Laser varifocal system synthesis for longitudinal Gaussian beam shifting

Pavel A. Nosov; Dmitry E. Piskunov; Alexander F. Shirankov

doi:10.1364/AO.58.003347

Applied Optics
Vol. 58,
Issue 13,
pp. 3347-3353
(2019)
•https://doi.org/10.1364/AO.58.003347

Laser varifocal system synthesis for longitudinal Gaussian beam shifting

Pavel A. Nosov, Dmitry E. Piskunov, and Alexander F. Shirankov

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Pavel A. Nosov, Dmitry E. Piskunov, and Alexander F. Shirankov, "Laser varifocal system synthesis for longitudinal Gaussian beam shifting," Appl. Opt. 58, 3347-3353 (2019)

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Abstract

This paper presents a modeling approach of a laser varifocal system for longitudinal Gaussian beam waist shifting. The developed approach is based on laser optics theory, which is considered to be classical optics generalized for coherent radiation. A basic optical system, consisting of two components, was developed using this theory. Components have fixed focal lengths and specific movements. An automated modeling algorithm was proposed, and an example was provided in text. Various systems shown in this paper form a Gaussian beam with required waist parameters and provide its longitudinal shifts, which exceed the length of the near-zone of a focused Gaussian beam. Such systems can be used in laser technologies, including micro- and nano-sized objects manipulation.

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Type of Dim. Limitation	RAV for $z_{w}$ , $z_{w}^{'}$ for Direct Constraints on $d_{1}$ , $d_{3}$	RAV for Constraint on Maximal Clear Aperture $2 h_{1, 2 ca sup}$ or Minimal $f$ number $N_{d 1, 2 \min}$ of First and Second Components	RAV, Range Number
$\| d_{1} \| \leq d_{1 sup}$ 1st type	$z_{w \min}^{(1)} = f_{1}^{'} - d_{1 \sup}$ , $z_{w \max}^{(1)} = f_{1}^{'} + d_{1 \sup}$	$d_{1 sup} = z_{c} \sqrt{{(\frac{2 h_{1 ca \sup}}{2 h_{w} K_{d}})}^{2} - 1}$ , $d_{1 sup} = z_{c} \sqrt{{(\frac{f_{1}^{'}}{2 h_{w} K_{d} N_{d 1 \min}})}^{2} - 1}$ , $z_{w \min}^{(1)} = f_{1}^{'} - d_{1 \sup}$ , $z_{w \max}^{(1)} = f_{1}^{'} + d_{1 \sup}$	(1) $z_{w \min}^{(1)} \leq z_{w} \leq z_{w \max}^{(1)}$
$\| d_{1} \| \geq d_{1 inf}$ 2nd type	$z_{w \min}^{(2)} = f_{1}^{'} + d_{1 \inf}$ , $z_{w \max}^{(2)} = f_{1}^{'} - d_{1 \inf}$	–	(1) $z_{w} \geq z_{w \min}^{(2)}$ , (2) $z_{w} \leq z_{w \max}^{(2)}$
$\| d_{3} \| \leq d_{3 sup}$ 1st type	$z_{w \min}^{' (1)} = - f_{2}^{'} - d_{3 \sup}$ , $z_{w \max}^{' (1)} = - f_{2}^{'} + d_{3 \sup}$	$d_{3 sup} = z_{c}^{'} \sqrt{{(\frac{2 h_{2 ca \sup}}{2 h_{w}^{'} K_{d}})}^{2} - 1}$ , $d_{3 sup} = z_{c}^{'} \sqrt{{(\frac{f_{2}^{'}}{2 h_{w}^{'} K_{d} N_{d 2 \min}})}^{2} - 1}$ , $z_{w \min}^{' (1)} = - f_{2}^{'} - d_{3 \sup}$ , $z_{w \max}^{' (1)} = - f_{2}^{'} + d_{3 \sup}$	(1) $z_{w \min}^{' (1)} \leq z_{w}^{'} \leq z_{w \max}^{' (1)}$
$\| d_{3} \| \geq d_{3 inf}$ 2nd type	$z_{w \min}^{' (2)} = - f_{2}^{'} - d_{3 \inf}$ , $z_{w \max}^{' (2)} = - f_{2}^{'} + d_{3 \inf}$	–	(1) $z_{w}^{'} \leq z_{w \min}^{' (2)}$ (2) $z_{w}^{'} \geq z_{w \max}^{' (2)}$

	First Component			Second Component			LVS Length Change
#	$f_{1}^{'}$ (mm)	$2 h_{1 ca \max}$ (mm)	$N_{d 2 \min}$	$f_{2}^{'}$ (mm)	$2 h_{2 ca \max}$ (mm)	$N_{d 2 \min}$	$L_{\min} \dots L_{\max}$ (mm)	$F z$
1	40.0	3.6	11.2	78.0	3.7	21.0	463.1 … 1257.5	3.16
2	30.0	2.7	11.2	60.0	2.8	21.3	195.3 … 772.9	2.31
3	40.0	3.6	11.0	80.0	3.9	20.6	240.0 …1288.0	3.27
4	30.0	2.7	11.2	61.5	2.9	21.2	201.6 … 790.4	2.33
5	35.0	3.0	11.6	71.7	3.2	22.5	254.2 … 887.1	2.55
6	40.0	3.7	10.9	82.0	4.1	20.2	266.2 … 1316.8	3.38

$z_{w}$ (mm)	$d_{1}$ (mm)	$d_{2}$ (mm)	$d_{3}$ (mm)	$L$ (mm)	$Δ L$ (mm)	$2 h_{1 ca}$ (mm)	$2 h_{2 ca}$ (mm)
−1.56	31.56	675.18	66.24	772.98	0.0	2.67	2.81
−1.37	31.37	671.23	65.47	768.07	4.9	2.65	2.78
−1.18	31.18	654.67	64.70	750.55	22.4	2.64	2.75
−0.98	30.98	621.05	63.93	715.96	57.0	2.62	2.71
−0.79	30.79	565.74	63.16	659.69	113.3	2.61	2.68
−0.60	30.60	485.13	62.39	578.11	194.9	2.59	2.65
−0.41	30.41	378.26	61.62	470.29	302.7	2.57	2.62
−0.21	30.21	248.71	60.85	339.77	433.2	2.56	2.59
−0.02	30.02	105.22	60.08	195.32	577.7	2.54	2.55

Type of Dim. Limitation	RAV for $z_{w}$ , $z_{w}^{'}$ for Direct Constraints on $d_{1}$ , $d_{3}$	RAV for Constraint on Maximal Clear Aperture $2 h_{1, 2 ca sup}$ or Minimal $f$ number $N_{d 1, 2 \min}$ of First and Second Components	RAV, Range Number
$\| d_{1} \| \leq d_{1 sup}$ 1st type	$z_{w \min}^{(1)} = f_{1}^{'} - d_{1 \sup}$ , $z_{w \max}^{(1)} = f_{1}^{'} + d_{1 \sup}$	$d_{1 sup} = z_{c} \sqrt{{(\frac{2 h_{1 ca \sup}}{2 h_{w} K_{d}})}^{2} - 1}$ , $d_{1 sup} = z_{c} \sqrt{{(\frac{f_{1}^{'}}{2 h_{w} K_{d} N_{d 1 \min}})}^{2} - 1}$ , $z_{w \min}^{(1)} = f_{1}^{'} - d_{1 \sup}$ , $z_{w \max}^{(1)} = f_{1}^{'} + d_{1 \sup}$	(1) $z_{w \min}^{(1)} \leq z_{w} \leq z_{w \max}^{(1)}$
$\| d_{1} \| \geq d_{1 inf}$ 2nd type	$z_{w \min}^{(2)} = f_{1}^{'} + d_{1 \inf}$ , $z_{w \max}^{(2)} = f_{1}^{'} - d_{1 \inf}$	–	(1) $z_{w} \geq z_{w \min}^{(2)}$ , (2) $z_{w} \leq z_{w \max}^{(2)}$
$\| d_{3} \| \leq d_{3 sup}$ 1st type	$z_{w \min}^{' (1)} = - f_{2}^{'} - d_{3 \sup}$ , $z_{w \max}^{' (1)} = - f_{2}^{'} + d_{3 \sup}$	$d_{3 sup} = z_{c}^{'} \sqrt{{(\frac{2 h_{2 ca \sup}}{2 h_{w}^{'} K_{d}})}^{2} - 1}$ , $d_{3 sup} = z_{c}^{'} \sqrt{{(\frac{f_{2}^{'}}{2 h_{w}^{'} K_{d} N_{d 2 \min}})}^{2} - 1}$ , $z_{w \min}^{' (1)} = - f_{2}^{'} - d_{3 \sup}$ , $z_{w \max}^{' (1)} = - f_{2}^{'} + d_{3 \sup}$	(1) $z_{w \min}^{' (1)} \leq z_{w}^{'} \leq z_{w \max}^{' (1)}$
$\| d_{3} \| \geq d_{3 inf}$ 2nd type	$z_{w \min}^{' (2)} = - f_{2}^{'} - d_{3 \inf}$ , $z_{w \max}^{' (2)} = - f_{2}^{'} + d_{3 \inf}$	–	(1) $z_{w}^{'} \leq z_{w \min}^{' (2)}$ (2) $z_{w}^{'} \geq z_{w \max}^{' (2)}$

	First Component			Second Component			LVS Length Change
#	$f_{1}^{'}$ (mm)	$2 h_{1 ca \max}$ (mm)	$N_{d 2 \min}$	$f_{2}^{'}$ (mm)	$2 h_{2 ca \max}$ (mm)	$N_{d 2 \min}$	$L_{\min} \dots L_{\max}$ (mm)	$F z$
1	40.0	3.6	11.2	78.0	3.7	21.0	463.1 … 1257.5	3.16
2	30.0	2.7	11.2	60.0	2.8	21.3	195.3 … 772.9	2.31
3	40.0	3.6	11.0	80.0	3.9	20.6	240.0 …1288.0	3.27
4	30.0	2.7	11.2	61.5	2.9	21.2	201.6 … 790.4	2.33
5	35.0	3.0	11.6	71.7	3.2	22.5	254.2 … 887.1	2.55
6	40.0	3.7	10.9	82.0	4.1	20.2	266.2 … 1316.8	3.38

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Equations (11)

Applied Optics