Material Properties

Gradient Index Profile Polynomials

The current Gradient Profiles have the following polynomial forms with associated coefficients.

Axial-Radial gradient

n(r, z) = n0 + nz1z + nz2z2 + nz3z3 + nz4z4 + nr1r2 + nr2r4 + nr3r6 + nr4r8

where… r2 = x2 + y2

Axial-Elliptical gradient

n(r, z) = n0 + nz1z + nz2z2 + nz3z3 + nz4z4 + nr1r2 + nr2r4 + nr3r6 + nr4r8

where…r2 = nrxx2 + nryy2

Axial-Sinusoidal gradient

n(r, z) = n0 + nz1z + nz2z2 + nz3z3 + nz4z4 + nr1r2 + nr2r4 + nr3r6 + nr4r8

where…r

2

=

1 + svasin

z

(x

2

+ y

2

)

-------- + svf

svp

TracePro 5.0 User’s Manual

7.9

Technical Reference

Axial-Tapered gradient

n(r, z) = n0 + nz1z + nz2z2 + nz3z3 + nz4z4 + nr1r2 + nr2r4 + nr3r6 + nr4r8

where…r2 = [1 + (tasz + tao)](x2 + y2)

SELFOC gradient

n2(r) = n02[1 – (nr1r)2 + nr2(nr1r)4 + nr3(nr1r)6 + nr4(nr1r)8]

where… r2 = x2 + y2

Wood Lens gradient

n(r) = n0 + nr1r2

where… r2 = x2 + y2

Spherical gradient

n(r) = n0 + nr1(sgc – r) + nr2(sgc – r)2 + nr3(sgc – r)3 + nr4(sgc – r)4

where… r2 = x2 + y2 + (z – sgc)2

7.10	TracePro 5.0 User’s Manual

Material Properties

Fisheye Lens gradient

	n(r)	=	n0		2
			--------------------------r
			1 + ---------
			nr1

where… r2 = x2 + y2 + (z – sgc)2

Luneburg Lens gradient

n(r) =

n 2 –

r 2

---------

nr1

where… r2

= x2 + y2 + (z – sgc)2

GRADIUM gradient

z'

z'

2

+ gnz3

z'

3

ngwv(z') = gnz0 + gnz1 gmz---------- + gnz2 gmz----------

gmz----------

+

gnz4

z'

4

+ gnz5

z' 5

+ gnz6

z'

6

gmz----------

gmz----------

gmz----------

+

gnz7

z'

7

+ gnz8

z' 8

+ gnz9

z'

9

gmz----------

gmz----------

gmz----------

+

gnz10

z' 10

z '

11

gmz----------

+ gnz11

gmz----------

where...z'

=

goz + z

TracePro 5.0 User’s Manual

7.11

Technical Reference

TracePro supports two formats for GRADIUM: Buchdahl and Sellmeier. Both use the above expression for ngwv(z’) as their index profile. The two GRADIUM

versions differ only in their dispersion formulae. The Buchdahl form is older and now obsolete. It is included for backward compatibility. The Sellmeier form is currently used by LightPath Technologies, the manufacturers of GRADIUM glass. Both of these forms are described in detail below.

The GRADIUM (Buchdahl) dispersion is described by 16 coefficients via a fourthorder expansion in Buchdahl’s chromatic coordinate ω.

At a wavelength other than gwv, the index is computed as follows.

nλ(z') = ngwv + V1ω + V2ω2 + V3ω3 + V4ω4

where...

	ω	=	λ – gwv
			λ – gwv
		1 +
			gwv – 0.187
	V1	= gra1ngwv3		+ grb1ngwv2
	V2	= gra2ngwv3		+ grb2ngwv2
	V3	= gra3ngwv3		+ grb3ngwv2
	V4	= gra4ngwv3		+ grb4ngwv2

+grc1ngwv + grd1

+grc2ngwv + grd2

+grc3ngwv + grd3

+grc4ngwv + grd4

The GRADIUM (Sellmeier) dispersion is described by 48 coefficients via a modified Sellmeier model with three resonances.

At a wavelength other than gwv, the index is computed as follows.

7.12	TracePro 5.0 User’s Manual

Material Properties

nλ2(z') – ngwv2

where...

8

	3	2		– gwv	2
	(z') = ∑Ki'(λ					)
			2
	i = 1	λ		– Li

K'i	=	∑K'ij ngwvj – 1 (z')
		j = 1
and
		8
L'i	=	∑Lij ngwvj – 1 (z')
		j = 1

TracePro 5.0 User’s Manual

7.13