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Klopfenstein has shown how to produce an optimal graded- impedance profile which is minimally reflective within the pass band for a given length of taper and minimum length of taper for a given maximum in-band reflection.¹

By converting impedance to index of refraction, we may produce an optimal graded-index profile for the transition between empty space and the material of our filter.²

This project focuses on the application of CMB detection, but this process may be used to carry out the design of a cryogenically robust ARC for a wide range of applications. MUSTANG2 is the first millimeter-wave detector to use a laser- ablated SWS ARC modeled off a Klopfenstein taper.

Designing Sub-Wavelength Structures as a Cryogenically Robust Anti-Reflection Coating

James Gómez Faulk,¹ Scott Cray,² Shaul Hanany²�¹Department of Physics, Case Western Reserve University, Cleveland, OH, USA

²School of Physics and Astronomy, University of Minnesota, Minneapolis, MN, USA

This work was supported partially by the Research Experiences for Undergraduates (REU) Program of the National Science Foundation under Award Number PHY-2348668

Background and purpose

Translating index of refraction to physical shape

Physical structure and predicted reflection

Conclusion and working SWS ARC example

Designing an optimal graded-index profile

This project outlines the process of designing sub-wavelength structures which behave as an optimal taper between media of different refractive indices to create a cryogenically robust anti-reflective coating (ARC).

The Cosmic Microwave Background (CMB) is relic radiation from the recombination era of the Big Bang, which we observe to glean insights about the history of the Universe.  

CMB detection requires a filter

which transmits light of relevant

wavelengths and reflects or absorbs light of irrelevant wavelengths.  To combat reflective loss of light relevant to the experiment, a well-designed filter includes an anti-reflection coating (ARC) which maximizes transmission of desirable wavelengths by providing a gradual transition of refractive index. 

Performing CMB experiments at cryogenic temperatures minimizes thermal emissions from equipment and increases detector sensitivity.  Because many anti-reflection coatings consist of layers that may expand and contract at different rates during heating and cooling, they are not suitable for cryogenic applications. 

Single-piece ARC are resistant to damage caused by thermal expansion. Moths’ eyes exhibit sub-wavelength structures (SWS) which are a natural example of a single-piece ARC. Wavelengths of light larger than the period of the SWS “see” the structures as a gradual transition of refractive index.�

References

¹R. W. Klopfenstein, "A Transmission Line Taper of Improved Design," in Proceedings of the IRE, vol. 44, no. 1, pp. 31-35 (Jan. 1956)

²Eric B. Grann, M. G. Moharam, and Drew A. Pommet, "Optimal design for antireflective tapered two-dimensional subwavelength grating structures," J. Opt. Soc. Am. A 12, p. 333-339 (1995)

³Ralf Bräuer and Olof Bryngdahl, "Design of antireflection gratings with approximate and rigorous methods," Appl. Opt. 33, 7875-7882 (1994)

⁵Ryota Takaku et al. "Large diameter millimeter-wave low-pass filter made of alumina with laser ablated anti-reflection coating," Opt. Express 29, 41745-41765 (2021)

​

There is no closed-form solution

which relates the physical shape

of SWS to effective refractive

index. However, two-dimensional effective medium theory (2-D EMT)³ provides an approximation which closely agrees with experimental results.³ The Klopfenstein taper yields optimal refractive index as a function of depth. 2-D EMT provides a relation between pyramid width and effective index, which enables us to design a physical structure that yields optimal behavior.

By increasing the resolution, we are able to design a smooth pyramid that displays the behavior of the Klopfenstein taper.

A sample pyramid designed for parameters specific to CMB detection

Reflection behavior of the Klopfenstein taper realized by the sample pyramid

Design and validation of MUSTANG2 filter transmission behavior

With the process shown in this project, one may design an ARC that behaves as a high-pass filter with any maximum in-band reflection coefficient and any lower bound of transmitted frequencies.

Acknowledgments

I would like to thank my advisor, Dr. Shaul Hanany, and my mentor, Scott Cray, for their guidance, support, and endless patience. Special thanks as well to Dr. Alex Kamenev, Alicia Canfield, and Dr. Foster Thompson for making the program possible. Thanks to the School of Physics and Astronomy at the University of Minnesota.