1402 L E T T E R S T O T H E E D I T O R Vol. 56

Diffraction Image of a Single Bar in the Presence of Off-Axis Aberrations

RICHARD BARAKAT AND AGNES HOUSTON Itek Corporation, Lexington, Massachusetts 02173

(Received 31 March 1966) INDEX HEADINGS: Diffraction; Aberrations.

W E have recently considered the diffraction images of in­finitely narrow incoherent line sources in the presence of

off-axis aberrations.1 The purpose of this note is to extend the analysis to cover the more general situation of incoherent line sources of finite width along with some typical numerical results. The notation is that of our previous paper.

The distribution of intensity in the object is:

so that the object-intensity spectrum becomes

NOW the image-illuminance spectrum I(ω) is related to 0(ω) via the equation

where T(ω,ø) is the transfer function of the optical system in question with respect to the azimuthal angle ø. The transfer func­tion is generally a complex-valued function

where Tr is an even function of ω and Ti is an odd function of ω. The distribution of illuminance i {v) in the image is simply the

Fourier transform of (3):

the finite limits occurring because T(ω,ø) vanishes for |ω|≥2. Equation (5) can be rewritten in the form

October 1966 L E T T E R S TO T H E E D I T O R 1403

FIG. 1. Distribution of illuminance in a bar target of half-width υo = 2 in the Fгaunhofer receiving plane in the presence of one wave of third-order coma (W131 = lλ) in the azimuths: ø=0°, ø=45°, — • —ø=90°. The heavy broken line represents the object when imaged through an aberration-free system.

FIG. 3. Distribution of illuminance in a bar target of half-width υ0 =6 in the Fraunhofer receiving plane in the presence of one wave of third-order coma (W131 = lλ) in the azimuths: ø =0°, ø=45°, - • - ø =90°. The heavy broken line represents the object when imaged through an aber­ration-free system.

using the parity properties of T, and Ti, which is in a convenient form for numerical calculation by gauss quadrature. The constant C2 was chosen so that when the optical system was aberration-free, the value of i(o) would be unity for all values of v and ø.When the imaging is confined to the on-axis situation, then Ti=0 and (6) simplifies to

When the object is very narrow (υ0г«1) then we can replace (sinv0ω)/ω in (6) by v0; this substitution yields

symmetric about v = 0. The heavy broken lines in the figures represent the object when imaged through an aberration-free system. I t is instructive to compare the aberration-free image and the coma-distorted image in the orientation ø = 0 ° . For one thing, the slope of the aberration-free image is much greater than the image with aberration; in addition the peak illuminance is greater. The image with aberrations in the other two orientations: ø = 45°, 90°, is highly asymmetric.

1 R. Barakat and A. Houston, J. Opt. Soc. Am. 35, 1132 (1965).

The integral is the line-spread function1 τ(v); thus the influence of a small width is to multiply the line-spread function by a constant but not alter its shape.

We have chosen to demonstrate the effects of off-axis aberrations using third-order coma W131 as an example. The results of the cal­culations are illustrated in Figs. 1-3 for a system with one wave of coma (W131 = lλ) in the paraxial receiving plane (W2 = 0). When the bar target is imaged in the orientation ø = 0°, the imaginary part of the transfer function vanishes and the image is

FIG. 2.'Distribution of illuminance in a bar target of half-width v0 =4 in the Fraunhofer receiving plane in the presence of one wave of third-order coma (W131=1λ) in the azimuths: ø = 0 ° , ø=45°, — • — ø=90° . The heavy broken line represents the object when imaged through an aberration-free system.