Refractive index profile optimization in diffused graded index lenses

W. E. Martin

doi:10.1364/AO.14.002427

Applied Optics
Vol. 14,
Issue 10,
pp. 2427-2431
(1975)
•https://doi.org/10.1364/AO.14.002427

Refractive index profile optimization in diffused graded index lenses

W. E. Martin

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W. E. Martin, "Refractive index profile optimization in diffused graded index lenses," Appl. Opt. 14, 2427-2431 (1975)

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Abstract

Refractive index profiles resulting from simple diffusion processes in glass are calculated, and optimum choices of diffusion parameters are chosen from minimum error in approximating focusing refractive index profiles. Time dependent boundary conditions are shown to result in considerably improved profiles.

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Slab diffusion parameters

Reference function	α²	Min. dev.	Dt(cm²)	N(0)/N₁
Sech	0.05	3.9 10⁻⁵	0.120	0.9990
	0.1	3.2 10⁻⁵	0.1254	0.9989
	0.2	2.0 10⁻⁵	0.138	0.9986
	0.3	1.2 10⁻⁵	0.152	0.9983
$0 ⩽ x ⩽ 0.8$	0.1	2.1 10⁻⁵	0.1274	0.9988
Helical	0.05	3.3 10⁻⁵	0.124	0.9989
	0.1	2.2 10⁻⁵	0.1344	0.9987
	0.2	9.1 10⁻⁶	0.162	0.9980
	0.3	2.7 10⁻⁶	0.224	0.9966
$0 ⩽ x ⩽ 0.8$	0.1	1.5 10⁻⁵	0.1364	0.9986
Parabolic	—	4.7 10⁻⁵	0.1154	0.9990
$0 ⩽ x ⩽ 0.8$	—	3.1 10⁻⁵	0.1173	0.9991
Cylinder diffusion parameters

Sech	0.05	1.6 10⁻⁴	0.077	0.9991
	0.1	1.5 10⁻⁴	0.080	0.9990
	0.2	1.2 10⁻⁴	0.084	0.9988
	0.3	9.3 10⁻⁵	0.089	0.9986
Helical	0.05	1.5 10⁻⁴	0.078	0.9991
	0.1	1.2 10⁻⁴	0.083	0.9988
	0.2	8.6 10⁻⁵	0.091	0.9985
	0.3	5.9 10⁻⁵	0.100	0.9981
Parabolic		1.8 10⁻⁴	0.076	0.9991

Slab boundary condition changes during diffusion

N₁, = 1.60, N₀ = 1.58, α² = 0.1

Reference function	N₀′	N₀″	Dt	t′(sec)	t″(sec)	Min. dev.	N(0)
Sech	1.579		0.162	1.5 10⁷		3.3 10⁻⁷	1.59684
Sech	1.5797	1.579	0.160	1.0 10⁷	1.5 10⁷	2.6 10⁻⁷	1.59691
Parabolic	1.579		0.140	1.3 10⁷		1.5 10⁻⁶	1.59765
Helical	1.579		0.166	1.0 10		2.6 10⁻⁶	1.59691
Cylinder diffusions

Reference function	N₀′	N₀″	Dt	t′	t″	Min. dev.	N(0)

Sech	1.5785		0.098	9.0 10⁶		3.4 10⁻⁶	1.59711
Sech	1.5799	1.5785	0.1033	3.0 10⁶	9.0 10⁶	2.9 10⁻⁶	1.59693

Slab diffusion parameters

Reference function	α²	Min. dev.	Dt(cm²)	N(0)/N₁
Sech	0.05	3.9 10⁻⁵	0.120	0.9990
	0.1	3.2 10⁻⁵	0.1254	0.9989
	0.2	2.0 10⁻⁵	0.138	0.9986
	0.3	1.2 10⁻⁵	0.152	0.9983
$0 ⩽ x ⩽ 0.8$	0.1	2.1 10⁻⁵	0.1274	0.9988
Helical	0.05	3.3 10⁻⁵	0.124	0.9989
	0.1	2.2 10⁻⁵	0.1344	0.9987
	0.2	9.1 10⁻⁶	0.162	0.9980
	0.3	2.7 10⁻⁶	0.224	0.9966
$0 ⩽ x ⩽ 0.8$	0.1	1.5 10⁻⁵	0.1364	0.9986
Parabolic	—	4.7 10⁻⁵	0.1154	0.9990
$0 ⩽ x ⩽ 0.8$	—	3.1 10⁻⁵	0.1173	0.9991
Cylinder diffusion parameters

Sech	0.05	1.6 10⁻⁴	0.077	0.9991
	0.1	1.5 10⁻⁴	0.080	0.9990
	0.2	1.2 10⁻⁴	0.084	0.9988
	0.3	9.3 10⁻⁵	0.089	0.9986
Helical	0.05	1.5 10⁻⁴	0.078	0.9991
	0.1	1.2 10⁻⁴	0.083	0.9988
	0.2	8.6 10⁻⁵	0.091	0.9985
	0.3	5.9 10⁻⁵	0.100	0.9981
Parabolic		1.8 10⁻⁴	0.076	0.9991

Slab boundary condition changes during diffusion

N₁, = 1.60, N₀ = 1.58, α² = 0.1

Reference function	N₀′	N₀″	Dt	t′(sec)	t″(sec)	Min. dev.	N(0)
Sech	1.579		0.162	1.5 10⁷		3.3 10⁻⁷	1.59684
Sech	1.5797	1.579	0.160	1.0 10⁷	1.5 10⁷	2.6 10⁻⁷	1.59691
Parabolic	1.579		0.140	1.3 10⁷		1.5 10⁻⁶	1.59765
Helical	1.579		0.166	1.0 10		2.6 10⁻⁶	1.59691
Cylinder diffusions

Reference function	N₀′	N₀″	Dt	t′	t″	Min. dev.	N(0)

Sech	1.5785		0.098	9.0 10⁶		3.4 10⁻⁶	1.59711
Sech	1.5799	1.5785	0.1033	3.0 10⁶	9.0 10⁶	2.9 10⁻⁶	1.59693

Abstract

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Applied Optics