Sample System Layout

As an example of the layout process, we use an infrared system which requires an external cold stop and also passive thermal compensa-tion. The specifications are listed below:

“First-order” specifications:

1. Focal length: 150 mm 2. Overall length: 260 mm

3. Back focal length: ≥23 mm (minimum clearance) 4. Cold stop location: 17 mm from detector array 5. Packaging: one mirror fold required

 /t  Tb

T_a Tb

n/t

(n 1)

 t

n/t

(n 1)





6. Thermal compensation: passive 7. Aperture diameter: 31 mm 8. Vignetting: none

9. Field of view: 3.0° (1.5°) 10. Wavelength: 4.8  0.3 m

11. Image quality: 50 percent within 50 m in central 1.5;°50 per-cent within 75 m in outer field

12. Distortion:≤2 percent

The requirement for an external cold stop (4) means that the exit pupil of the system must coincide with the cold stop, which is a refrigerated aperture that prevents the infrared-sensitive detector from “seeing” any of the warm, infrared-emitting “structure” of the system. In order to get an external pupil, the system must have at least two separated compo-nents, with the second acting as a relay lens. The focal length of a sys-tem of this type will be negative (as discussed in Sec. 2.4, where the com-bined focal length of the objective and erector of the terrestrial telescope was noted to have a negative focal length). Figure 2.7 also shows a pupil at the “glare stop” external to the objective-erector combination.

Five first-order requirements are listed above, and, in a two-compo-nent system, we have only two powers and a spacing as variable para-meters with which to satisfy these specifications. We approach this optimistically, electing to control the three characteristics which have definite specifications, namely, the focal length, the length, and the cold stop position. Our optimism resides in the hope that when we have these three in hand, the other two (which are the minimal clearance distance and enough space somewhere for a mirror fold) will fall into place. Should this turn out to be overoptimistic, we have to resort to compounding the components, that is, making a component out of two separated subcomponents in the form of a telephoto or retrofocus (Sec.

2.9). Obviously, this is the same as using more than two components.

Going back to Chap. 1, Sec. 1.10, which dealt with systems of two separated components, Eqs. (1.27) and (1.29) give us expressions for the focal length and back focus of a two-component system. Thus, with reference to Fig. 5.5, we have for the focal length


a
b D
a
b (5.17) and for the back focus

B (1  D
)EFL L  D (5.18)

1 EFL

Our cold stop must be located at the image of the objective aperture which is formed by component b. The object distance from b is (D) and Eq. (1.4) can be solved for the pupil distance to get

S B  17  (5.19)

Thus we have three equations in three unknowns, and a simultane-ous solution would give us the values of
a,
b, and D necessary to satisfy the requirements for the focal length EFL 150 mm, the system length (L D  B)  260 mm, and the cold stop (pupil) posi-tion (BS)  17.

But we already have a simultaneous solution of Eqs. (5.17) and (5.18) in the form of Eqs. (1.31) and (1.32), which are easily converted to

a (5.20)

b (5.21)

where L B  D and F  EFL. If we determine
b from Eq. (5.21), we can then determine the stop position from Eq. (5.19).

Using L 260 and F  150 per the specification list, and realiz-L F

Figure 5.5 Schematic of the optical system for the sample layout of Sec. 5.8. In longer-wavelength infrared systems the detector must not “see” the structure of the device, because even at ordinary temperatures a significant amount of infrared radiation is emitted and can affect the performance. The cold stop is a refrigerated aperture stop which shields the detector.

select a few reasonable values for D, evaluate Eqs. (5.21) and (5.19), and tabulate the results as follows.

D 230 B 260D  30
b 0.0594203 S 18.16 BS  11.84

 220 B 260D 40
b 0.0465909 S 23.78 BS 16.22

 210 B 260D 50
b 0.0390476 S 29.17 BS 20.83

Interpolating between D 220 and D  210, we get

D 218.3 B 41.7
b 0.0450396 S 24.72 BS  16.98

which gives us a value of 16.98 mm for the pupil/cold stop to detector distance, in good agreement with the 17 mm required by specification 4, and finally we get
a +0.0058543 from Eq. (5.20).

Tracing the axial and principal rays through the system gives us the component diameters required for the specified zero vignetting as 2(|y|+|y_p|). For component a, specification 7 sets the clear aperture at 31 mm, and the raytrace yields 20 mm as the necessary clear aper-ture for component b. These seem reasonable for the component pow-ers we have arrived at.

Silicon is a reasonable material for a system in the specified wave-length region. It has an index of n 3.427 and a v-value of 1511 over our spectral bandpass (specification 10). Using Eq. (5.8) we can calcu-late the image blur due to chromatic aberration as

TAchA   0.014 mm

Given the 50-m-diameter blur specified for 50 percent of the energy in the image (in specification 11), we arrive at a preliminary conclu-sion that we may not need to achromatize the system.

However, we are not so lucky with regard to the thermal change in focus. Using silicon, with n 3.427,  2.62e06, and n/t  159e06, we calculate T for Eq. (5.12) as 6.289e05. Thus the power of an element when the temperature is raised by 100°C is given by

100
(1  100T)  1.00629

If the mounting structure of the system is aluminum with a CTE of

 0.000024, we have as the parameters for the nominal system, and for the system at t  +100°C:

y²

Vu′k

Using the 100° powers and spacing, and calculating the back focus from Eqs. (5.17) and (5.18), we get B₁₀₀ 40.0858, indicating a ther-mal focus shift of (40.085841.8006)  1.715 mm away from the detector, which is at a distance of 41.8006 mm. Our system, with an aperture of 31 mm and a focal length of 150 mm, has an f-number of 150/31 or f/4.8, so the blur resulting from the thermal defocus is 1.715/4.8 0.35 mm, many times larger than the 50-m size indicat-ed in specification 11.

As indicated in Eq. (5.13), we can athermalize a component by com-bining two materials with different T numbers, much as we achroma-tize by combining materials with different v-values. Ideally, we would like to find a pair of materials where such a doublet would be both achromatized and athermalized. Some materials suitable for our spectral region are tabulated below.

Silicon T 6.29e  05 V 1511 1/V 6.62e  04

Germanium T13.22e 05 V 673 14.86e 04

Amtir T 2.93e 05 V 642 15.04e 04

Zinc selenide T 3.03e 05 V 342 28.42e 04

Zinc sulfide T 3.57e 05 V 915 11.15e 04

When T is plotted vs. 1/V, as in Fig. 5.6, a line drawn between the points for two materials and extended to the T-axis will indicate the equivalent T-value of an achromatic doublet made from the two mate-rials. The dashed line in the figure indicates that silicon and germani-um make an interesting pair. Since the chromatic aberration of the silionly system calculated above seemed acceptable, we can con-sider the possibility of athermalizing the entire system by adding a single negative germanium element, rather than achromatizing and athermalizing both of the components separately. Component a is the bigger contributor to our problems, so we should probably add the germanium element there.

An algebraic solution is possible, but proceeding numerically, we elect to add a germanium element of power
cto the first component and adjust
ato maintain the total power of the first component at

Nominal Att  100°C

a 0.0058543 1.00629  0.0058911

D 218.3 1.00240  218.82392

b 0.0450396 1.00629  0.0453229

B (the space) 41.7 1.00240  41.8006

the original value of0.0058453. We make a number of trials using values for
c of 0.005, 0.010, etc., and find that a value of
c

0.011599 (combined with
a +0.017453, to maintain the first com-ponent power at 0.0058453) gives us a system with zero thermal focus shift, and a chromatic aberration blur of only 0.008 mm, which is an improvement over the uncorrected value of 0.014 mm which we previously calculated.

The final (lens-designed) configuration is shown in Fig. 5.7.

Component b, the relay, was split into three elements by the lens designer. This was necessary to achieve the specified image quality

Figure 5.6 A plot of T [(n/t)/(n1)] vs. 1/V for the materials of a system can be used to assess the combined requirement for achromatism and athermalism. A line drawn between the points representing two materials and extended to the T axis indicates the thermal power change  /t  T for an achromat made of the two materials. If extended to the 1/V axis, the intersection can indicate the chro-matic aberration of an athermal doublet.

and (especially) to correct the distortion and the pupil aberration. The 50 percent blur spot was less than 27 m over the entire field, and the distortion was less than 1.5 percent, easily meeting the specifica-tions and leaving ample room for fabrication tolerances.

Figure 5.7 The final lens-designed optics of the sample layout exercise of Sec. 5.8. The objective is a doublet of silicon and germanium which athermalizes the system and par-tially corrects the chromatic aberration. The relay component was split into three ele-ments of silicon in order to control the distortion and the aberrations of the pupil. The plano elements are the Dewar window and a filter.

6

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