One of the modern techniques of optical system design entails the treatment of an optical system as a two-dimensional linear system that is linear in intensity and can be characterized by a two-dimensional transfer function (17). Consider the linear optical system of Figure 2.4-1. The system input is a spatial light distribution obtained by passing a constant-intensity light beam through a transparency with a spatial sine-wave transmittance. Because the system is linear, the spatial output intensity distribution will also exhibit sine-wave intensity variations with possible changes in the amplitude and phase of the output intensity compared to the input intensity. By varying the spatial frequency (number of intensity cycles per linear dimension) of the input transparency, and recording the output intensity level and phase, it is possible, in principle, to obtain the optical transfer function (OTF) of the optical system.
Let rny) represent the optical transfer function of a two-dimensional linear system where rnx = 2n/Tx and rny = 2n/Ty are angular spatial frequencies with spatial periods Tx and Ty in the x and y coordinate directions, respectively. Then, with ¡¡(x, y) denoting the input intensity distribution of the object and Io(x, y)
representing the output intensity distribution of the image, the frequency spectra of the input and output signals are defined as
I(<x> <y) = J"°° J°° x y)exp{-i(axx + <yy)} dxdy (2.4-1)
Io(<x-<y) = J1J1 Jo(x, y)exp{-i(raxx + <yy)} dxdy (2.4-2)
The input and output intensity spectra are related by
Io(<x, <y) = H(<x, <y)I(<x, <y) (2.4-3)
The spatial distribution of the image intensity can be obtained by an inverse Fourier transformation of Eq. 2.4-2, yielding
Jo(x, y) = ¡Z,jZ,Io(<x' <y) exp {i(<xx + <yy)} d<xd<y (2.4-4)
In many systems, the designer is interested only in the magnitude variations of the output intensity with respect to the magnitude variations of the input intensity, not the phase variations. The ratio of the magnitudes of the Fourier transforms of the input and output signals, is called the modulation transfer function (MTF) of the optical system.
Much effort has been given to application of the linear systems concept to the human visual system (18-24). A typical experiment to test the validity of the linear systems model is as follows. An observer is shown two sine-wave grating transparencies, a reference grating of constant contrast and spatial frequency and a variable-contrast test grating whose spatial frequency is set at a value different from that of the reference. Contrast is defined as the ratio max---min max + min where max and min are the maximum and minimum of the grating intensity, respectively. The contrast of the test grating is varied until the brightnesses of the bright and dark regions of the two transparencies appear identical. In this manner, it is possible to develop a plot of the MTF of the human visual system. Figure 2.4-2a is ahypothetical plot of the MTF as a function of the input signal contrast. Another indication of the form of the MTF can be obtained by observation of the composite sine-wave grating of Figure 2.4-3, in which spatial frequency increases in one
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