Analytic HG-mode propagation through circular apertures with Zernike phase offset

Abstract

Mode expansion methods (MEMs) have been used in both the scientific and commercial world to accurately propagate monochromatic light fields. Compared to classical methods for light propagation from scalar diffraction theory, it is often less costly and much more easily implemented into optical chains. We provide an analytic expression for the overlap between incoming and outgoing Hermite Gauss (HG) TEM modes over an aperture with potential wavefront error represented by Zernike terms for the purpose of propagating light through apertures. We show agreement between our results and those of prior studies of the topic, as well as an application of this method to an ongoing interferometry mission.

Wavefront error based tilt-to-length noise analysis for the LISA transmitted beam

“…The mission DFACS system constrains DC pointing offset to 10 nrad and in band misalignment jitter J to 10 nrad √ Hz −1 [1,21], so that this roll term would be suppressed by an order of 10 −8 relative to other terms above. The calculation of these components in the FF given some initial WFE written as a sum of Zernike polynomials (orthogonal polynomials over the unit disk ideal for representing phase aberrations over an aperture [15]), is performed via Hermite Gauss (HG) modal decomposition [16]. Appendix B more thoroughly explains how this, and the Fresnel diffraction integrals of 2.2, are utilized in TTL calculations.…”

“…As mentioned, TTL estimates of this paper are all initially extracted from fields found with an HG modal decomposition based propagation scheme [16]. These utilize an expansion for the initial outgoing beam:…”

“…We tested for 1123 randomly generated 10 nm RMS WFE 8 the linear and second order TTL approximations against actual TTL obtained from FF propagation based on HG modal decomposition [16,17]. These simulations are binned in histograms of figure 4 by greatest error in the magnitude of TTL θ components recovered (φ component errors are nearly identical) over the FF grid of receiving SC positions.…”

“…The majority of beam recreations in this paper are based on the HG modal decomposition based propagation described in [16]. This exploits that for sufficiently paraxial light, such as the LISA transmit beams we are concerned with, the projection to each HG mode is unchanging (for more on HG modes and the parameters they depend on such as waist size and waist size location, see [16,17,23]).…”

“…The majority of beam recreations in this paper are based on the HG modal decomposition based propagation described in [16]. This exploits that for sufficiently paraxial light, such as the LISA transmit beams we are concerned with, the projection to each HG mode is unchanging (for more on HG modes and the parameters they depend on such as waist size and waist size location, see [16,17,23]). Thus if the amplitude of the initial outgoing LISA beam in the (0, 0) or fundamental Gaussian HG mode in some basis is a 0,0 , the amplitude of the FF in the propagated form of this mode is still a 0,0 .…”