Aspheres and Diffractive Surfaces
The Confusion with Inches/Millimeters and the
Phase Coefficient Relation Between Code V/Zemax

By Max J. Riedl

1. Introduction
Sometimes, lens data are given in inches. This by itself is of course no problem. However, it is frequently overlooked that higher term coefficients for an aspheric surface change as well when the design data are changed from inches to metric (usually mm). Furthermore, if a surface is a diffractive one, there is much of a struggle to convert the phase coefficients from one program to another. In addition, there is occasionally a doubt about the direction of the phase structure, which can be devastating if the wrong one is taken.

These relations become clear on hand of a numerical example for which a 2 element LWIR Petzval objective has been chosen with a diffractive phase profile superimposed on the aspheric rear surface of the front element. The front surface of the second element is an asphere. The other surfaces of the lens elements are spheres. The objective is shown in Fig.1

2. The Objective

Fig.1. A typical LWIR objective with filter in front of the detector.

For our analysis here, we concern ourselves only with the dimensions and coefficients of the front element. These are presented in Table 1.

2. Prescription of the front element in mm.
Phase coefficients as expressed in Code V program

Table 1

Item	Front	Rear
Surface type	Sphere	Asphere with diffractive phase profile
Radius R	174.529 mm	209.925 mm
Conic Constant K	0 (Sphere)	0 (Base surface is a sphere)
Aspheric coefficients
A4 or A	-	0.180817 E-08 mm -3
A6 or B	-	0.392629 E-13 mm -5
A8 or C	-	0.356891 E-18 mm-7
A10 or D	-	0.369300 E-22 mm-9
Phase coefficients
Power of 2 (S2)	-	-0.581441 E-05
Power of 4 (S4)	-	-0.176653 E-10
Design wavelength 0	10 µm	10 µm
Index of refraction N10	4.00312	4.00312

3. Conversion of phase coefficients for Zemax program
The general conversion equation for the phase coefficients from Code V to Zemax is

(1)

where P2n is the Zemax phase coefficient, the design wavelength, R0 the normalized radius, a feature used with Zemax. S2 is the Code V phase coefficient. n indicates the order. This will become clear in the following calculation for our lens.

Important remark: It is customary, but not necessary, to set R = 100, when the lens dimensions are given in mm. The case for R0 100 is discussed in section 10.

With that,
	(Notice, for this calculation, n = 1)

For P4 we use n = 2, and obtain

It is also well to remember to state the wavelength in mm if the lens dimensions are in mm. Therefore we inserted 0.01 mm for 10 µm.

4. The zone radii
The zone-or transition radii can be calculated from the phase equation, which is

(2)

with being the zone radius, and i = 1,2,3,….. the zone number.

Inserting the values from our example, we find for the first zone radius

which leads to r1 = 41.364 mm.

Solving for the rest of the zone radii yields:

2 = 58.348 mm,
3 = 71.282 mm,
4 = 82.106 mm, and
5 = 91.573 mm.

5. Prescription of the front element in inches.
Phase coefficients as expressed in Code V program

Table 2

Item	Front	Rear
Surface type	Sphere	Asphere with diffractive phase profile
Radius R	6.8712 inch	8.2648 inch
Conic Constant K	0 (Sphere)	0 (Base surface is a sphere)
Aspheric coefficients
A4 or A	-	0.296306 E-04 inch-3
A6 or B	-	0.415098 E-06 inch-5
A8 or C	-	0.243429 E-08 inch-7
A10 or D	-	0.162511 E-09 inch -9
Phase coefficients
Power of 2 (S2)	-	-0.147686 E-03
Power of 4 (S4)	-	-0.289483 E-06
Design wavelength 0	10 µm	10 µm
Index of refraction N10	4.00312	4.00312

6. Conversion of phase coefficients for Zemax program
This time we have to insert the dimensions into Eq.(1) in inches. Therefore,

Notice, the phase coefficient does not change for Zemax. It is the same as for the lens dimensioned in mm. Of course the wavelength as well as the normalized radius have to be inserted in inches.

7. The zone radii
For the first zone radius, Eq.(2) becomes

which yields 1 = 1.62850 inch.

For the rest we find:
2 = 2.29717 inch,
3 = 2.80638 inch,
4 = 3.232521 inch, and
5 = 3.60525 inch.

Obviously, the conversion could have been done simply by multiplying the values from section 4 by 25.4. The purpose for the long way around was to demonstrate the relationship and correct use of the coefficients. To convert the aspheric coefficients A2 through A10 (or A through D as they are sometimes called) from the mm to the inch format, the following rule is applied

(3)

For our example,

, therefore

This confirms the number in Table 2.

8. The step size at the zone transition
The relation among the step size, the design wavelength, and the index of refraction of the lens material is given by

(4)

The material for the front element was Germanium with N = 4.00312. Therefore, for the design wavelength of =10 µm,

9. The direction of the phase profile and its effects
The sagittal height of an asphere is expressed by the well known relation

(5)

The sagittal height of a diffractive profile is stated by

(6)

The symbols used in these expressions have been identified above, with the exception of c, which is 1/R, the curvature of the surface.

The total sagittal height is simply the sum of both,

(7)

To demonstrate all this with numbers, we analyze the second surface of the front element of our Petzval objective. As a radial coordinate we pick = 75 mm.

- 0.011077 + 0.009990 = - 0.00109 mm.

Therefore,

= 26.86005 - 0.00109 = 26.85896 mm.

Here is where the mistake is sometimes made. The sign of is incorrectly applied. The effect for the correct and incorrect direction of the diffractive structure is shown in Fig. 2.

Fig. 2. Encircled energy for system with right and wrong direction of phase profile.

This completes the analysis of the 200 mm Petzval front element.

10. Normalized radius for the Zemax program

Eq.(1)

states clearly that the normalized radius R0 needs to be known for the correct coefficient conversion from Code V to the Zemax or vice versa.

To incorporate the relationship in a general way into the standard phase expression, Eq.(2) changes to

Notice, the wavelength does not appear in this expression.

Let us demonstrate with an example. For a particular design the Zemax coefficient P2 was given as -141.56377. The aperture radius was chosen as the normalized radius. It was R = 1.825 inches. The design wavelength was 10 µm (0.0003937 inch).

Using Eq.(1), and solving for the Code V coefficient, yields

To calculate the first zone radius, we use Eq.(5). Since there is only a quadratic term we can write

With this, all the zone radii can be quickly calculated.

Table 3

Conclusion
It has been demonstrated that much care has to be taken when coefficients are to be converted from one program to another. This includes also the change from metric lens units to inches. Furthermore, close attention has to be paid to the direction of the phase profile.

©2002 Precitech Inc.