Intrinsic and induced aberration sensitivity to surface tilt Gavriel Catalan

Technion—Israel Institute of Technology, Physics De­partment, Haifa 32000, Israel. Received 11 July 1987. 0003-6935/88/010022-02$02.00/0. © 1988 Optical Society of America. Analysis of wave aberration errors due to small changes in

the system parameters was performed by several authors1–3

using formulas based on the finite ray trace data of the unperturbed system.

In this Letter, the axial coma resulting from a surface tilt around its center will be given in an approximated formula, expressing this aberration explicitly in terms of the construc­tion parameters and ray trace quantities.

The exact formula for the change of the optical path along a ray is given by1,2

sinI. The resulting formula will be expressed for the jth surface:

If the system is also corrected for the sine condition, the first term is an exact transversal focal shift term δω11 = x'ξ'/R', where x' is the incident height of the marginal ray on the exit pupil sphere, R' is its radius of curvature, and ξ' is the focal shift given by

if x = h at the entrance pupil (h' is the Gaussian height at the exit pupil sphere). Therefore, the formula for the axial coma is

where δr is the displacement vector of the incident point of the ray resulting from the parameter change; g is the unit vector along the gradient; I,I' are the incident angles before and after refraction; n,n' are the refractive indices before and after refraction. The derivation is based on the Fermat principle.

If the system is corrected for spherical aberration, for a tilt around the surface center the change in optical path along a marginal ray gives the wave aberration.

The second term includes construction parameters and Gaussian ray trace quantities of the jth surface only. On the other hand, the first term also includes the effect of the aberrations of the previous surfaces (on xj – hj).

Comparison of these two terms shows that the ratio be­tween them may be written as

Let us substitute quantities up to the third order in the aperture. If the surface is tilted by an angle t around the y axis and z is along the optical axis, δr is given by

and from the surface equation

where Q is the conic constant, it may be shown that

and its sign therefore depends on the sign of [(x – h)lh]nn'. One may see that if the induced part—the first term— cancels the intrinsic contribution, the finite incident height should be smaller than the Gaussian height for a refracting surface and larger for a reflecting surface.

The above results are illustrated for some simple systems. Their construction parameters are presented in Table I, including the values of x — h for the marginal ray (object at infinity).

The two first systems are a 100-mm F/2.5 Cassegrain, and a 100-mm F/2.5 doublet. The stop is in front of the system and therefore x — h = 0 at the first surfaces of the systems.

Table I. System Construction Parameters

and therefore one obtains

The expansion of the second term is given by

which leads to

where h and i are the Gaussian quantities replacing x and

22 APPLIED OPTICS / Vol. 27, No. 1 / 1 January 1988

Table II. Wave Aberration Change for 1-mrad Tilt (in µm)

Both systems are corrected for spherical aberration and sine condition at the margin of the aperture.

The results shown in Table II represent changes resulting from a 1-mrad tilt. The values of the axial coma from the approximated formula (9) are shown along with the intrinsic contribution (columns 5 and 6). The accuracy of the ap­proximation may be seen from the difference between the exact and approximated expressions for the total wave aber­ration change [Eqs. (2) and (8)]. Also included is the value for the axial coma obtained from Eq. (2) after subtracting the transverse focal shift term:

and comparison to Eq. (9) should include both the approxi­mation error and the error originating in the small residual OSC (offense against sine condition).

One may notice that the axial coma is small relative to the focal shift term (in the range of 1.5-15% in systems 1 and 2). The approximation errors are small relative to the total wave aberration, but the axial coma values are less accurate. It was also found that the residual OSC leads to an error, which is only a small part of the error in the axial coma.

In the second surface of the Cassegrain the axial coma is about one-third the intrinsic contribution, while in lens sys­tem (2) it is larger than the intrinsic contribution by 50-80% in the three surfaces (2,3,4).

This difference between the two systems is in accordance with the above result concerning the sign of [x – h/h] nn'.

An additional check was performed using ray aberrations. Three aperture rays were traced at heights h,-h,o on the entrance pupil through the system with the tilted surface. From their incident heights at the Gaussian image plane, xu ,xD,x0, the third-order coma wave aberration is given by

and the values of this coefficient are presented in the last column of Table II. It is seen that they are close to the accurate coefficients [from Eq. (11)] previously calculated.

Results are also shown for system 1 stopped down to F/5 (the system is also corrected for that aperture). The relative errors are much smaller, as expected.

From the results obtained so far, one concludes that it is possible to try to obtain a reduced sensitivity to tilt for an internal surface during the optimization process. For a front surface there is the possibility to design a corrector.

For that purpose a singlet was added in front of system 1. The conic constant of its second surface was optimized to annul the axial coma at the large mirror (the approximate value). Then the conical constants of the Cassegrain were reoptimized to correct the system at the marginal aperture.

From Table II (system 3, surface 3) we see that almost the whole value of the exact axial coma is an approximation error (the approximate value was reduced to zero). Yet it is only 5% of the aberration (intrinsic) of the large mirror in system 1. This reduction of sensitivity is important in the tolerance budgeting of an optical system.

The author is on sabbatical leave from MOD the Ministry of Defense.

References 1. H. H. Hopkins and H. Tiziani, "A Theoretical and Experimental

Study of Lens Centring Errors and Their Influence on Optical Image Quality," Br. J. Appl. Phys. 17, 33 (1966).

2. M. P. Rimmer, "Analysis of Perturbed Lens Systems," Appl. Opt. 9, 533 (1970).

3. T. B. Andersen, "Optical Aberration Functions: Derivatives with Respect to Surface Parameters for Symmetrical Systems," Appl. Opt. 24, 1122 (1985).

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