Statistical Optics Wiley Pure And Applied Optics 2nd Edition Joseph W Goodman - Solutions

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Show that any two statistically independent random variables have a correlation coefficient that is zero.
Given the random variables\[ \begin{aligned} & U=\cos \Theta \\ & V=\sin \Theta \end{aligned} \]with\[ p_{\Theta}(\theta)= \begin{cases}1 / \pi & -\frac{\pi}{2}
Prove the following properties of characteristic functions:(a) Every characteristic function has value unity at the origin.(b) The second-order characteristic function \(\mathbf{M}_{U V}\left(\omega_{U}, 0\right)\) is equal to the characteristic function \(\mathbf{M}_{U}(\omega)\) of the random
A random variable \(U\) is uniformly distributed on the interval \((-A, A)\) and its probability density function is zero otherwise.(a) Find an expression for \(A\) in terms of the variance \(\sigma_{U}^{2}\) of \(U\).(b) Find an expression for the characteristic function of \(U\) as a function of
Show that if the moment \(\overline{u^{n} v^{m}}\), if it exists, can be found from the joint characteristic function \(\mathbf{M}\left(\omega_{U}, \omega_{V}\right)\) by the formula\[ \overline{u^{n} v^{m}}=\left.\frac{1}{j^{n+m}} \frac{\partial^{n+m}}{\partial \omega_{U}^{n} \partial
Find the probability density function of the random variable \(Z\) in terms of the known density function \(p_{U}(u)\) when(a) \(z=a u+b\), where \(a\) and \(b\) are real constants.(b) \(z=\left\{\begin{array}{cc}|u| & -1
Using the method given in Eq. (2.5-25), find the joint probability density function \(p_{W Z}(w, z)\) when\[ \begin{align*} & w=u^{2} \\ & z=u+v \tag{p.2-1} \end{align*} \]and \(p_{U V}(u, v)=\operatorname{rect} u\) rect \(v\), where rect \(x=1\) for \(|x| \leq 1 / 2\) and is zero otherwise.
Consider two independent, identically distributed random variables \(\Theta_{1}\) and \(\Theta_{2}\), each of which obeys a probability density function\[ p_{\Theta}(\theta)=\left\{\begin{array}{cc} \frac{1}{2 \pi} & -\pi
Consider two identically distributed and independent random variables \(X_{1}\) and \(X_{2}\) with common probability density function \(p_{X}(x)\). Show that the probability density function of the difference of these two random variables, \(X_{1}-X_{2}\), is given by the autocorrelation function
Consider the random phasor sum of Section 2.9 with the single change that the phases \(\phi_{k}\) are uniformly distributed on \((-\pi / 2, \pi / 2)\). Find the following quantities: \(\bar{r}, \bar{i}, \sigma_{r}^{2}, \sigma_{i}^{2}\), and \(ho\). Make a rough plot of the contours of constant
Let the random variables \(U_{1}\) and \(U_{2}\) be jointly Gaussian, with zero means, equal variances, and correlation coefficient \(ho eq 0\). Consider the new random variables \(V_{1}\) and \(V_{2}\) defined by a rotational transformation about the origin of the \((u, v)\) plane,\[
Consider \(n\) independent random variables \(U_{1}, U_{2}, \ldots, U_{n}\), each of which obeys a Cauchy density function,\[ p_{U}(u)=\frac{1}{\pi \beta\left[1+\left(\frac{u}{\beta}\right)^{2}\right]} \]where \(\beta\) is real and positive.(a) Show that this density function does not have a
A certain computer contains a random number generator that generates random numbers with uniform relative frequencies (or probability density) on the interval \((0,1)\). Suppose, however, that it is desired to simulate trials of a random variable \(Z\) with density function \(p_{Z}(z)\) that is not
Let the real-valued random process U(t)U(t) be defined byU(t)=Acos(2πvt−Φ)U(t)=Acos⁡(2πvt−Φ)where vv is a known constant, ΦΦ is a random variable uniformly distributed on (−π,π),A(−π,π),A is a random variable that takes on the values 1 and 2 with equal probabilities of 1/21/2,
Consider the random process \(U(t)=A\), where \(A\) is a random variable uniformly distributed on \((-1,1)\).(a) Sketch some sample functions of this process.(b) Find the time autocorrelation function of \(U(t)\).(c) Find the statistical autocorrelation function of \(U(t)\).(d) Is \(U(t)\)
An ergodic real-valued random process \(U(t)\) with autocorrelation function \(\Gamma_{U}(\tau)=\left(N_{0} / 2\right) \delta(\tau)\) is applied to the input of a linear, time-invariant filter with impulse response \(h(t)\). The filter output \(V(t)\) is multiplied by a delayed version of \(U(t)\),
Consider the random process \(Z(t)=U \cos \pi t\), where \(U\) is a random variable with probability density function\[ p_{U}(u)=\frac{1}{\sqrt{2 \pi}} \exp \left(-\frac{u^{2}}{2}\right) \](a) What is the probability density function of the random variable \(Z(0)\) ?(b) What is the joint density
Find the statistical autocorrelation function of the random process\[ U(t)=a_{1} \cos \left(2 \pi v_{1} t-\Phi_{1}\right)+a_{2} \cos \left(2 \pi v_{2} t-\Phi_{2}\right) \]where \(a_{1}, a_{2}, v_{1}\), and \(v_{2}\) are known constants, while \(\Phi_{1}\) and \(\Phi_{2}\) are independent random
A certain random process \(U(t)\) takes on equally probable values +1 or 0 with changes occurring randomly in time. The probability that \(n\) changes occur in time \(\tau\) is known to be\[ P_{N}(n)=\frac{1}{1+\alpha|\tau|}\left(\frac{\alpha|\tau|}{1+\alpha|\tau|}\right)^{n}, \quad n=0,1,2, \ldots
A certain random process \(U(t)\) consists of a sum of (possibly overlapping) pulses of the form \(p\left(t-t_{k}\right)=\operatorname{rect}\left(\left(t-t_{k}\right) / b\right)\) occurring with mean rate \(\bar{n}\) pulses per second. The times of occurrence are completely random, with the number
Assuming that the random process \(U(t)\) is wide-sense stationary, with mean \(\bar{u}\) and variance \(\sigma^{2}\), which of the following functions represent possible structure functions for \(U(t)\) ? Explain each answer.(a) \(D_{U}(\tau)=2 \sigma^{2}[1-\exp (-\alpha|\tau|)]\),(b)
Prove that the Hilbert transform of the Hilbert transform of a function \(u(t)\) is \(-u(t)\), up to a possible additive constant.
(a) Show that for an analytic signal representation of a real-valued, narrowband random process, the autocorrelation function of the resulting complex process \(\mathbf{U}(t)\) (assumed wide-sense stationary) can be written in the form\[ \boldsymbol{\Gamma}_{U}(\tau)=\mathbf{g}(\tau) e^{-j 2 \pi
Let \(\mathbf{V}(t)\) be a linearly filtered complex-valued, wide-sense stationary random process with sample functions given by\[ \mathbf{v}(t)=\int_{-\infty}^{\infty} \mathbf{h}(t-\tau) \mathbf{u}(\tau) d \tau \]where \(\mathbf{U}(t)\) is a complex-valued input process and \(\mathbf{h}(t)\) is
Find the power spectral density of a doubly stochastic Poisson impulse process having a rate process \(\Lambda(t)\) described by\[ \Lambda(t)=\lambda_{0}[1+\cos (2 \pi \bar{v} t+\Phi)] \]where \(\Phi\) is a random variable uniformly distributed on \((-\pi, \pi)\), while \(\lambda_{0}\) and
Starting with Eq. (4.1-10), show that if \(\Delta v \ll \bar{v}\) and \(r \ll 2 c / \Delta v\) for all \(P_{1}\), then\[ \mathbf{u}\left(P_{0}, t\right) \approx \iint_{\Sigma} \frac{e^{j 2 \pi(r / \bar{\lambda})}}{j \bar{\lambda} r} \mathbf{u}\left(P_{1}, t\right) \chi(\theta) d S \]can be used
(a) Show that the Jones matrix of a polarization analyzer set at angle \(\alpha\) to the \(X\)-axis is given by\[ \underline{\mathbf{L}}(\alpha)=\left[\begin{array}{cc} \cos ^{2} \alpha & \sin \alpha \cos \alpha \\ \sin \alpha \cos \alpha & \sin ^{2} \alpha \end{array}\right] \](b) Show that
By finding the trace of the appropriate coherency matrix, show that the average intensity transmitted by a polarization analyzer set at \(+45^{\circ}\) to the \(X\)-axis can be expressed as\[ \bar{I}=\frac{1}{2}\left[\mathbf{J}_{x x}+\mathbf{J}_{y y}\right]+\operatorname{Re}\left\{\mathbf{J}_{x
By finding the trace of the appropriate coherency matrix, show that the average intensity transmitted by a quarter-wave plate followed by a polarization analyzer set at \(+45^{\circ}\) to the \(X\)-axis can be expressed as\[ \bar{I}=\frac{1}{2}\left[\mathbf{J}_{x x}+\mathbf{J}_{y
Consider a light wave that has \(X\) - and \(Y\)-polarization components of its electric field at point \(P\) given by\[ \begin{aligned} & \mathbf{u}_{X}(t)=\exp \left[-j 2 \pi\left(\bar{v}-\frac{\Delta v}{2}\right) t\right] \\ & \mathbf{u}_{Y}(t)=\exp \left[-j 2 \pi\left(\bar{v}+\frac{\Delta
Show that the characteristic function of the intensity of polarized thermal light is given by\[ \mathbf{M}_{I}(\omega)=\frac{1}{1-j \omega \bar{I}} \]
Show that the standard deviation \(\sigma_{I}\) of the instantaneous intensity of partially polarized thermal light is\[ \sigma_{I}=\sqrt{\frac{1+\mathcal{P}^{2}}{2}} \bar{I} \]
When light falls on a balanced detector (i.e., a detector pair, with one detector output subtracted from the other's output), the output current is proportional to the difference of the intensities incident on the two detectors. Find the probability density function of the difference of the
Let the field emitted by a multimode laser oscillating in \(N\) equals strength and independent modes be represented by\[ \mathbf{u}(t)=\sum_{n=1}^{N} \exp \left[-j\left(2 \pi v_{n} t-\theta_{k}(t)\right)\right] \]where the \(\theta_{k}(t)\) are statistically independent and uniformly distributed
Consider a single-mode laser emitting light described by the analytic signal\[ \mathbf{u}(t)=\exp (-j[2 \pi \bar{v} t-\theta(t)]) \](a) Assuming that \(\Delta \theta=\theta\left(t_{2}\right)-\theta\left(t_{1}\right)\) is an ergodic random process, show that the autocorrelation function of
Show that the second moment \(\overline{I^{2}}\) of the intensity of a wave is not equal to the fourth moment \(\overline{\left[u^{(r)}\right]^{4}}\) of the real amplitude of that wave, the difference being due to a low-pass filtering operation and a scaling by a factor of 2 implicit in the
An idealized model of the normalized power spectral density of a gas laser oscillating in \(N\) equal-intensity axial modes is\[ \widehat{\mathcal{G}}(v)=\frac{1}{N} \sum_{n=-(N-1) / 2}^{(N-1) / 2} \delta(v-\bar{v}+n \Delta v) \]where \(\Delta v\) is the mode spacing (equal to the speed of
The gas mixture in a helium-neon laser (end mirrors removed) emits light at \(633 \mathrm{~nm}\) with a Doppler-broadened spectral width of about \(1.5 \times 10^{9} \mathrm{~Hz}\). Calculate the coherence time \(\tau_{c}\) and the coherence length \(l_{c}=c \tau_{c}(c=\) the speed of light) for
(Lloyd's mirror) A point source of narrowband light is placed at distance \(s\) above a perfectly reflecting planar mirror. At distance \(d\) away, the interference fringes are observed on a screen, as shown in Fig. 5-4p. The complex degree of (self-) coherence of the light is\[
Consider a Young's interference experiment performed with broadband light.(a) Show that the field incident on the observing screen can be expressed as\[ \mathbf{u}(Q, t)=\tilde{K}_{1} \frac{d}{d t} \mathbf{u}\left(P_{1}, t-\frac{r_{1}}{c}\right)+\tilde{K}_{2} \frac{d}{d t} \mathbf{u}\left(P_{2},
As shown in Fig. 5-7p, a positive lens with focal length \(f\) is placed in contact with a pinhole screen in a Young's experiment. The lens and pinhole plane are at distance \(z_{1}\) from the source and the observation screen is at distance \(z_{2}\)from the lens and pinhole screen. For
Consider a Michelson interferometer that is used in a Fourier spectroscopy experiment. To obtain high resolution in the computed spectrum, the interferogram must be measured out to large pathlength differences, where the interferogram has fallen to very small values.(a) Show that under such
In the Young's interference experiment shown in Fig. 5-5-9pp, the normalized power spectral density \(\widehat{\mathcal{G}}(v)\) of the light is measured at point \(Q\) by a spectrometer. The mutual coherence function of the light is known to be separable,\[ \boldsymbol{\Gamma}\left(P_{1}, P_{2},
A monochromatic, unit-amplitude plane wave falls normally on a "sandwich" of two diffusers. The diffusers are moving in opposite directions with equal speeds, as shown in Fig. 5-5-10pp. The amplitude transmittance of the diffuser pair is expressible as\[ \mathbf{t}_{A}(\xi,
Show that under quasimonochromatic conditions, mutual intensity \(\mathbf{J}\left(P_{1}, P_{2}\right)\) obeys a pair of Helmholtz equations\[ \begin{aligned} & abla_{1}^{2} \mathbf{J}\left(P_{1}, P_{2}\right)+\bar{k}^{2} \mathbf{J}\left(P_{1}, P_{2}\right)=0 \\ & abla_{2}^{2}
Consider an incoherent source radiating with spatial intensity distribution \(I(\xi, \eta)\).(a) Using the Van Cittert-Zernike theorem, show that the coherence area of the light (mean wavelength \(\bar{\lambda}\) ) at distance \(z\) from the source can be expressed as\[ A_{c}=(\bar{\lambda} z)^{2}
A Young's interference experiment is performed in the geometry shown in Fig. 5-14p. The pinholes are circular and have finite diameter \(\delta\) and spacing \(s\). The source has bandwidth \(\Delta v\) and mean frequency \(\bar{v} . f\) is the focal length of the lens. The following two effects
The sun subtends an angle of about \(32 \mathrm{~min}\) of arc ( 0.0093 radians) on Earth. Assuming a mean wavelength of \(550 \mathrm{~nm}\), calculate the coherence diameter and coherence area of sunlight observed on Earth (assume quasimonochromatic conditions).
Show that for quasimonochromatic, stationary thermal light, the fourth-order coherence function\[ \boldsymbol{\Gamma}_{1234}\left(t_{1}, t_{2}, t_{3}, t_{4}\right)=E\left[\mathbf{u}\left(P_{1}, t_{1}\right) \mathbf{u}\left(P_{2}, t_{2}\right) \mathbf{u}^{*}\left(P_{3}, t_{3}\right)
The output of a single-mode, well-stabilized laser is passed through a spatially distributed phase modulator (or a phase-only spatial light modulator that is changing with time). The field observed at point \(P_{k}\) at the output of the spatial light modulator is of the form\[
Show that for light with a Lorentzian spectral profile, the parameter \(\mathcal{M}\) is given by\[ \mathcal{M}=\left[\frac{\tau_{c}}{T}+\frac{1}{2}\left(\frac{\tau_{c}}{T}\right)^{2}\left(e^{-2 T / \tau_{c}}-1\right)\right]^{-1} \]
Examination of Fig. 6.5 shows that a relatively abrupt threshold in values of \(\tilde{\lambda}_{n}\) occurs as \(n\) is varied. In particular, as a rough approximation,\[ \tilde{\lambda}_{n} \approx \begin{cases}1 & nn_{\text {crit }}\end{cases} \]where \(n_{\text {crit }}=2 c / \pi\). Show
Suppose that we wish to know the standard deviation of the phase of \(\mathbf{J}_{12}(T)\) under the condition that the measurement time \(T\) is sufficiently long compared with the correlation time \(\tau_{c}\) of the wave incident on the points \(P_{1}\) and \(P_{2}\) that the noise cloud
Under the same conditions described in the previous problem, the fluctuations of the length of \(\mathbf{J}_{12}(T)\) are caused mainly by the fluctuations in the real part of the noise. Making this assumption, show that the rms signal-to-(self-) noise ratio associated with the measurement of
Compare the rms signal-to-(self) noise ratios achievable using amplitude interferometry and intensity interferometry to measure \(\mu_{12}\) under the assumption that \(T / \tau_{c} \gg 1 / \mu_{12}^{2}\). Which method is more sensitive?
Find what modification of Eq. (6.3-32) must be made if the light incident on the detectors is unpolarized thermal light. SM rms M/2 1+2 B/b - b 4B (6.3-32)
Given that the condenser lens of Fig. 7.4 is circular with diameter \(D\), specify the diameter required of a circular incoherent source to assure that the approximation of Eq. (7.2-7) is valid. (a,) Source Condenser lens (x,y) (5.1) Object 22 Figure 7.4 Illumination optics.
In the optical system shown in Fig. 7-4p, a square incoherent source ( \(L \times L\) meters) lies in the source plane. The object consists of two pinholes separated in the \(\xi\) direction by distance\[ X=\frac{\bar{\lambda} f}{D} \]where \(\bar{\lambda}\) is the mean wavelength, \(f\) is the
A partially polarized thermal light wave is incident on a photodetector. The total incident integrated intensity can be regarded to consist of two statistically independent components, \(W_{1}\) (mean \(\bar{W}_{1}\) ) and \(W_{2}\left(\right.\) mean \(\bar{W}_{2}\) ). Thus the probability density
Light from a partially polarized pseudothermal source is found to have a coherence matrix of the form\[ \underline{\mathbf{J}}=\bar{I}\left[\begin{array}{ll} 1 / 2 & -1 / 6 \\ -1 / 6 & 1 / 2 \end{array}\right] \]This light falls on a photoevent counting photodetector, for which the counting
Assuming factorability of the complex degree of coherence of polarized thermal light, demonstrate that when the photodetector area is much larger than the coherence area of the incident light, the number of spatial degrees of freedom reduces to the ratio of the photodetector area to the coherence
Given the assumption that the energy levels of an harmonic oscillator can take on only the values \(n h v\) and given the Maxwell-Boltzmann distribution of occupation numbers, show that the probability distribution \(P(n)\) of occupation number \(n\) being realized is a Bose-Einstein distribution
Show that when two identical detectors in an intensity interferometer have read noise with the same variance \(\sigma_{\text {read }}^{2}\) and when the average intensities incident on the two detectors are identical, the signal-to-noise ratio obtained in a single counting interval becomes\[

Statistical Optics Wiley Pure And Applied Optics 2nd Edition Joseph W Goodman - Solutions

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