Questions and Answers of Statistical Optics

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
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
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\[
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)
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,
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}
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
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},
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
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
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
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
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
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
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
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},
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\[
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
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
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
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)
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
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 /
(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
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\[
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
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
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
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}
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
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
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}
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
(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,
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}
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
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
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
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
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},
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
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
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
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},
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
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 /
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
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
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
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 /
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,)
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
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, \(W_{1}\)
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
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
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
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