Analytical expressions for the paraxial parameters of a single lens with a spherical distribution of refractive index

Sead Dorić; Nathalie Renaud

doi:10.1364/AO.31.005197

Applied Optics
Vol. 31,
Issue 25,
pp. 5197-5200
(1992)
•https://doi.org/10.1364/AO.31.005197

Analytical expressions for the paraxial parameters of a single lens with a spherical distribution of refractive index

Sead Dorić and Nathalie Renaud

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Sead Dorić and Nathalie Renaud, "Analytical expressions for the paraxial parameters of a single lens with a spherical distribution of refractive index," Appl. Opt. 31, 5197-5200 (1992)

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Abstract

The concept of isoindicial surfaces and position variables is used to develop analytical expressions for the effective focal length and back focal length of a single gradient lens with a spherical distribution of refractive index, where the square of the refractive index is a quadratic function of the distance from the center of symmetry.

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Equations (31)

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			Paraxial Derivatives of the Position Variable

Name of Distribution	Exact Position Variable	Paraxial Position Variable	$\frac{\partial p}{\partial x}$	$\frac{\partial p}{\partial y}$	$\frac{\partial p}{\partial z}$
Plane	p = z	p = z	0	0	1
Cylindrical	p² = x² + y²	p = 0	x/p	y/p	0
Spherical	p² = x² + y² + (z − z_c)²	p = \|z − z_c\|	$\frac{x}{∣ z - z_{c} ∣}$	$\frac{y}{∣ z - z_{c} ∣}$	$\frac{z - z_{c}}{∣ z - z_{c} ∣}$ or ∂(\|z − z_c)/∂z

No.	Curvature/Radius		Separation	Glass
1	c₁	r₁	0	1
2	c₂	r₂	d	n(r)
3	0	∞	?	1

			Paraxial Derivatives of the Position Variable

Name of Distribution	Exact Position Variable	Paraxial Position Variable	$\frac{\partial p}{\partial x}$	$\frac{\partial p}{\partial y}$	$\frac{\partial p}{\partial z}$
Plane	p = z	p = z	0	0	1
Cylindrical	p² = x² + y²	p = 0	x/p	y/p	0
Spherical	p² = x² + y² + (z − z_c)²	p = \|z − z_c\|	$\frac{x}{∣ z - z_{c} ∣}$	$\frac{y}{∣ z - z_{c} ∣}$	$\frac{z - z_{c}}{∣ z - z_{c} ∣}$ or ∂(\|z − z_c)/∂z

No.	Curvature/Radius		Separation	Glass
1	c₁	r₁	0	1
2	c₂	r₂	d	n(r)
3	0	∞	?	1

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Applied Optics