Far-field optical path noise coupled with the pointing jitter in the space measurement of gravitational waves

Ya Zhao; Ya Zhao; Jia Shen; Jia Shen; Chao Fang; Zhi Wang; Zhi Wang; Ruihong Gao; Ruihong Gao; Wei Sha

doi:10.1364/AO.405467

Applied Optics
Vol. 60,
Issue 2,
pp. 438-444
(2021)
•https://doi.org/10.1364/AO.405467

Far-field optical path noise coupled with the pointing jitter in the space measurement of gravitational waves

Ya Zhao, Jia Shen, Chao Fang, Zhi Wang, Ruihong Gao, and Wei Sha

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Ya Zhao, Jia Shen, Chao Fang, Zhi Wang, Ruihong Gao, and Wei Sha, "Far-field optical path noise coupled with the pointing jitter in the space measurement of gravitational waves," Appl. Opt. 60, 438-444 (2021)

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Abstract

Space-based gravitational wave detection programs, like the Laser Interferometer Space Antenna or the Taiji program, aim to detect gravitational waves in space with interferometric arms of millions of kilometers. In the process of far-field propagation, the exit wavefront error of the transmitting telescope will couple with the unavoidable pointing between two spacecraft to generate an optical path noise. In this paper, we firstly build a comprehensive theoretical model concerning the effect of 3rd ${\sim}7{\rm th}$ aberrations on this coupling and compare the coupling coefficients of different aberrations. Then, we use this model to analyze the far-field optical path noise of the exit wavefront of a built prototype telescope for the Taiji program. Finally, we take advantage of the Monte Carlo algorithm of the model for investigating the effect of the different wavefront qualities on optical path noise. The results can provide meaningful guidance for the construction of the subsequent telescope.

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Supplementary Material (1)

Name	Description
Code 1	The code can be used to calculate the far-field phase nosie

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$i$	$n, m$	$j$	$Z_{j} (ρ, θ)$	Aberration Name
2,3	$1, \pm 1$	1	$ρ \cos (θ - θ_{Ti})$	Tilt
4	2,0	2	$\sqrt{3} (2 ρ^{2} - 1)$	Defocus
5,6	$2, \pm 2$	3	$\sqrt{6} ρ^{2} \cos (2 θ - (θ_{PA}))$	Primary astigmatism
7,8	$3, \pm 1$	4	$\sqrt{8} (3 ρ^{3} - 2 ρ) \cos (θ - (θ_{PC}))$	Primary coma
9	4,0	5	$\sqrt{5} (6 ρ^{4} - 6 ρ^{2} + 1)$	Primary spherical
10,11	$3, \pm 3$	6	$\sqrt{8} ρ^{3} \cos (3 θ - θ_{PTR})$	Primary trefoil
12,13	$4, \pm 2$	7	$\sqrt{10} (4 ρ^{4} - 3 ρ^{2}) \cos (θ - θ_{SA})$	Secondary astigmatism
14,15	$5, \pm 1$	8	$\sqrt{5} (10 ρ^{5} - 12 ρ^{3} + 3 ρ) \cos (2 θ - θ_{SC})$	Secondary coma
16	$6, 0$	9	$\sqrt{5} (20 ρ^{6} - 30 ρ^{4} + 12 ρ^{2} - 1)$	Secondary spherical
17,18	$4, \pm 4$	10	$\sqrt{5} ρ^{4} \cos (4 θ - θ_{PTE})$	Primary tetrafoil
19,20	$5, \pm 3$	11	$\sqrt{8} (5 ρ^{5} - 4 ρ^{3}) \cos (3 θ - θ_{STR})$	Secondary trefoil
21,22	$6, \pm 2$	12	$\sqrt{5} (15 ρ^{6} - 20 ρ^{4} + 6 ρ^{2}) \cos (2 θ - θ_{TA})$	Tertiary astigmatism
23,24	$7, \pm 1$	13	$\sqrt{5} (35 ρ^{7} - 60 ρ^{5} + 30 ρ^{3} - 4 ρ) \cos (2 θ - θ_{TC})$	Tertiary coma
25	$8, 0$	14	$\sqrt{5} (70 ρ^{8} - 140 ρ^{6} + 90 ρ^{4} - 20 ρ^{2} + 1)$	Tertiary spherical

Mag/ori	$A_{2}^{D E}$	$A_{3}^{P A} / θ_{P A}$	$A_{4}^{P C} / θ_{P C}$	$A_{5}^{P S}$	$A_{6}^{P T R} / θ_{P T R}$	$A_{7}^{S A} / θ_{S A}$
nm/rad	26.6	23.8/2.62	44.9/4.6	47.2	22.3/0.95	18.4/1.06
$A_{8}^{S C} / θ_{S C}$	$A_{9}^{S S}$	$A_{10}^{P T E} / θ_{P T E}$	$A_{11}^{S T R} / θ_{S T R}$	$A_{12}^{T A} / θ_{T A}$	$A_{13}^{T C} / θ_{T C}$	$A_{14}^{T S}$
22.4/0.57	21.03	5.49/5.33	1.02/4.67	9.71/5.67	0.81/1.82	8.26

$δ_{W} [λ]$	Mean (pm/nrad)	Pro ( $\leq 0.1 p m / n r a d$ )	Pro ( $\leq 1 p m / n r a d$ )	Range (pm/nrad)
0.01	0.03	100%	100%	0–0.09
0.02	0.10	56.4%	100%	0–0.32
0.03	0.20	23.9%	100%	0–0.69
0.04	0.33	10.7%	99.6%	0–1.32
0.05	0.51	5.8%	90.5%	0–2.10
0.06	0.71	3.9%	75.2%	0–3.10
0.07	0.96	1.9%	59.0%	0–4.29
0.08	1.24	1.7%	46.7%	0–5.12
0.09	1.57	1.1%	36.8%	0–6.85
0.1	1.89	0.9%	30.1%	0–7.54

$i$	$n, m$	$j$	$Z_{j} (ρ, θ)$	Aberration Name
2,3	$1, \pm 1$	1	$ρ \cos (θ - θ_{Ti})$	Tilt
4	2,0	2	$\sqrt{3} (2 ρ^{2} - 1)$	Defocus
5,6	$2, \pm 2$	3	$\sqrt{6} ρ^{2} \cos (2 θ - (θ_{PA}))$	Primary astigmatism
7,8	$3, \pm 1$	4	$\sqrt{8} (3 ρ^{3} - 2 ρ) \cos (θ - (θ_{PC}))$	Primary coma
9	4,0	5	$\sqrt{5} (6 ρ^{4} - 6 ρ^{2} + 1)$	Primary spherical
10,11	$3, \pm 3$	6	$\sqrt{8} ρ^{3} \cos (3 θ - θ_{PTR})$	Primary trefoil
12,13	$4, \pm 2$	7	$\sqrt{10} (4 ρ^{4} - 3 ρ^{2}) \cos (θ - θ_{SA})$	Secondary astigmatism
14,15	$5, \pm 1$	8	$\sqrt{5} (10 ρ^{5} - 12 ρ^{3} + 3 ρ) \cos (2 θ - θ_{SC})$	Secondary coma
16	$6, 0$	9	$\sqrt{5} (20 ρ^{6} - 30 ρ^{4} + 12 ρ^{2} - 1)$	Secondary spherical
17,18	$4, \pm 4$	10	$\sqrt{5} ρ^{4} \cos (4 θ - θ_{PTE})$	Primary tetrafoil
19,20	$5, \pm 3$	11	$\sqrt{8} (5 ρ^{5} - 4 ρ^{3}) \cos (3 θ - θ_{STR})$	Secondary trefoil
21,22	$6, \pm 2$	12	$\sqrt{5} (15 ρ^{6} - 20 ρ^{4} + 6 ρ^{2}) \cos (2 θ - θ_{TA})$	Tertiary astigmatism
23,24	$7, \pm 1$	13	$\sqrt{5} (35 ρ^{7} - 60 ρ^{5} + 30 ρ^{3} - 4 ρ) \cos (2 θ - θ_{TC})$	Tertiary coma
25	$8, 0$	14	$\sqrt{5} (70 ρ^{8} - 140 ρ^{6} + 90 ρ^{4} - 20 ρ^{2} + 1)$	Tertiary spherical

Mag/ori	$A_{2}^{D E}$	$A_{3}^{P A} / θ_{P A}$	$A_{4}^{P C} / θ_{P C}$	$A_{5}^{P S}$	$A_{6}^{P T R} / θ_{P T R}$	$A_{7}^{S A} / θ_{S A}$
nm/rad	26.6	23.8/2.62	44.9/4.6	47.2	22.3/0.95	18.4/1.06
$A_{8}^{S C} / θ_{S C}$	$A_{9}^{S S}$	$A_{10}^{P T E} / θ_{P T E}$	$A_{11}^{S T R} / θ_{S T R}$	$A_{12}^{T A} / θ_{T A}$	$A_{13}^{T C} / θ_{T C}$	$A_{14}^{T S}$
22.4/0.57	21.03	5.49/5.33	1.02/4.67	9.71/5.67	0.81/1.82	8.26

Abstract

Supplementary Material (1)

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Tables (3)

Equations (25)

Applied Optics