the spot diagrams in Fig. 6.

_{R}## 3. Design and implementation of anamorphic zoom lens

An anamorphic zoom lens is designed for a 0.4 mm square core fiber with a 0.2 NA. The design required the square to have 9× magnification and demonstrate a change in aspect ratio of 10×. The completed design of the lens is shown in Fig. 7. The rotation of the cylindrical lens pairs changes the sph and cyl power of the lens in the Galilean beam expander resulting in a change in lens spacing and lens working distance. In effect the spherical power of the collimator and negative cylinder group are added together. The same effect takes place for the positive cylinder group and the focusing lens. This generates the total magnification for the system. The change in the cylindrical power due to the rotation of the cylinders results in a change in the aspect ratio anamorphic or magnification depending on one’s preference in terminology.

An image simulation of a uniform 0.4 mm square is shown for various anamorphic zoom ratios (Fig. 8). The simulated images show sharp corners and edges, indicating there is not a significant roll off effect due to aberrations induced by the movement and rotation of the lenses.

The lens in Fig. 7 was constructed using stock optics to demonstrate proof of concept. The pilot beam (visible aiming beam) for a high power laser is used. With careful and painstaking alignment and much patience we obtained images of various aspect ratios shown in Fig. 9. System boresight error and mismatches in rotation angles of the cylindrical lens resulted in a parallelogram shape rather than a perfect rectangle.

## 4. Conclusion

The combination of two simple optical systems allows for the design and construction of an anamorphic zoom lens for laser beam shaping. Rotation of the cylinder allows for the changing of *x* and *y* magnifications while maintaining image focus, which is important in laser welding and machining applications. The concept has been successfully constructed and has demonstrated the ability to change magnification and aspect ratio of laser beams.

## Acknowledgments

The author would like to thank the following people: Scott Cannon of Sakar Technologies who expressed the need and importance of an anamorphic zoom lens for laser beam shaping for the laser welding/ machining industry; Jim Schwiegerling PhD of University of Arizona College of Optical Sciences for verification of the cylindrical and spherical equivalent power calculations as a function of angle; and Elise Hastie, intern for Optical Engineering & Analysis, who painstakingly aligned the zoom lens to generate experimental data for this publication.

## Disclosures

The author declares no conflicts of interest.

## Data Availability

No data were generated or analyzed in the presented research.

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