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Transfer Function of an Optical System

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Transfer Function of an Optical System Appendix A Transfer Function of an Optical System A.1 Linear System A linear system is characterized by its response to a delta function, known as impulse response, for example, the term, LŒı.x/, is the impulse response of an optical system. Invariant impulse response is an important condition that can be imposed upon. In a time- and/ or shift-invariant linear system, the eigen functions are exponentials eiux. Such a system responds to an harmonic input by an harmonic output at the same frequency u and these responses specify the properties of the sys- tem. Many physical processes may be approximated as being linear shift-invariant systems. An optical element described by an operator L, that maps an input function onto an output function, yields an output g1 for any input g0, g1.x1/ D L Œg0.x0/ : (A.1) The Fourier transform of the convolution of a single aperture function with the Dirac delta function, ı, array is equal to the product of the individual transform. The input is decomposed as a sum of impulse function, Z 1 g0.x0/ D g0./ı.x0 /d; (A.2) 1 where the coefficient g0./ is the weighting factors of the decomposition. Suppose for a rectangular slit, 1 for; jja=2Ijjb=2; g0./ D (A.3) 0 otherwiseI i.e., for an optical system, this term g0./ denotes the transmittance pattern, D .; / the 2-D position vector, and therefore, 411 412 A Transfer Function of an Optical System ÄZ 1 g1.x1/ D L g0./ı.x0 /d 1 Z 1 D g0./L Œı.x0 / d: (A.4) 1 A system is linear if the principle of superposition applies. This principle states that the system response to the sum of two inputs is equal to the responses to the each of the two inputs as expressed in (A.4). In order to ensure the shift-invariance, the isoplanatism condition imposes that the supports of the functions g0 and g1 is restricted to the isoplanatic patch. In the presence of atmospheric turbulence, the time-invariance condition for the impulse response is needed. If these conditions are met, the impulse response is constant, and L Œı.x0 / D h.x1 /; (A.5) hence, Z 1 g1.x1/ D g0./h.x1 /d (A.6a) 1 D g0.x/?h.x/; (A.6b) where denotes 2-D convolution. The (A.6) states that if the image formation system is space-invariant, the point spread function (PSF) depends on the differences of the corresponding coordinates. The output of a shift-invariant system is the convolution of the input and the impulse response of the system. In this case, a displacement of the object would result in a displacement of the image but not a change of image configuration. The convolution of two functions results in a function which is broader than both, the implication of which is that all optical systems decrease the resolution of an image. In a per- fect image formation system, the image of a point source is not blurry. A linear and space-invariant image formation system is represented in the spatial frequency domain and taking the Fourier transform (FT) of both sides of (A.6), one expresses, b bg1.u/ D bg0.u/h.u/; (A.7) in which bh.u/ is the transfer function of the system, A cosinusoidal input to a real time invariant linear system produces a cosinu- soidal output. The (A.7) states that each of the elementary sinusoids is filtered by the system with a complex gain bh; the output is the sum of the filtered sinusoids. The properties of L are determined either by the impulse response h.x/ or by the transfer function bh.u/. A.2 Measures of Coherence 413 A.2 Measures of Coherence (Table A1) Table A.1 Definitions of various measures of coherence Symbols Definitions Names Coherence J.r1; r2/ hU.r1;t/U .r1;t/i Mutual intensity Spatial quasi-monochromatic D .r1; r2;0/ .r1; r1;/ hU.r1;t C /U .r1;t/i Self-coherence Temporal function .r1; r1;/ .r1; r1;/ Complex degree of Temporal .r1; r1;0/ self-coherence .r1; r2;/ hU.r1;t C /U .r2;t/i Mutual coherence Spatial & temporal function .r1; r2;/ .r1; r2;/ p Complex degree of Spatial & temporal .r1; r1;0/.r2; r2;0/ coherence J.r1; r2/ .r1; r2/ p Complex coherence Spatial quasi-monochromatic J.r1; r1/J.r2; r2/ factor D .r1; r2;0/ Appendix B Fourier Optics B.1 Fourier Transform A field with general time dependence may be thought of as a linear superposition of fields varying harmonically with time at different frequencies. This relationship is known as Fourier transform (FT; J. B. J. Fourier, 1768–1830). The Fourier transform is used extensively to analyze the output from optical, as well as radio telescopes. It is based on the discovery that it is possible to take any periodic function of time f.t/ and transform it to a function of frequency f./b , in which the notation,bdenotes the Fourier transform (FT) of a particular physical quantity, the frequency ,andthe time t are Fourier dual coordinates. The linear addition of various components of a quantity is carried out by summing up the phasor vectors. Adding two components with identical phasors but with opposite signs of the angular frequency, ! D 2, provides the real quantity. In the analysis of time signal, the one-dimensional Fourier transform plays an important role, which can be evaluated as a sum of exponential frequency terms, each weighted by the corresponding monochromatic response f./b . A waveform of a given signal, f.t/, on a time axis is related to the spectrum (on a frequency axis) by FT and is represented as a superposition of harmonic oscillations ei2t. In the case of electrical filters, transfer functions depend on frequency, while in the case of spectroscopy where the scanning function varies with frequency, the trans- fer function depends on time. The Fourier equation associated with the harmonic function of frequency (Bracewell 1965), Z 1 f./b D f.t/ei2tdt; (B.1) 1 Z 1 f.t/ D f./eb i2td: (B.2) 1 in which f.t/relates to the spectral distribution function f./b by the dual transfor- mation theorem. 415 416 B Fourier Optics a 1 1 0.5 0.5 0 0 −0.5 −0.5 −5 0 5−50 5 TIME (sec) FREQUENCY (rad/s) b 1 1 0.5 0.5 0 0 −0.5 −0.5 −5 0 5 −5 0 5 TIME (sec) FREQUENCY (rad/s) c 1 1 0.5 0.5 0 0 −0.5 −0.5 −5 0 5 −5 0 5 TIME (sec) FREQUENCY (rad/s) Fig. B.1 1-D Fourier transform of (a) sinc t,(b) sinc2t,and(c) et2 The spectral distribution function, f./b , is the Fourier transform of f.t/.There- ciprocal relation between such pairs of functions can be expressed as, f.t/ $ f./b . A set of examples of Fourier transform pairs may be found (see Fig. B.1)in: 2 2 et • e ; (B.3a) ı.t/ • 1; (B.3b)  à  à 1 1 1 1 cos.t/ • ı C ı C ; (B.3c) 2 2 2 2  à  à i 1 i 1 sin.t/ • ı ı C ; (B.3d) 2 2 2 2 sinc t • …./: (B.3e) The Fourier components are the elementary sinusoidal (or cosinusoidal) waves which contribute to a complex waveform. Any complex wave may be built up or broken down from a number of simple waves of suitable amplitude, frequency and phase. The conditions for the existence of these integrals are: R 1 1. f.t/is absolutely integrable, i.e., j 1 f.t/dtj < 1 and 2. f.t/ has no infinite discontinuities, but has a finite number of discontinuities and/or extrema in any finite interval. B.1 Fourier Transform 417 Time domain signal consisting of 50 Hz and 120 Hz signals 2 1 0 Amplitude −1 −2 0 50 100 150 200 250 Time (milliseconds) Spectrum of the time domain signal 60 50 40 30 Power 20 10 0 0 50 100 150 200 250 Frequency (Hz) Fig. B.2 Fourier transform and power of two added sinusoidal waves The Fourier transform of the addition of two sinusoidal waves, x.t/ D sin 2f1t C sin 2f2t; in which f1 D 50 Hz and f2 D 120 HzisshowninFig.B.2. The Fourierh transformi 2 of x.t/;Œbx.f / is two delta functions peaked at f1 and f2. The power, jbx.f /j , would again be two delta functions positioned at 50 Hz and 120 Hz. In the second example (see Fig. B.3), the random Gaussian noise has been added (normally dis- tributed with mean zero variance 2 D 1 and standard deviation D 1)tox.t/, so that x1.t/ D x.t/ C noise: The Fourier transform and the power of x.t/ would again be two delta functions at 50 Hz and 120 Hz. However, this spectrum is not as clean as the FT of x.t/ because of the presence of noise. The Fourier transform is important in dealing with the problems involving lin- ear time and space-invariant systems. The eigen functions eit are respondent to an harmonic input by an harmonic output at the same frequency . These responses specify the properties of the system. The transform of a real function f.t/ has f.b / D fb./,inwhich stands for complex conjugate.
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