Highly linear sub-nanoradian tilt measurement based on dual-beam interferometry

Hao Yan; Qiuli Mao; Siyuan Xie; Shuai Liu; Jun Luo; Hsien-chi Yeh

doi:10.1364/OL.392856

Optics Letters
Vol. 45,
Issue 10,
pp. 2792-2795
(2020)
•https://doi.org/10.1364/OL.392856

Highly linear sub-nanoradian tilt measurement based on dual-beam interferometry

Hao Yan, Qiuli Mao, Siyuan Xie, Shuai Liu, Jun Luo, and Hsien-chi Yeh

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Hao Yan, Qiuli Mao, Siyuan Xie, Shuai Liu, Jun Luo, and Hsien-chi Yeh, "Highly linear sub-nanoradian tilt measurement based on dual-beam interferometry," Opt. Lett. 45, 2792-2795 (2020)

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Abstract

The tilt measurement method based on dual-beam interferometry is presented in this Letter. Due to symmetric property of the nonlinear errors of two displacement measurement arms, the composite nonlinearity of the tilt measurement is counterbalanced, so that a small nonlinearity over a large dynamic range can be obtained. According to the theoretical analysis with a Gaussian beam model, the second-order nonlinear error is dominant. The experimental results show that a measurement noise of $0 . 4 \;{\rm nrad/}\sqrt {\rm Hz}$ at 1 Hz with a nonlinearity of less than 60 nrad for a measurement range of $\pm {500}\;\unicode{x00B5} {\rm rad}$ has been achieved.

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Parameter	Description
$k$	Laser wave number, $k = 2 π / λ$
$ω$ , $ω_{1}$ , $ω_{2}$	Laser waist radius
$ω_{e f f}$	Effective waist radius, $2 / ω_{e f f}^{2} = 1 / ω_{1}^{2} + 1 / ω_{2}^{2}$
$z_{_{R}}$ , $z_{_{R 1}}$ , $z_{_{R 2}}$	Rayleigh range, $z_{_{R}} = π ω^{2} / λ$
$z$ , $z_{1}$ , $z_{2}$	Propagation distance from waist
$z_{p}$ , $z_{p 1}$ , $z_{p 2}$ , $z_{pm}$	Pivot point of beam rotation
$R,$ $R_{1}$ , $R_{2}$ , $R_{m}$	Radius of the wavefront curvature, $R = z [1 + (z_{_{R}} / z)^{2}]$
$R_{Q}$	Radius of the QPD detection area
$R_{r e l}$	Relative beam radius of the curvature, $1 / R_{r e l} = 1 / R_{1} + 1 / R_{2}$
$Δ ω$ , $Δ z_{_{R}}$ , $Δ z$ , $Δ R$	Parameters’ corresponding deviations, $Δ ω = ω_{1} - ω_{2}$ , $Δ z_{_{R}} = z_{_{R 1}} - z_{_{R 2}}$ , $Δ z = z_{1} - z_{2}$ , $Δ R = R_{1} - R_{2}$
$L$	Distance between two measuring points
$l$	Length of the OL
$x_{0 s}$	Static displacement between the two beams

Parameter	Value
Laser wavelength: $λ$	1064 nm
Laser waist radius: $ω$	1 mm
Propagation distance from waist: $z$	$\sim 10 c m$
Pivot point of beam rotation: $z_{p}$	$\sim 2 c m$
Effect waist radius of the DWS: $ω_{e f f}$	1.8 mm
Distance between the points $L$	2 cm
Arm length of the OL: $l$	0.3 m
Radius of the QPD detection area: $R_{Q}$	4 mm
Heterodyne frequency $Δ f$	10 kHz
Size of the cube test mass $a$	40 mm

Parameter	Description
$k$	Laser wave number, $k = 2 π / λ$
$ω$ , $ω_{1}$ , $ω_{2}$	Laser waist radius
$ω_{e f f}$	Effective waist radius, $2 / ω_{e f f}^{2} = 1 / ω_{1}^{2} + 1 / ω_{2}^{2}$
$z_{_{R}}$ , $z_{_{R 1}}$ , $z_{_{R 2}}$	Rayleigh range, $z_{_{R}} = π ω^{2} / λ$
$z$ , $z_{1}$ , $z_{2}$	Propagation distance from waist
$z_{p}$ , $z_{p 1}$ , $z_{p 2}$ , $z_{pm}$	Pivot point of beam rotation
$R,$ $R_{1}$ , $R_{2}$ , $R_{m}$	Radius of the wavefront curvature, $R = z [1 + (z_{_{R}} / z)^{2}]$
$R_{Q}$	Radius of the QPD detection area
$R_{r e l}$	Relative beam radius of the curvature, $1 / R_{r e l} = 1 / R_{1} + 1 / R_{2}$
$Δ ω$ , $Δ z_{_{R}}$ , $Δ z$ , $Δ R$	Parameters’ corresponding deviations, $Δ ω = ω_{1} - ω_{2}$ , $Δ z_{_{R}} = z_{_{R 1}} - z_{_{R 2}}$ , $Δ z = z_{1} - z_{2}$ , $Δ R = R_{1} - R_{2}$
$L$	Distance between two measuring points
$l$	Length of the OL
$x_{0 s}$	Static displacement between the two beams

Parameter	Value
Laser wavelength: $λ$	1064 nm
Laser waist radius: $ω$	1 mm
Propagation distance from waist: $z$	$\sim 10 c m$
Pivot point of beam rotation: $z_{p}$	$\sim 2 c m$
Effect waist radius of the DWS: $ω_{e f f}$	1.8 mm
Distance between the points $L$	2 cm
Arm length of the OL: $l$	0.3 m
Radius of the QPD detection area: $R_{Q}$	4 mm
Heterodyne frequency $Δ f$	10 kHz
Size of the cube test mass $a$	40 mm

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Equations (14)

Optics Letters