Modulation transfer function degradation in segmented windows James I. Gimlett

The author is with the McDonnell Douglas Electronic Systems Company, 1801 East St. Andrew Place, P.O. Box 35020, Santa Ana, California 92705-6520. Received 11 December 1990. 0003-6935/92/162981-02$05.00/0. © 1992 Optical Society of America.

Segmented windows will degrade the modulation transfer function of high-acuity sensors unless the optical thick­nesses of the individual panes are carefully matched.

Many applications of airborne sensors call for either a large window or an open port. The open-port solution is normally deemed unfeasible for a variety of reasons, such as contam­ination, organ piping, turbulent airflow across the cavity, or a decrease in structural integrity. Unfortunately, window size, which is a function of the aperture, placement of the sensor relative to the window, and the field of regard can be a problem in and of itself.

Maintenance of the optical quality that is required for high-acuity sensors dictates window aspect ratios (length: thickness) of 20:1 and smaller. Large, thick panes evoke weight, thermal gradient, and material problems. One solution, often suggested, is to use a mosaic window composed of two or more panes. Each of the small panes can be thinner than the corresponding single pane. Even if the individual panes are separated by mullions, the mosaic window can be much lighter than a single pane covering the same area.

Effects caused by diffraction at the pane edges and mullions, and the air wedge, which is produced by a mosaic window with tilted panes and a pressurized sensor cavity, are well understood. Not so well recognized are the phase effects, which are caused by the individual panes having different optical thicknesses. (Piston errors in distributed aperture systems produce the same effects.) A CODE v (see Ref. 1) evaluation, performed by D. Oinen of Eastman Kodak, of a high-acuity sensor viewing through a two-pane window brought this phenomenon to the attention of this author rather forcibly. The brief analysis below verifies the drop in the modulation transfer function (MTF) that CODE V (see Ref. 1) predicted.

Assume diffraction-limited optics centered on the split in a two-pane window located at the entrance pupil. The two panes have thicknesses t1 and t2, and index n. The ƒ-number of the perfect lens is F# and the wavelength is λ. Then for the geometry of Fig. 1 we can write the optical transfer function,2 in the direction that is transverse to the window septum, in the form

The pupil function F(β, γ) is the complex split pillbox function:

Fig. 1. Pupil function convolution.

where the phase change across the window panes φk. and half the cutoff frequency α0 are given by

respectively. The bracketed term is the optical path be­tween two planes separated by a thickness T. The two window panes lie between, and are parallel to, the two planes.

The convolution can be expressed in closed form in terms of the monochromatic MTF of a perfect lens, namely

as

In Eqs. (4) and (5) w and its component in the x direction wx are angular spatial frequencies. Notice that the system is diffraction limited at frequencies greater than half the cutoff frequency, as it is in the direction parallel to the septum at all spatial frequencies.

Figure 2 is a plot of the modulus of the monochromatic optical transfer function for phase differences of Π and π/2, which correspond to path differences of 1/2 and 1/4 waves, respectively. Notice that in the pathological (the former) case there is an MTF null at the spatial frequency where M0(W) = 2M0(2w). At this frequency the convolution inte­gral is over a region that can be divided into two subregions of equal area, but whose contributions are exactly out of phase, i.e., when φ2 - φ1 = Π. To avoid this problem the two

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Fig. 2. Monochromatic MTF for a split-pane window. Fig. 4. MTF degradation versus delta thickness.

Fig. 3. Polychromatic MTF for a split-pane window.

panes have to be within a small fraction of a wavelength (or within a fraction of an integer number of wavelengths) of each other in optical thickness (nt), not just in physical thickness t.

The dramatic case could be devastating for monochro­matic systems. There is no null in the polychromatic MTF, which has the same general form as the π/2 monochromatic curve. This is not so surprising since the average phase difference over even a small wavelength range is close to π/2. (There is frequency rollover about ~ φ = Π so that the MTF's for the phase differences of π/2 and at 3π/2 are identical.) The polychromatic MTF for the 500-700-nm spectral region is plotted in Fig. 3 assuming equal weighting across the band. The two panes, which are assumed to be

from the same BK7 melt, differ in thickness by 0.001 in. (0.00254 cm). The relative frequency of 1.0 on the horizon­tal axis corresponds to the diffraction limit at 500 nm.

In Fig. 3 the MTF loss at a relative spatial frequency of 0.2 is some 0.25. This is for the 0.001-in. thickness differ­ence between the two panes. A logical question is how closely do the pane thicknesses have to match to produce negligible degradation? The loss in the polychromatic MTF at the 0.2 point is plotted in Fig. 4 as a function of the delta thickness. Minima occur at optical thickness differences that roughly correspond to integer wavelengths (~600 nm). Under the Fig. 3 conditions the delta thickness would have to be less than 142 nm (0.22λ at λ = 632.8 nm) for the MTF drop to be less than 0.05. Unless the panes can be cut from the same finished black, this requirement would be difficult, if not impossible, to meet by following normal optical shop practice.

In summary, whenever a high-acuity sensor must simul­taneously view through two or more panes of a segmented window, care must be taken to closely match the optical thickness of the individual panes. Otherwise, there can be a serious loss in MTF over an appreciable range of spatial frequencies. The range and loss depend on the position of the septa in the pupil. If the panes are tilted with respect to each other, then the loss must be accepted as there is no way to match optical thicknesses for all angles of incidence.

References 1. CODE v is a registered trademark of Optical Research Associates,

Pasadena, Calif. 2. E. L. O'Neill, Introduction to Statistical Optics (Addison-

Wesley, Reading, Mass., 1963).

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