Analysis of Geometric Tolerance for Monolithic Nonplanar Ring Oscillators

Fig. 1. Schematic diagrams of the eigen optical path solution. (a) The optical path reflected among the planar mirrors and the virtual rays; (b) the corresponding monolithic NPRO. The point $O$ is the reference origin. $A$ , $B$ , $C$ , and $D$ are the reflecting points on the four planes $h_{0}$ , $h_{1}$ , $h_{2}$ and $h_{3}$ along the eigen optical path, respectively. $D_{(k + 1)}$ is the image point of $D_{(k)}$ about the plane $h_{k}$ , ( $k = 0,1, 2,3$ ). $D_{(0)}$ is $D$ ,and $n_{k}$ represents the normal unit vector of the plane $h_{k}$ (the unit vector points towards the outside of the resonator)

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Fig. 2. Analysis of the geometric tolerance of a monolithic NPRO

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Fig. 3. Geometry design of a monolithic NPRO. (a) Tetrahedral cavity and the optical path inside; (b) cutting the tetrahedron into a monolithic NPRO; (c) the final geometry of a monolithic NPRO

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Fig. 4. A Gaussian beam with fundamental transverse mode passes through a tilted polygonal aperture. (a) A three-dimensional schematic diagram; (b) projection onto X-Y plane

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Fig. 5. The matching of the external optical path to a monolithic NPRO

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Fig. 6. Two NPROs with different sizes. (a) $l_{h z} = 5.0 m m$ ; (b) $l_{h z} = 2.2 m m$

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$Changes of the reflecting point offset (a), roundtrip diffraction loss (b), and output-axis offset (c) caused by normal translations of the surfaces (h0‒h3) on a monolithic NPRO (lhz=5.0 mm) with normal offset of each reflective plane, where the horizontal lines indicate the corresponding thresholds$

Fig. 7. Changes of the reflecting point offset (a), roundtrip diffraction loss (b), and output-axis offset (c) caused by normal translations of the surfaces ( $h_{0}$ ‒ $h_{3}$ ) on a monolithic NPRO ( $l_{h z} = 5.0 m m$ ) with normal offset of each reflective plane, where the horizontal lines indicate the corresponding thresholds

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Fig. 8. Offset tolerance ranges of the optical surfaces for a monolithic NPRO. (a) $l_{h z} = 2.2 m m$ ; (b) $l_{h z} = 5.0 m m$ . The blue horizontal lines and values represent the actual facets’ deviations of the NPRO after manufacturing, as detailed in Section 4

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Fig. 9. Designed and finished monolithic NPROs with different sizes. (a) Four NPROs with different sizes; (b) pump light path fluorescence ( $l_{h z} = 10 m m$ ); (c) a Nd∶YAG monolithic NPRO with $l_{h z} = 2.2 m m$ emitting 1064 nm laser (beam spot indicated with a circle)

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Fig. 10. Correction scheme for a monolithic NPRO ( $l_{h z} = 5.0 m m$ ) whose $h_{2}$ -plane offset exceeds the tolerance. (a) Front view; (b) side view; (c) top view; (d) 3D view

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Table 1. Geometric parameters of the monolithic NPRO and the related optical path

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Table 1. Geometric parameters of the monolithic NPRO and the related optical path

Parameter	Value		Description
Parameter	$l_{h z} = 5.0 m m$	$l_{h z} = 2.2 m m$	Description
$α_{A O P}$ /（°）	45	45	Incident angle of pump beam
$α_{A O L}$ /（°）	45	45	Refraction angle of output beam
$l_{n p}$ /mm	1.5	0.66	Span of non-horizontal optical path
$β_{n p}$ /（°）	15	15	Nonplanar angle
$l_{z}$ /mm	6.8	3.0	Length of resonator
$w_{x}$ /mm	5.5	3.3	Width of resonator
$t_{y}$ /mm	3.0	1.5	Thickness of resonator

Table 2. Constraints used in the analysis of the tolerance of a monolithic NPRO

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Table 2. Constraints used in the analysis of the tolerance of a monolithic NPRO

Parameter	Value	Description
$f_{h}$ /mm	200 mm	Focal length of thermal lens^［14］
$Δ τ_{d i f f r}$	$1 \times 10^{- 5}$	Threshold of roundtrip diffraction loss
$Δ d_{p, l}$ /mm	0.5 mm	Maximum transverse shift allowed for pump and output beams

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Jiankang Peng, Yuan Ru, Zhenqian Li, Lisheng Chen, Liufeng Li. Analysis of Geometric Tolerance for Monolithic Nonplanar Ring Oscillators[J]. Chinese Journal of Lasers, 2025, 52(10): 1001005

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