Guohao Ju, Changxiang Yan, Zhiyuan Gu, and Hongcai Ma, "Computation of astigmatic and trefoil figure errors and misalignments for two-mirror telescopes using nodal-aberration theory," Appl. Opt. 55, 3373-3386 (2016)

In active optics systems, one concern is how to quantitatively separate the effects of astigmatic and trefoil figure errors and misalignments that couple together in determining the total aberration fields when wavefront measurements are available at only a few field points. In this paper, we first quantitatively describe the impact of mount-induced trefoil deformation on the net aberration fields by proposing a modified theoretical formulation for the field-dependent aberration behavior of freeform surfaces based on the framework of nodal aberration theory. This formulation explicitly expresses the quantitative relationships between the magnitude of freeform surfaces and the induced aberration components where the freeform surfaces can be located away from the aperture stop and decentered from the optical axis. On this basis, and in combination with the mathematical presentation of nodal aberration theory for the effects of misalignments, we present the analytic expressions for the aberration fields of two-mirror telescopes in the presence of astigmatic primary mirror figure errors, mount-induced trefoil deformations on both mirrors, and misalignments. We quantitatively separate these effects using the analytical expressions with wavefront measurements at a few field points and pointing errors. Valuable insights are provided on how to separate these coupled effects in the computation process. Monte Carlo simulations are conducted to demonstrate the correctness and accuracy of the analytic method presented in this paper.

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XDE and YDE are in mm, ADE and BDE are in degrees, and the Fringe-Zernike coefficients are in $\lambda $($\lambda =500\text{\hspace{0.17em}}\mathrm{nm}$).

Table 4.

RMS Deviations Between the Introduced and Computed Values Using NAT and NLLSFA for Different Cases^{a}

Case 1

Case 2

Case 3

Case 4

NAT

NLLSFA

NAT

NLLSFA

NAT

NLLSFA

NAT

NLLSFA

${\mathrm{XDE}}_{\mathrm{SM}}$

$6.1\times {10}^{-5}$

$1.7\times {10}^{-3}$

$1.4\times {10}^{-4}$

$1.2\times {10}^{-2}$

$2.1\times {10}^{-4}$

$4.1\times {10}^{-2}$

$3.2\times {10}^{-2}$

$3.8\times {10}^{-2}$

${\mathrm{YDE}}_{\mathrm{SM}}$

$4.2\times {10}^{-5}$

$2.8\times {10}^{-5}$

$1.3\times {10}^{-4}$

$7.2\times {10}^{-4}$

$2.1\times {10}^{-4}$

$2.4\times {10}^{-3}$

$3.1\times {10}^{-2}$

$6.9\times {10}^{-3}$

${\mathrm{ADE}}_{\mathrm{SM}}$

$4.4\times {10}^{-9}$

$2.7\times {10}^{-6}$

$2.2\times {10}^{-7}$

$7.0\times {10}^{-5}$

$1.3\times {10}^{-6}$

$2.3\times {10}^{-4}$

$3.5\times {10}^{-4}$

$5.9\times {10}^{-4}$

${\mathrm{BDE}}_{\mathrm{SM}}$

$5.0\times {10}^{-9}$

$1.6\times {10}^{-4}$

$2.3\times {10}^{-7}$

$3.1\times {10}^{-3}$

$1.1\times {10}^{-6}$

$3.9\times {10}^{-3}$

$3.5\times {10}^{-4}$

$3.5\times {10}^{-3}$

${}_{F}C_{5}^{(\mathrm{PM})}$

$1.4\times {10}^{-5}$

$2.6\times {10}^{-4}$

$4.3\times {10}^{-5}$

$3.1\times {10}^{-3}$

$7.1\times {10}^{-5}$

$9.0\times {10}^{-3}$

$2.8\times {10}^{-3}$

$8.5\times {10}^{-3}$

${}_{F}C_{6}^{(\mathrm{PM})}$

$1.3\times {10}^{-5}$

$1.8\times {10}^{-4}$

$4.0\times {10}^{-5}$

$2.1\times {10}^{-3}$

$7.2\times {10}^{-5}$

$6.7\times {10}^{-3}$

$3.0\times {10}^{-3}$

$7.2\times {10}^{-3}$

${}_{F}C_{10}^{(\mathrm{PM})}$

$6.0\times {10}^{-4}$

$8.1\times {10}^{-4}$

$1.9\times {10}^{-3}$

$8.3\times {10}^{-3}$

$3.3\times {10}^{-3}$

$2.0\times {10}^{-2}$

$3.5\times {10}^{-3}$

$2.2\times {10}^{-2}$

${}_{F}C_{11}^{(\mathrm{PM})}$

$6.9\times {10}^{-4}$

$1.4\times {10}^{-5}$

$1.8\times {10}^{-3}$

$3.6\times {10}^{-4}$

$2.7\times {10}^{-3}$

$1.2\times {10}^{-3}$

$3.7\times {10}^{-3}$

$1.4\times {10}^{-2}$

${}_{F}C_{10}^{(\mathrm{SM})}$

$2.3\times {10}^{-5}$

$9.7\times {10}^{-4}$

$5.1\times {10}^{-5}$

$1.0\times {10}^{-2}$

$7.2\times {10}^{-5}$

$2.4\times {10}^{-2}$

$1.8\times {10}^{-2}$

$2.6\times {10}^{-2}$

${}_{F}C_{11}^{(\mathrm{SM})}$

$1.6\times {10}^{-5}$

$1.6\times {10}^{-5}$

$4.5\times {10}^{-5}$

$4.2\times {10}^{-4}$

$7.0\times {10}^{-5}$

$1.4\times {10}^{-3}$

$1.7\times {10}^{-2}$

$1.7\times {10}^{-2}$

XDE and YDE are in mm, ADE and BDE are in degrees and the Fringe-Zernike coefficients are in $\lambda $ ($\lambda =500\text{\hspace{0.17em}}\mathrm{nm}$).

XDE and YDE are in mm, ADE and BDE are in degrees, and the Fringe-Zernike coefficients are in $\lambda $($\lambda =500\text{\hspace{0.17em}}\mathrm{nm}$).

Table 4.

RMS Deviations Between the Introduced and Computed Values Using NAT and NLLSFA for Different Cases^{a}

Case 1

Case 2

Case 3

Case 4

NAT

NLLSFA

NAT

NLLSFA

NAT

NLLSFA

NAT

NLLSFA

${\mathrm{XDE}}_{\mathrm{SM}}$

$6.1\times {10}^{-5}$

$1.7\times {10}^{-3}$

$1.4\times {10}^{-4}$

$1.2\times {10}^{-2}$

$2.1\times {10}^{-4}$

$4.1\times {10}^{-2}$

$3.2\times {10}^{-2}$

$3.8\times {10}^{-2}$

${\mathrm{YDE}}_{\mathrm{SM}}$

$4.2\times {10}^{-5}$

$2.8\times {10}^{-5}$

$1.3\times {10}^{-4}$

$7.2\times {10}^{-4}$

$2.1\times {10}^{-4}$

$2.4\times {10}^{-3}$

$3.1\times {10}^{-2}$

$6.9\times {10}^{-3}$

${\mathrm{ADE}}_{\mathrm{SM}}$

$4.4\times {10}^{-9}$

$2.7\times {10}^{-6}$

$2.2\times {10}^{-7}$

$7.0\times {10}^{-5}$

$1.3\times {10}^{-6}$

$2.3\times {10}^{-4}$

$3.5\times {10}^{-4}$

$5.9\times {10}^{-4}$

${\mathrm{BDE}}_{\mathrm{SM}}$

$5.0\times {10}^{-9}$

$1.6\times {10}^{-4}$

$2.3\times {10}^{-7}$

$3.1\times {10}^{-3}$

$1.1\times {10}^{-6}$

$3.9\times {10}^{-3}$

$3.5\times {10}^{-4}$

$3.5\times {10}^{-3}$

${}_{F}C_{5}^{(\mathrm{PM})}$

$1.4\times {10}^{-5}$

$2.6\times {10}^{-4}$

$4.3\times {10}^{-5}$

$3.1\times {10}^{-3}$

$7.1\times {10}^{-5}$

$9.0\times {10}^{-3}$

$2.8\times {10}^{-3}$

$8.5\times {10}^{-3}$

${}_{F}C_{6}^{(\mathrm{PM})}$

$1.3\times {10}^{-5}$

$1.8\times {10}^{-4}$

$4.0\times {10}^{-5}$

$2.1\times {10}^{-3}$

$7.2\times {10}^{-5}$

$6.7\times {10}^{-3}$

$3.0\times {10}^{-3}$

$7.2\times {10}^{-3}$

${}_{F}C_{10}^{(\mathrm{PM})}$

$6.0\times {10}^{-4}$

$8.1\times {10}^{-4}$

$1.9\times {10}^{-3}$

$8.3\times {10}^{-3}$

$3.3\times {10}^{-3}$

$2.0\times {10}^{-2}$

$3.5\times {10}^{-3}$

$2.2\times {10}^{-2}$

${}_{F}C_{11}^{(\mathrm{PM})}$

$6.9\times {10}^{-4}$

$1.4\times {10}^{-5}$

$1.8\times {10}^{-3}$

$3.6\times {10}^{-4}$

$2.7\times {10}^{-3}$

$1.2\times {10}^{-3}$

$3.7\times {10}^{-3}$

$1.4\times {10}^{-2}$

${}_{F}C_{10}^{(\mathrm{SM})}$

$2.3\times {10}^{-5}$

$9.7\times {10}^{-4}$

$5.1\times {10}^{-5}$

$1.0\times {10}^{-2}$

$7.2\times {10}^{-5}$

$2.4\times {10}^{-2}$

$1.8\times {10}^{-2}$

$2.6\times {10}^{-2}$

${}_{F}C_{11}^{(\mathrm{SM})}$

$1.6\times {10}^{-5}$

$1.6\times {10}^{-5}$

$4.5\times {10}^{-5}$

$4.2\times {10}^{-4}$

$7.0\times {10}^{-5}$

$1.4\times {10}^{-3}$

$1.7\times {10}^{-2}$

$1.7\times {10}^{-2}$

XDE and YDE are in mm, ADE and BDE are in degrees and the Fringe-Zernike coefficients are in $\lambda $ ($\lambda =500\text{\hspace{0.17em}}\mathrm{nm}$).