2. Classical Coherence¶
PDF pages 63–96
Chapter 2 Classical Coherence Coherence theory is concerned with characterizing light, especially its fluctuation properties that influence how it will act in an interference-type experiment. This is covered to some extent by the spectral profile of the field, which is one of the themes here, but isn’t the full story. Obviously things will get more interesting in the quantum case, but it is important to establish some classical results immediately so we can characterize atomic radiation. Let’s now consider an interference experiment that involves a range of frequencies. Recalling the interference of two monochromatic plane waves, we can write the superposition of the two waves as E(+)
20 ei(kz−\omegat)ei\phi. (2.1) Here, \phi is a relative phase difference between the two waves. Recall that we are just writing the positive- frequency components, so that the physical fields also must include the negative-frequency parts. The intensity of the superposition is
\eta D (Esum)2E . (2.2) Writing this out explicitly in terms of the component fields, Isum = 1 \eta E(+)
20 e−i(kz−\omegat)e−i\phi2 . (2.3) The optical terms of the form exp(\pmi2\omegat) vanish in the time average, so we obtain Isum = 2 \eta E(−) 10 E(+) 10 + 2 \eta E(−) 20 E(+) 20 + 2 \eta E(−) 10 E(+)
\eta E(−) 20 E(+) 10 e−i\phi
2 \eta E(−) 10 E(+)
. (2.4) Again, the interference is in the terms with the relative phase \phi. Suppose that the phase difference represents a difference in optical path length, in the form of a time
2 \eta E(−) 10 E(+)
. (2.5) Now let’s handle the case of multiple frequencies. To simplify things, we’ll assume that the two waves have equal amplitude and come from a common source. Then the intensity density at frequency \omega is
2
. (2.6)
2.1 Wiener-Khinchin Theorem¶
Phase Space and Operator Representation¶
Definition. The set of pairs \((q, p)\) with \(-h < q < h\)¶
and \(-\infty<p<\hbar(2\pi/h)^{1/2}\) is the configuration space for this system (the phase space). A function \(\phi(\alpha,\beta): \mathbb{R}^2 \to
\mathbb{C}, (\alpha, \beta) \mapsto \langle q|\psi\rangle\) and a function \(f(p), p\in \mathbb{R}\) is the momentum representation of state \(\phi\). We have:
$$|a|^ = |b^\rangle\langle b|. $$
Definition. The expectation value in any pure states, for all operators A that exist and well-defined on it (as do their spectral decomposition), can be written as \(w_A(\alpha,\beta)= \mathrm{Tr} (\rho w_{A})\), where \(\rho = |\psi\rangle\langle\psi|\) is the density operator or statistical state. We know:¶
Definition. The trace of a bounded linear operator T on \(S(\mathbb{R}^n)\), denoted \(\mathrm{Tr}(T), T = (x,y)\). This has the important property:¶
We have two main ways to compute expectation values for operators \(A\):
1. Find eigenfunctions and use spectral decomposition of \(\rho\) with respect to A. The result is that any function on space \(S(\mathbb{R}^{2n})\), defined by a bounded self-adjoint operator (e.g., Hamiltonian), has Fourier transform:
Given operators in phase-space representation, the Weyl quantization maps functions from classical phase space to quantum operators. The correspondence is established through an integral formula involving creation and annihilation operators:
$\(H = \langle H(\alpha,\beta)|\psi\rangle.\)$
The expectation value of any operator \(A\) can be expressed in terms of the Wigner function \(W_{|\psi}(q,p)\) via its spectral decomposition. Any self-adjoint operator on \(\mathcal{Q}_1\), including bounded observables, can be represented through functional calculus:
$\(w_A(0) = \frac{\partial}{\partial t} |a(t)|^*.\)$
where \(A(\alpha,\beta)\) corresponds to operators in phase space. The trace formula gives the expectation as \(\langle A\rangle_{|\psi}\rangle = \mathrm{Tr}(A)\phi\). Any operator-valued function can be expanded using eigenfunctions and their associated spectral measures:
$\(\sum_k a_n |a|_b^2 + c,\)$
with \(C^\infty(\mathbb{T}^{6n})\), \(W(q,t)\). This leads to the Moyal product for quantum mechanics with classical phase space. The Wigner transform of any density operator \(\rho \geq 0, \mathrm{Tr}\,\rho=1\):
$$\omega_{|\psi}(p|a|^*), $$
where \(S^*\) is a bounded linear functional on operators (e.g., expectation value). In particular:
For all pure states and their spectral decompositions exist. If the state has discrete spectrum with eigenvalues \(\lambda_k\) for any density operator, then its Wigner function can be written as
$\(p_0(t) = \int p(\theta') dt.\)$
This gives another representation of phase-space observables in quantum mechanics through functional calculus and spectral decomposition:
Definition. The canonical commutation relation (CCR\(_n\))\(\left[ Q_i, P_j \right]_\star^{\dagger}(h), f(Q,P)\) where \(f(g(\alpha,\beta))\). This defines a map from classical observables to quantum operators with noncommutative product structure:¶
Definition. The algebraic relation for position and momentum is¶
where \(\mathcal{W}_{n}(g(\cdot))\) represents Weyl quantization of the operator \(A\). The algebra generated by operators satisfying CCR\(_{2m}\) forms a noncommutative structure with spectral decomposition and functional calculus:
Definition. Operators on phase space form an associative *-algebra where commutation relations hold for bounded self-adjoint operators (Hamiltonian observables). Any observable can be expressed through its Wigner transform \(W(q,t)\), which is positive semidefinite density function. The correspondence between classical observables and their quantum counterparts is given by¶
The spectral decomposition of any self-adjoint operator \(S\) yields eigenfunctions \(\psi_k, k=1,n\). For bounded operators with discrete spectrum \(\sigma(S)\) where all moments and expectation values are well-defined. The trace formula gives the Wigner function as a weighted sum over density matrix elements:
$$|\langle\alpha| - \beta\rangle|^ |b^\rangle\langle b|. $$
This completes our discussion of phase-space representations, including operator algebras with appropriate domains and spectral properties for quantum observables.
2.2 Optical Wiener-Khinchin Theorem¶
2.2 Optical Wiener–Khinchin Theorem as the square of a signal amplitude. Thus, the Wiener–Khinchin theorem states that the Fourier transform of the autocorrelation function gives the energy spectrum. There is one subtle point to these definitions: for some signals, such as steady optical signals, the correlation integral diverges: Z \infty −\infty
(2.15) In this case, we should consider a time average instead of the normal integral. For an averaging time of T,
T Z T /2 −T /2
(2.16) For bounded signals, this integral is guaranteed to converge. To be physically sensible, T should be a suitably long observation time (e.g., long enough to resolve the frequency spectrum). For such signals, we can write the Wiener–Khinchin theorem as
(2.17) Here we have defined a finite-time Fourier transform by FT [f] := 1 T Z T /2 −T /2 f(t) ei\omegat dt. (2.18) Defining this seems a bit funny, but it avoids problems with singular spectra. Now the Wiener–Khinchin theorem says that the Fourier transform of the (time-averaged) correlation function is the power spectral density, or the energy spectral density per unit time. For a stationary process, the correlation function is independent of t (generally for a sufficiently long averaging time T). Then we can extend the averaging time T −\rightarrow \infty. Denoting this long-time averaging limit as
(2.19) we can thus write the Wiener–Khinchin theorem as
T \rightarrow \infty T
Z T /2 −T /2 f(t) ei\omegat dt
. (Wiener–Khinchin theorem, time-average form) (2.20) Again, the right-hand side is the power spectral density, and in this form it is more clear that this is the energy density per unit time. 2.2 Optical Wiener–Khinchin Theorem In terms of stationary optical fields, the Wiener–Khinchin theorem (2.20) becomes Z \infty
(optical Wiener–Khinchin theorem) (2.21)
field. Note that from the inverse Fourier transform, there is conventionally a factor of 1/2\pi, but it is missing
which gives the total intensity: Z \infty
\eta D |E(+)(t)|2E . (2.22)
2.2.1 Michelson Interferometer¶
Chapter 2. Classical Coherence In optics, as in statistics and other fields, it is conventional to define a normalized correlation function
, (degree of first-order temporal coherence) (2.23) so that
Z \infty
g(1)(\tau). (2.24)
That is, a correlation value of unity indicates perfect correlation. Note that we could just as well have written the correlation function as \langle E(+)(t + \tau)E(−)(t)\rangle , but it turns out that the order of E(+) and E(−) matters in quantum mechanics, so we’ll be careful to stick to the form in Eq. (2.23). Notice also that from the definition of g(1)(\tau), the correlation function is subject to the constraint
(2.25) (time symmetry of correlation function) In quantum optics, g(1)(\tau) is called the degree of first-order temporal coherence. The light is said to
This function can be generalized to include spatial correlation by
p
, (degree of first-order coherence) (2.26) which is the degree of first-order coherence. We will focus on the case (2.23) of temporal coherence. Returning to the interference result of Eq. (2.7), we find Itotal = 2 Z \infty
2 \eta Z \infty |E(−)
= Z \infty
n 2 + h
io , (2.27) and thus Itotal = 2 Z \infty
n 1 + Re h g(1)(\tau) io . (interferometer signal in terms of g(1)(\tau)) (2.28) It is worth keeping in mind that the form (2.13) is still relevant for pulsed fields, in which case the result here is modified so that the coefficient in front is the integrated power rather than intensity. 2.2.1 Michelson Interferometer One example where the interference result (2.28) arises directly—and thus where the optical Wiener–Khinchin theorem is very useful—is in the Michelson interferometer. The interferometer splits and interferes the beam with itself, and the path-length difference \tau varies with the displacement of one of the mirrors. Eo(+)o(t) detector
vary t
2.2.3 Spectrum of Atomic Radiation¶
2.2 Optical Wiener–Khinchin Theorem The photodetector at the output measures the time average of the product of the fields, up to an overall constant, and thus measures g(1)(\tau), just as in Eq. (2.28). The Michelson interferometer thus gives a method for measuring the spectrum of an optical field. The idea is then to digitize the output of the photodetector as a function of the mirror displacement, effectively recording Re[g(1)(\tau)]. Computing the Fourier transform of the signal on the computer gives the spectrum I(\omega). (Note that I(\omega) is real, and so the imaginary parts of g(1)(\tau) don’t contribute to the spectrum.) This is the technique behind, for example, Fourier-transform infrared (FTIR) spectroscopy. 2.2.2 Example: Monochromatic Light As a simple example, let’s compute the correlation function for monochromatic light and verify that the Wiener–Khinchin theorem makes sense. Let’s take a monochromatic wave of the form
e−i\omega0t. (2.29) The correlation function is D
E = E(+)
(2.30) In normalized form, this function becomes
(2.31) Thus, a wave with harmonic time dependence leads to a harmonic correlation function. This is true inde- pendent of the phase of the input field; that is, the correlation function does not reflect any extra phase in E(+) . The power spectrum is easy to calculate via the Wiener–Khinchin theorem: F hD
Ei = E(+)
F
= E(+)
(2.32) Again, the harmonic time dependence produces a single frequency in the power spectrum. Of course, with our convention of positive and negative frequencies, there would be a matching \delta(\omega + \omega0) component, but this is already included in the one-sided spectrum I(\omega). Let’s now compute the power spectrum directly, using Eq. (2.17). The normal spectrum is F h E(+)(t) i
(2.33) The finite-time transform is FT h E(+)(t) i = 1 T Z T /2 −T /2 E(+)
(2.34) Note that the value of the sinc function is 1 at \omega = \omega0. Thus, the spectrum is F ∗h E(+)(t) i FT h E(+)(t) i
E(+)
(2.35) This result is consistent with Eq. (2.32). Note that without being careful with finite-time transforms, we would run into something bad involving the square of a \delta-function. 2.2.3 Spectrum of Atomic Radiation With this formalism, it is straightforward to analyze the spectrum of the scattered light from the Lorentz atom. If the atom is briefly excited, then the amplitude of the field decays exponentially with time. The
2.2.4 Normalized One- and Two-Sided Spectra¶
Chapter 2. Classical Coherence autocorrelation function does the same, and the Fourier transform of a decaying exponential is a Lorentzian lineshape (see Problem 2.1). Thus, the radiated spectrum has the form
(2.36) (Actually, this turns out to be the spectrum s+(\omega) that we will define below, corresponding to a coherence
dipole oscillates sinusoidally in steady state at the driving frequency \omegaL (say of the driving laser), rather than
The above broadened spectrum is a transient effect that is swamped by this ‘‘elastic’’ peak (elastic since the scattered light has the same frequency as the incident light). 2.2.4 Normalized One- and Two-Sided Spectra Now we will be a bit more precise about the nature of the spectrum, to avoid confusion with different possible conventions for the power spectrum. First, it’s convenient to define a normalized spectral density (lineshape function)
I(\omega) Z \infty
. (2.37) (normalized spectral density) Note that this spectrum extends only over positive frequencies, as the intensity spectrum I(\omega) corresponded to physical frequency components. Examining the inverse Fourier transform, we combine Eqs. (2.21) and (2.24) to obtain
Z \infty
(first-order coherence in terms of normalized, one-sided spectral density) (2.38) However, the usual inverse Fourier transform has an integral extending over both positive and negative frequencies. We can obtain something of this form by considering the real part of the correlation function, Re g(1)(\tau)
g(1)(\tau) ∗
= 1 Z \infty
Z \infty
(2.39) so that if we define a two-sided spectrum s↔(\omega) (for all \omega \in R) via
(\omega > 0) s(\omega)
(\omega < 0), (two-sided spectrum in terms of one-sided spectrum) (2.40) which satisfies
(2.41) (symmetry of two-sided spectrum) we obtain Re g(1)(\tau) = Z \infty −\infty
(first-order coherence in terms of two-sided spectrum) (2.42) We have lost the imaginary part, but only the real part can contribute to a physical result: g(1)(\tau) must always be accompanied by its conjugate, as we saw in the interference experiment.
2.2 Optical Wiener–Khinchin Theorem Inverting this last relation, we have¶
2\pi Z \infty −\infty Re h g(1)(\tau) i
= 1 2\pi Z \infty −\infty
= 1 2\pi Z \infty
(\omega \in R). (two-sided spectral density in terms of first-order coherence) (2.43) The one-sided spectrum can then be written in terms of the double-sided spectrum as
(\omega > 0) s↔(\omega)
(\omega < 0), (two-sided spectrum in terms of one-sided spectrum) (2.44) so that the one-sided spectrum simply concentrates all the power at positive and negative frequencies on the positive side. Of course, this cannot be done for any spectrum, but the power spectral density—the Fourier transform of the coherence function—has a special structure due to the symmetry of the correlation function. Thus, we can write the one-sided spectrum as
\pi Z \infty −\infty Re h g(1)(\tau) i
= 1 \pi Z \infty −\infty
= 1 \pi Z \infty
(one-sided spectral density in terms of first-order coherence) (2.45) in terms of the coherence function. Combining Eqs. (2.42) and (2.44), we then find Re g(1)(\tau) = Z \infty
(first-order coherence in terms of one-sided spectrum) (2.46) for the reverse transformation. Finally, note that it can be more convenient to associate spectra separately with g(1)(\tau) and its con- jugate, by defining
4\pi Z \infty −\infty
4\pi Z \infty
(\omega \in R), (component spectral densities in terms of first-order coherence) (2.47) so that
(2.48) (symmetry of component spectra) Then the relations
(\omega \in R)
(component spectral densities in terms of first-order coherence) (2.49) recover the total spectra (2.43) and (2.45), respectively.
2.4 Coherence Time, Coherence Length, and Uncertainty Measures¶
$\(\nabla^2 \xi - M^2 c^{8/7} = 0,\)$ (469)
The equation of state for the fluid follows from conservation of particle number:
where \(n\) is the baryon density. For a perfect fluid with pressure \(p\), energy density \(\mathcal{E}\), and four-velocity normalization \(u_\nu u^\nu = -1\):
The pressure is a function of energy density alone: \(p = p(\mathcal{E})\). For the fluid rest frame at each point, these relations hold with appropriate interpretation using local inertial coordinates \((t,x,y,z)\) where \(\rho\) measures mass per unit coordinate time squared near infinity.
2.4 Coherence Time, Coherence Length, and Uncertainty Measures and thus see that the ‘‘widths’’ of g(1)(\tau) and s↔(\omega) are inversely related. The problem is that for some useful functions in optics, these uncertainties can diverge (e.g., for Lorentzian functions) or vary by orders of magnitude. The uncertainty inequality is always satisfied, of course, but as a practical relation for the temporal and frequency widths, this is less useful. Thus we will adopt other uncertainty conventions in time and frequency1 We can define the coherence time as the power-equivalent width of the correlation function:
Z \infty −\infty Re h g(1)(\tau) i d\tau. (coherence time, power-equivalent width) (2.57)
with the same area as |Re[g(1)(\tau)]|2. That is, \delta\tau is the width of a unit-height box signal with the same power as |Re[g(1)(\tau)]|2. t Reo[go(1)o(t)]2 dt same area We’ll take a slightly different convention for the width of the frequency spectrum. Let’s define the average value of the one-sided normalized spectrum as ¯s = Z \infty
Z \infty −\infty s 2
(2.58) That is, we’re regarding s(\omega) as a probability distribution (because it has the appropriate normalization), and then we’re calculating the expectation value of s(\omega) with respect to itself. Then we’ll define the effective frequency width by the reciprocal of the average value:
Z \infty
−1 . (2.59) (spectral width) Thus, \delta\omega is the width of the box function of unit area and height ¯s. w so(w) dw s- unit area Note that we have constructed the diagram as if s(\omega) is two-sided, but the argument carries through for the one-sided case as well, being careful to keep track of factors of 2. The definitions for \delta\tau and \delta\omega look like inverses, but they’re really quite different because of the different normalizations of the two functions. The big advantage of these definitions is that they are related in a simple way. We can write
Z \infty −\infty Re h g(1)(\tau) i d\tau Z \infty −\infty s 2
. (2.60) 1as in Max Born and Emil Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge, 1999).
2.5 Interference Between Two Partially Coherent Sources¶
Chapter 2. Classical Coherence We can use Parseval’s theorem to evaluate this ratio, which states that the signal power is equivalently measured in either the time or frequency basis: Z \infty −\infty
2\pi Z \infty −\infty ˜f(\omega)
d\omega. (2.61) (Parseval’s theorem) Noting again that Re[g(1)(\tau)] and 2\pis↔(\omega) form a Fourier-transform pair, and recalling the factor of two for using the one-sided spectrum, we can use this to write
(2.62) (uncertainty relation) This ‘‘uncertainty relation’’ is a strict equality, valid for any functions as long as the measures exist and are finite. Neat, eh? The point of all this is, for time (optical path length) delays larger than the coherence time \delta\tau, the fringe visibility is mostly gone. This coherence time corresponds to a physical path length difference
(2.63) (coherence length) which is called the coherence length. Here are some examples. For a He-Ne laser, the laser line width is \delta\nu = \delta\omega/2\pi ∼1 GHz. This corresponds to a coherence time of \delta\tau ∼1 ns, or a coherence length ℓc ∼30 cm. On the other hand for a light bulb that spans the visible wavelength range of 400-700 nm, the line width is
\lambda
(2.64) This gives a coherence time \delta\tau ∼3 fs and a coherence length ℓc ∼1 µm. So in fact it is possible to see interference of white light in a Michelson, but it’s very difficult because the path lengths must be matched to µm accuracy. On the other hand, it’s much easier to observe interference or record a hologram with light from a He-Ne laser, because it remains coherent on the scale of about a foot. 2.5 Interference Between Two Partially Coherent Sources In general, we can now look at the interference pattern between two partially coherent sources, represented by the two fields E(+) (t) and E(+) (t). The second field has an adjustable time delay of \tau. Then the intensity of the superposition of these waves is I = 2 \eta D E(+)
E = 2 \eta D E(+) (t)
E + 2 \eta D E(+)
E + 2 \eta D E(−) (t)E(+)
E + c.c.
p I1I2 Re h g(1) 12 (\tau) i , (2.65) where g(1) 12 (\tau) is the normalized cross-correlation function: g(1)
D E(−) (t)E(+)
E D E(−) (t)E(+) (t)
E . (2.66) (normalized cross-correlation) Again, the visibility is
I1 + I2 g(1) 12 (\tau) . (2.67) (visibility for two different fields) This tells us that very different beams, resulting in little correlation, don’t interfere very well.
2.6 Second-Order Coherence¶
2.6 Second-Order Coherence 2.6 Second-Order Coherence The degree of second-order coherence is the autocorrelation function for the intensity, rather than the field. We can define this function as
. (degree of second-order coherence) (2.68) Classically, this expression becomes
\langle I\rangle 2 . (2.69)
E(+)(t) as with E(+)(t + \tau), but this constraint is somewhat different for g(2)(\tau), because the normalization convention is somewhat different. First let’s consider the variance of I(t), which is manifestly nonnegative:
2 \ge 0. (2.70) Multiplying this out, this constraint becomes
(2.71) which implies the boundary condition
(2.72) (classical bunching constraint) for the second-order coherence. There is no corresponding upper bound, and this argument fails for \tau̸ = 0. This inequality is classical, and we will see that it is possible for a quantum field to violate this condition (see Section 5.7.5 or Problem 8.12). This inequality effectively provides a boundary between quantum and classical statistical behavior. We can also start with the inequality h
i2 \ge 0, (2.73) which when multiplied out yields
(2.74) Taking the time average of this relation then gives
(2.75) This implies
(2.76) (classical constraint) so the second-order coherence is a nonincreasing function, at least in the vicinity of \tau = 0. The second-order coherence is related to how ‘‘concentrated’’ or ‘‘bunched’’ the intensity function is.
is coherent at second order. Note that we can generalize the correlation functions in an obvious way to arbitrarily high order, where it turns out that a monochromatic wave always takes on the value unity and is thus coherent to all orders. However, the first- and second-order degrees of coherence are the most important and relevant for experiments.
2.6.1 Thermal Light¶
Chapter 2. Classical Coherence For the opposite extreme of a periodic train of short pulses,
X n \delta(t −nT), (2.77) we can see that
T (2.78) and
T X n
(2.79) so that
X n
(2.80) Note that in deriving Eq. (2.79), we have used the relation Z
m\rightarrow \infty Z
(2.81) where we have used the definition of the delta function as a limit of a sequence of normalized Gaussians of decreasing width. This relation also makes sense as a convolution, since the delta function is the identity kernel of the convolution operation. Thus, we see that very large values of g(2)(\tau) correspond to temporally concentrated intensity. This is the classical manifestation of photon bunching. 2.6.1 Thermal Light To model light from a thermal source, we will assume that E(+)(t) fluctuates as a stationary Gaussian random process. That is, E(+)(t) is a complex Gaussian random variable whose statistics are time-independent, and E(+)(t) is correlated with the itself at other times as required by the power spectrum. Gaussian random processes are fundamental in modeling noisy systems, as they allow for tremendous simplifications. We will call Z a complex Gaussian random variable if Z = X + iY , where X and Y are independent and identically distributed Gaussian random variables. The joint probability density of X and Y is
(2.82) if we assume that \langle X\rangle = \langle Y \rangle = 0. Then we can write the probability density of Z as
= \pi\sigma 2 z
z , (2.83) where \sigma 2
(2.84) is the variance of Z. Generalizing this to N complex Gaussian variables Z1, . . . , ZN, the probability density becomes
e−z∗
(2.85) where summations are implied by repeated indices, and
\beta\rangle (2.86)
2.6.2 Experiment of Hanbury Brown and Twiss¶
2.6 Second-Order Coherence is the covariance matrix. For such complex Gaussian variables, one can show that high-order moments factor as2 \langle Z∗ \alpha1Z∗
X all N! pairings \langle Z∗
(Gaussian moment factorization) (2.87) if M = N, with the moment vanishing otherwise. We can always factor high-order moments because the Gaussian distribution (2.85) itself only depends on the quadratic moments. In particular, we need the factorization for the fourth-order moment \langle Z∗ 1Z∗
2Z1\rangle
1Z2\rangle |2. (2.88) Applying this to the second-order coherence function (2.68), we find the important relation
(2.89) (thermal-field coherence constraint)
t go(2)o(t) The correlation function thus drops from 2 to 1 with \tau, as shown here for light with a Gaussian power spectrum (in addition two the Gaussian fluctuations). Thus, thermal light exhibits some degree of bunching. 2.6.2 Experiment of Hanbury Brown and Twiss One important feature of the second-order coherence function is that it can be measured with a reasonably simple setup, the famous Hanbury-Brown–Twiss apparatus.3 A simplified schematic of this type of apparatus is shown here. delay t Eo(+)o(t) detector detector An input field is divided by a beam splitter, and the two components are monitored by two photodetectors. The two detector signals are fed into a signal multiplier (mixer), though only after a variable time delay is added to one of the signals. The mixer signal is fed through a low-pass filter, which can be thought of as a integrator with a running time average. 2see Leonard Mandel and Emil Wolf, Optical Coherence and Quantum Optics (Cambridge, 1995), p. 38. 3R. Hanbury Brown and R. Q. Twiss, ‘‘Correlation Between Photons in Two Coherent Beams of Light,’’ Nature 177, 27 (1956).
2.7.1 Spectra of Phase and Frequency Fluctuations¶
Chapter 2. Classical Coherence This setup, because it effectively correlates the two intensities, seems to give the g(2)(\tau) function as its output signal directly. But the detectors are generally ac-coupled, so that the detectors monitor I(t) −\langle I\rangle . Then the output signal of the mixer is
(2.90) and the output signal of the low-pass filter is
(2.91) After proper normalization, we can see that the output is just g(2)(\tau) −1. Thus, for thermal light this setup should give some signal for \tau = 0 that decays to zero for large delays. The setup here makes it somewhat intuitive how the quantum field will violate the inequality (2.72). If the quantum field arrives as a stream of separated, individual photons, then at any given time, only one of the photodetectors can ‘‘click.’’ That is, a single photon causes a detection event on one detector or the
phenomenon of antibunching. The treatment here is simplified, since taking into account the response time of the detectors compli- cates things considerably. The original experiment used the same idea to measure the spatial correlation of intensities, which can be used to measure the size of the optical source. This has important applications, for example, in astronomy, where the signals from two separated telescopes can be mixed in the same way. In such a stellar interferometry arrangement, the measured correlation function gives a measure of the diameter of the distant star. For our purposes, the quantum-mechanical version of this experiment that we will return to later demonstrates some of the uniquely quantum-mechanical features of light. 2.7 Phase Noise 2.7.1 Spectra of Phase and Frequency Fluctuations An important form of noise in oscillators and lasers is phase noise, for example where the time dependence of a signal has the form
(2.92)
frequency is just the time derivative of the phase:
dt
dt . (2.93) Thus, the phase noise translates into frequency noise as well. Given a signal with time dependence of the form exp[−i\phi(t)], let us define an unnormalized, one-sided spectral density of phase fluctuations via
Z \infty −\infty
(spectral density of phase fluctuations (one-sided)) (2.94) in analogy with Eq. (2.45). Correspondingly, if we are instead interested in the spectral density of frequency fluctuations
Z \infty −\infty D
E
(spectral density of frequency fluctuations (one-sided)) (2.95) These spectra are related by
(phase to frequency spectrum conversion) (2.96)
2.7.3 Spectrum of the Signal¶
2.7 Phase Noise (see Problem 2.4). Thus, for example, if \phi(t) is a random-walk process, then ˙\phi is a white-noise process, so that S\omega(\omega) is independent of frequency (i.e., \phi(t) ∝W(t) is a Wiener process, whose derivative gives white noise, as in Chapter 17). Inverting the above relations, we find
\pi Z \infty
(phase correlation in terms of phase spectral density) (2.97) as we find by analogy to Eqs. (2.45) and (2.46), noting the difference in the 2\pi factor. Also,
\pi Z \infty
\omega2 d\omega, (phase correlation in terms of freqency spectral density) (2.98) as follows from Eq. (2.96) 2.7.2 Variance of Phase Fluctuations A useful quantity to consider is the variance of the phase fluctuation in a time \tau. Thus, we define
(2.99) which we may regard as independent of t in a statistical sense, under the assumption of a stationary process. Then D
= 2
\phi2(t)
= 2 \pi Z \infty
\omega2 d\omega, (2.100) where we used Eq. (2.98). Thus, we have the result D
= 4 \pi Z \infty
\omega2 d\omega. (variance of phase fluctuations related to frequency-noise spectrum) (2.101) Note that this integral gives a finite result even for white noise, which has an unnormalized spectrum, due to the 1/\omega2 factor, but it diverges for a ‘‘1/f’’ spectrum. 2.7.3 Spectrum of the Signal Supposing the phase of a signal fluctuates, with a particular spectrum of frequency fluctuations. What is the effect on the spectrum of the signal itself?4 Suppose that we have an optical signal of the form
(2.102) (technically, we could be mixing positive- and negative-frequency components, but this doesn’t really matter). We then want the correlation function g(1)(\tau) ∼ D
E , (2.103) which when normalized looks like
. (2.104) Making the assumption that ∆\phi(\tau) represents Gaussian noise (which, from the central-limit theorem, is guaranteed essentially provided that the variance of ∆\phi(\tau) is finite), we can rewrite this as
/2. (2.105) 4D. S. Elliott, Rajarshi Roy, and S. J. Smith, ‘‘Extracavity laser band-shape and bandwidth modification,’’ Physical Review A 26, 12 (1982) (doi: 10.1103/PhysRevA.26.12).
2.8.1 Photodetection Spectrum¶
Chapter 2. Classical Coherence (See Problem 2.5 for the intermediate steps here.) Now using Eqs. (2.45) for the one-sided, normalized spectrum,
\pi Z \infty −\infty
= 1 \pi Z \infty −\infty
/2 = 2 \pi Z \infty
/2, (2.106) and so
\pi Z \infty
−2 \pi Z \infty
\omega′2 d\omega′
(spectrum of the signal) (2.107) Thus, we have the spectral density for the signal itself, in terms of the spectral density for the frequency fluctuations. This is the spectrum that is more likely to be analyzed, e.g., on a spectrum analyzer (the phase- or frequency-fluctuation spectra would be observed directly only with a phase or frequency detector, for example, using a phase-locked loop to track the signal). 2.7.3.1 Example: White Noise Suppose we take the simple case of a flat, white-noise frequency-fluctuation spectrum:
(2.108) Then the exponent in Eq. (2.107) is −2 \pi Z \infty
\omega′2
\pi Z \infty
\omega′2
2 . (2.109) So
\pi Z \infty
= 1 \pi Z \infty
= 1 2\pi Z \infty
= 1 2\pi
(2.110) and thus
(heterodyne spectrum, white frequency noise) (2.111) which is a properly normalized, one-sided Lorentzian spectrum with a full width at half maximum of \gamma (the second term being the negative-frequency ‘‘mirror image’’ of the positive-frequency Lorentzian). 2.8 Optical Linewidth Measurements 2.8.1 Photodetection Spectrum The spectrum analyzer measures the power spectrum of the detector photocurrent Idet(t), which by the Wiener–Khinchin theorem is the Fourier transform of the autocorrelation function
Zin
(2.112)
2.8.2 Heterodyne Spectroscopy¶
2.8 Optical Linewidth Measurements where Zin is the input impedance of the spectrum analyzer (typically 50 Ω), which yields the appropriate units of power. The photocurrent is related to the optical intensity at the detector by
(2.113) where I(t) is the optical intensity, averaged over the detector surface, Adet is the detector area, and \etaI is the
I A 2 det \eta 2 0 Zin D E(−) det (t) E(−)
det (t) E , (2.114) where E(+) det (t) is the optical field at the detector. Thus, we find
I A 2 det \pi\eta 2 0 Zin Z \infty −\infty d\tau D E(−) det (t) E(−)
det (t) E
(2.115) for the spectrum-analyzer signal, where we introduce the factor of \pi to match the normalization convention of Eq. (2.45). This ensures that the integrated spectrum gives the total measured electrical power (at least
I A 2 det \pi\eta 2 0 Zin Z \infty d\tau D E(−) det (t) E(−)
det (t) E
(detected signal on spectrum analyzer) (2.116) due to the even symmetry of the correlation function. Spectrum analyzers typically use normal (not angular) frequencies, so we can write
I A 2 det \eta 2 0 Zin Z \infty −\infty d\tau D E(−) det (t) E(−)
det (t) E
(detected signal on spectrum analyzer) (2.117)
2.8.2 Heterodyne Spectroscopy In a heterodyne measurement, we beat two independent lasers together, but assume they are statistically identical—otherwise, it is not possible to attribute unbalanced fluctuations to the proper laser. laser 1 laser 2 detector Then the signal field is the superposition of the two fields, E(+)
(t), (2.118) where \eta1 and \eta2 are the intensity transmission coefficients through the beam splitter and any other optical components from the respective laser source to the detector. Then we may write the fields as E(+)
E(+)
(2.119)
Chapter 2. Classical Coherence where again the phase noises on the two lasers are independent but statistically identical. We also assume the two lasers have nearly the same center frequency, so we define the detuning
(2.120) However, we also assume that the frequency width induced by the phase fluctuations is small compared to |∆|. Now we must substitute this expression into the correlation function (2.114), noting that of the 16 total terms, the 10 that rotate in time with frequencies that are optical or |∆| will simply average to zero in the relatively slow spectrum-analyzer response time. Thus, we have Ganalyzer(\tau) ∝\eta 2 1 |E(+)
2 |E(+)
01 E(+)
01 E(+) 02 |2D
E , (2.121)
dc components to be blocked (not to mention far enough away from the spectrum of interest that we can ignore them, since we have assumed ∆to be a sufficiently large radio frequency), so that Ganalyzer(\tau) ∝\eta2
D
E , (2.122) where I1,2 are the (stationary) output intensities of the two lasers. We can again use the relation for Gaussian phase increments, D
= e−
/2, (2.123) as in Eq. (2.105), so that Ganalyzer(\tau) ∝\eta2
e−i∆\taue−
= \eta2
e−i∆\taue−
- c.c. = \eta2
cos ∆\tau e−
, (2.124) where we have used the independence of ∆\phi1(\tau) and ∆\phi2(\tau), and used the statistical identity to write
I A 2 detI1I2 \piZin Z \infty
−4 \pi Z \infty
\omega′2 d\omega′ , (heterodyne beat signal on spectrum analyzer) (2.125) where we have used Eq. (2.101) for the mean-square phase increment. Note that this spectrum has the same form as the spectrum of the optical signal in Eq. (2.107), except that \omega0 is replaced by ∆, and there is a factor of two in the exponent, meaning that the spectral part of the Fourier transform is squared. This means that the spectrum on the analyzer is the actual optical spectrum, but convolved with itself. This broadens the apparent line to a degree that depends on the precise form of the line shape. We can rewrite the main result here as
I A 2 detI1I2 Zin Z \infty
−1 \pi3 Z \infty
\nu′2 d\nu′ , (heterodyne beat signal on spectrum analyzer) (2.126) in terms of the more experimentally relevant frequency \nu. Here, we have rewritten the spectrum of frequency
I A 2 detI1I2/Zin (again, after removing dc terms).
2.8.3 Self-Heterodyne Spectroscopy¶
2.8 Optical Linewidth Measurements 2.8.2.1 Example: White Noise in Heterodyne Spectroscopy Returning to the white-noise example in Section 2.7.3.1,
(2.127) the exponent in Eq. (2.125) is −4 \pi Z \infty
\omega′2
\pi Z \infty
\omega′2
(2.128) This is the same result as in Section 2.7.3.1, with the replacement \gamma −\rightarrow \gamma/2. Then the normalized form of Eq. (2.125) is
\pi Z \infty
(2.129) carries through to the same result with the same rescaling of \gamma, so that
(heterodyne spectrum, white frequency noise) (2.130) which is a properly normalized, one-sided Lorentzian spectrum with a full width at half maximum of 2\gamma. This is what we expect for a self-convolution of a Lorentzian distribution. 2.8.3 Self-Heterodyne Spectroscopy Heterodyne spectroscopy is conceptually simple, but it may be that two identical copies of a laser are not available. An alternative approach in this case is to beat the output of a laser with a time-delayed and frequency-shifted version of itself.5 If the delay is long compared to the coherence time, then this is something like a heterodyne of two independent sources. The setup is shown here; the beam is split by an acousto-optic modulator (AOM), with the zeroth order beam coupling into a fiber for a delay, and the first order being shifted by the AOM frequency \omegam. laser AOM detector w0 w0 w0o+owm fiber delay td The fiber is assumed to cause a well-defined time delay of the undiffracted beam, of around 5 µs for a 1 km fiber. (A fiber delay of the first-order beam just amounts to taking a negative \taud.) For delays achievable in 5The results derived here were developed, e.g., in P. Gallion, F. J. Mendieta, and R. Leconte, ‘‘Single-frequency laser phase- noise limitation in single-mode optical-fiber coherent-detection systems with correlated fields,’’ Journal of the Optical Society of America 72, 1167 (1982) (doi: 10.1364/JOSA.72.001167); Philippe B. Gallion and Guy Debarge, ‘‘Quantum Phase Noise and Field Correlation in Single Frequency Semiconductor Laser Systems,’’ IEEE Journal of Quantum Electronics QE-20, 343 (1984) (doi: 10.1109/JQE.1984.1072399); Linden B. Mercer, ‘‘1/f Frequency Noise Effects on Self-Heterodyne Linewidth Measurements,’’ Journal of Lightwave Technology 9, 485 (1991) (doi: 10.1109/50.76663); Hanne Ludvigsen, Mika Tossavainen, and Matti Kaivola, ‘‘Laser linewidth measurements using self-homodyne detection with short delay,’’ Optics Communications
Chapter 2. Classical Coherence a fiber (∼km), though, there will in general be some residual level of correlation to deal with, which will be reflected in the spectrum. The combined field at the detector is E(+)
(2.131) where \taud is the delay due to the fiber (we will ignore the other small delays in the optical system). We will assume the undelayed field to be shifted in frequency by the acousto-optic modulator by an amount \omegam, and the delayed field to be unshifted. Thus, writing out the explicit field phases, we have E(+)
(2.132) Comparing to Eq. (2.119), we see that this problem is equivalent to the heterodyne case, but with the replacements E(+) 01,02 −\rightarrow E(+)
Thus, adapting Eq. (2.122) for the correlation signal after the dc block, we have for the present case Ganalyzer(\tau) ∝\eta2
D
E . (2.133) Again using Eq. (2.123), we have Ganalyzer(\tau) ∝\eta2
. (2.134) The phase expectation value can then be transformed (Problem 2.6) so that Ganalyzer(\tau) ∝\eta2
−
+
= \eta2
−
+
/2. (2.135) Using Eq. (2.101), we then find (see Problem 2.7) Ganalyzer(\tau)∝\eta2
cos \omegam\tau exp −8 \pi Z \infty
\omega′2 d\omega′ . (2.136) Thus, the spectrum (2.116) is
I A 2 detI 2 \piZin Z \infty
−8 \pi Z \infty
\omega′2 d\omega′ , (self-heterodyne beat signal on spectrum analyzer) (2.137)
factor of 2 sin2(\omega\taud/2) in the integrand, due to the extra coherence in beating the signal with a time-delayed version of itself. Note that this factor reduces to unity if it is replaced by its average value, which we expect if this coherence is lost. For example, in the limit of large \taud, this factor is a rapidly oscillating function of \omega. So long as the oscillations are rapid on the scale of the structure of the spectrum and of the relevant values of \tau (i.e., \taud is much larger than the coherence time of the signal), it is a good approximation to replace this factor by unity, so that this expression collapses to the pure heterodyne result. Finally, we can again write the main result as
I A 2 detI 2 Zin Z \infty
−2 \pi3 Z \infty
\nu′2 d\nu′ , (self-heterodyne beat signal on spectrum analyzer) (2.138) in terms of the more experimentally relevant frequency \nu = \omega/2\pi.
2.8 Optical Linewidth Measurements 2.8.3.1 Example: White Noise in Self-Heterodyne Spectroscopy Returning once again to the white-noise example in Section 2.7.3.1,
(2.139) the exponent in Eq. (2.137) is −8 \pi Z \infty
\omega′2
\pi Z \infty
\omega′2
. (2.140) which follows most easily by comparing Eq. (2.135) with Eq. (2.124), noting that we adapt the result simply by making several time-offset copies of the heterodyne-exponent result. Then the normalized form of Eq. (2.137) is
\pi Z \infty
−\gamma
. (2.141) Now since \tau \ge 0 and the exponent is an even function of \taud, we can write
=
(2.142) Thus,
\pi Z |\taud|
Z \infty |\taud|
= 2 \pi Z |\taud|
- 2
Z \infty
= 1 \pi Z \taud
- 1
Z \infty
= 1 \pi Z \taud
- 1
Z \infty
(2.143)
Chapter 2. Classical Coherence Then using the integral representation of the delta function for the second integral and evaluating the first integral,
" 2\pi Z |\taud| d\tau
- c.c. #
= 1 2\pi "
−
- c.c. #
=
2\pi
- c.c.
=
2\pi
=
\gamma
(2.144) and so
\gamma
(self-heterodyne spectrum, white frequency noise) (2.145) The first term here is a modified Lorentzian spectrum, where oscillations in frequency are superimposed on the Lorentzian envelope. These oscillations become finer and smaller with increasing |\taud|, and vanish in the limit |\taud| −\rightarrow \infty, when we recover the heterodyne result (Lorentzian with a full width at half maximum of 2\gamma). The other terms likewise vanish in this limit. The delta-function term is the most obvious result of residual correlations between the optical signal and its time-delayed copy. Recalling that the correlation goes away exponentially with the delay, this is equivalent to an ensemble of optical signals that are exactly monochromatic and phase coherent, except for a phase jump at a time \tau to a completely random phase, where \tau is a random time with exponential probability distribution. Then the light tends to correlate perfectly at short time delays, and not at all for long ones; the exponential function here is just what we expect in the ensemble average, so the correlation decays away exponentially on the time scale of the coherence time. The spectrum is plotted below for several coherence times, including the heterodyne (\taud −\rightarrow \infty) Lorentzian limit. The oscillations in the tails are only slightly visible, but what is clearly evident is a strong impact on the apparent width of the line.
2.8 Optical Linewidth Measurements gtdoáo" gtdo=o4 gtdo=o3 gtdo=o2 gtdo=o1
-20 pgso(w) On a logarithmic vertical scale, the effects near the line center are mitigated, but the oscillations in the tails are more apparent. gtdoáo" gtdo=o4 gtdo=o3 gtdo=o2 gtdo=o1
-20 pgsoo(w) 10-3 Both plots were made under the assumption of a narrow line, \gamma ≫\omegam, so that we can ignore the contribution of the component centered at −\omegam (\gamma should also implicitly be small compared to the optical frequency). 2.8.3.2 Calculation of General Self-Heterodyne Spectra In both the heterodyne and self-heterodyne setups, the Fourier-tranform integral to obtain the measured spectrum cannot be done analytically for arbitrary frequency-noise spectra, and thus they must be carried out numerically. For narrow spectral (laser) lines, the width of the line is much smaller than the optical frequency, and thus we can ignore any counterrotating terms in the spectrum (that is, any terms in the self-heterodyne spectrum centered about −\omegam). For example, we can then rewrite the (normalized) self- heterodyne spectrum (2.137) as
\pi Z \infty
−8 \pi Z \infty
\omega′2 d\omega′ = 1 2\pi Z \infty −\infty
−8 \pi Z \infty
\omega′2 d\omega′ , (2.146)
Chapter 2. Classical Coherence
ically, in the ideal case), and then the resulting exponential function must be transformed in the remaining integral, which is just a Fourier transform, giving a centered spectrum about \omega = \omegam. 2.8.3.3 Self-Heterodyne Spectrum of 1/f Noise As an example, consider a frequency-noise spectrum consisting of white-noise and 1/f-noise components:
\omega . (model for semiconductor-laser frequency noise) (2.147) This spectrum can accurately model phase noise in semiconductor lasers, for example.6 Let’s start by calculating the mean-square phase increment, from Eq. (2.101): D
= 4 \pi Z \infty
\omega2 d\omega = 4 \pi Z \infty
\omega
\omega2 d\omega. (2.148) We have already seen [Eq. (2.109)] that the white-noise part leads to \gamma|\tau|. Now we still need to evaluate the 1/f part, however: D
\pi Z \infty
\omega3 . (2.149) Unfortunately, though, this integral has a 1/\omega divergence at \omega = 0. This essentially means we have non- Gaussian statistics, and our assumption of Gaussian noise has broken down. Nevertheless, we will proceed with the self-homodyne spectrum, which involves an extra factor of sin2(\omega\taud/2) and thus has no divergence. Recall from Eq. (2.135) that the presence of this extra factor is equivalent to considering the combination
+
−
−
. (2.150) Note in particular that the delta function we saw in the Lorentzian spectrum is a general feature of self- heterodyne spectra: as \tau −\rightarrow \infty, this combination tends to reduce to
, which is a constant offset that yields a delta function at \omega = \omegam in the Fourier transform. In the self-heterodyne case, the integral in the exponential can be carried out, with the result \pi Z \infty
\omega′2 d\omega′
−k 4\pi
d log \tau 2
. (2.151) 6Linden B. Mercer, ‘‘1/f Frequency Noise Effects on Self-Heterodyne Linewidth Measurements,’’ Journal of Lightwave Technology 9, 485 (1991) (doi: 10.1109/50.76663).
2.8 Optical Linewidth Measurements Here, we have used Z ∞ x¶
\omega3 = 1 Z \infty x
\omega3 = 4x2 −\tau 2 Z \infty x\tau
\omega3 = 4x2 −cos x\tau 4x2 + \tau 2 Z \infty x\tau
\omega2 = 4x2 −cos x\tau 4x2
4x + \tau 2 Z \infty x\tau
\omega = 4x2 −cos x\tau 4x2
4x −\tau 2 4 Ci(x\tau)
(2.152) where Ci(x) is the cosine integral [see Eq. (13.27)], and c is a constant whose value is unimportant. In the combination of terms we have in (2.150), both the c terms and the x-dependence of the logarithms cancel, avoiding any dc divergences. Putting the exponent (2.151) into Eq. (2.146), we have
2\pi Z \infty −\infty
(self heterodyne signal for white and 1/f noise) (2.153) This is a Fourier transform, which is straightforward to calculate numerically. Note again that as \tau −\rightarrow \infty, the correlation function becomes the constant value e−\gamma\taud|\taud|k\tau 2 d/\pi. It may be convenient to subtract this away before the Fourier transform to avoid dealing with the delta function in the spectrum. But note that this amplitude diverges with \taud—the heterodyne limit is problematic, as we have already discussed. Physically, the 1/f spectrum must be cut off at low frequencies, as we will discuss in the next section. An alternative strategy for handling the delta function is to include an explicit cutoff of the tail of the integrand: sanalyzer(\omega) = 1 2\pi Z \infty −\infty
(self heterodyne signal for white and 1/f noise, with Lorentzian resolution bandwidth) (2.154) Here, \delta\omegaRB is the resolution bandwidth (full width at half maximum) of the spectrum analyzer, assuming a Lorentzian bandpass filter. For a Gaussian filter shape to model the resolution bandwidth, we instead have sanalyzer(\omega) = 1 2\pi Z \infty −\infty
(self heterodyne signal for white and 1/f noise, with Gaussian resolution bandwidth) (2.155) where now \delta\omegaRB is the Gaussian resolution bandwidth (full width at half maximum). In the frequency domain, these cutoffs convolve the spectrum with a Lorentzian or Gaussian, respectively, of width \delta\omegaRB, eliminating the delta function and most closely emulating the results of a physical measurement. 2.8.3.4 Observation Time and Linewidth of 1/f Noise For 1/f noise, we have seen that the self-heterodyne spectrum can be calculated, but there is a divergence in the heterodyne limit. In fact, the width of the self-heterodyne line increases with the delay time \taud. Effectively, this is the time over which the self-heterodyne apparatus monitors the frequency fluctuations of the laser, and this is why the 1/f divergence is cut off in the self-heterodyne spectrum (2.137), where there is an extra factor of sin2(\omega′\taud/2) in the exponent, compared with the heterodyne spectrum (2.125). The point
Chapter 2. Classical Coherence is that over any finite observation time Tobs, the lowest (problematic) frequencies in the 1/f spectrum cannot contribute; roughly speaking, only frequencies above the cutoff frequency \omegac = 2\pi/Tobs can contribute to the observed line width. To understand the problem more specifically here, consider again the case of white frequency noise,
(2.156) Recall that this noise causes the frequency of the laser to fluctuate about its center value \omega0 with a Lorentzian distribution. But recall that the spectrum of phase fluctuations diverges at zero frequency,
\omega2 . (2.157) The phase, being the integral of the frequency, diffuses in a random walk, and does not remain near a particular phase, and is thus not a stationary noise process. The arbitrarily large dc noise essentially means that the time-averaged phase is not a well-defined quantity. In the same way, for 1/f noise, the divergence indicates that the time-averaged frequency is not a well-defined quantity: the center frequency itself can wander, and it wanders farther the longer it is observed. Physically, this is the case for lasers whose frequencies can drift over long times due to pressure and temperature fluctuations, relaxing material stresses, and mechanical vibration and creep. Thus, it may not be surprising that the spectral linewidth of a laser depends on the time scale over which it is observed, and we are seeing that this is the case when there is a 1/f component to the frequency-noise spectrum. To account for the time of observation,7 we must revise the calculation of Section 2.7.2 of the variance of the phase fluctuations. There, we computed the mean-square phase fluctuation \langle [∆\phi(\tau)]2\rangle , with the
Here, we will take the angle brackets to denote a time average taken over a finite time T. We will also now explicitly compute the variance, since \langle ∆\phi(\tau)\rangle is not necessarily zero when the average is taken over a finite time interval:
D
T = 1 T Z T /2 −T /2
T Z T /2 −T /2
!2 . (2.158)
subscript t indicates that the time-averaging interval (measurement interval) is centered around time t. The subscript T on the angle brackets denote the finite-time average, which we will write out explicitly below (the absence of this subscript still denotes the limit T −\rightarrow \infty). However, this finite-time average should be averaged over all time (or equivalently, averaged over the ensemble of all possible noise realizations), to obtain the finite-time statistical variance:
T
t−T /2
- −1 T 2
t−T /2
!2+ = 1 T
t−T /2 dt′ D
−1 T 2
t−T /2 dt′
t−T /2
- = D
−1 T 2 Z \infty −\infty dt′ Z \infty −\infty
. (2.159)
7L. S. Cutler and C. L. Searle, ‘‘Some aspects of the theory and measurement of frequency fluctuations in frequency stan- dards,’’ Proceedings of the IEEE 54, 136 (1966) (doi: 10.1109/PROC.1966.4627).
2.8 Optical Linewidth Measurements transform of this pulse function, we find Z \infty −\infty dt′ fT (t′ −t) T
T
t−T /2 dt′ ei\omegat′ = i\omegaT
\omegaT/2
(2.160)
fT (t′ −t) T = 1 2\pi Z \infty −\infty
(2.161) Then we can use this result twice in Eq. (2.159):
D
− (2\pi)2 Z \infty −\infty d\omega Z \infty −\infty d\omega′ Z \infty −\infty dt′ Z \infty −\infty
= D
− (2\pi)2 lim T ′\rightarrow \infty T ′ Z \infty −\infty d\omega Z \infty −\infty d\omega′ Z \infty −\infty dt′ Z \infty −\infty dt′′ Z T ′/2 −T ′/2
= D
− (2\pi)2 lim T ′\rightarrow \infty T ′ Z \infty −\infty d\omega Z \infty −\infty d\omega′ Z T ′/2 −T ′/2 dt′ Z \infty −\infty dt′′ Z \infty −\infty
(2.162) We now get a factor of 2\pi\delta(\omega + \omega′) from the t integration, which takes care of the \omega′ integral:
D
−1 2\pi lim T ′\rightarrow \infty T ′ Z \infty −\infty d\omega Z T ′/2 −T ′/2 dt′ Z \infty −\infty
= D
−1 2\pi Z \infty −\infty d\omega lim T ′\rightarrow \infty
T ′ Z T ′/2 −T ′/2
(2.163) Now we use the Wiener–Khinchin theorem in the form of Eq. (2.20),
D
−1 2\pi Z \infty −\infty d\omega Z \infty −\infty
(2.164) where S∆\phi(\omega) is the (one-sided) power spectral density corresponding to the signal ∆\phi(\tau). We can work out the phase expectation value here as
= 1 \pi Z \infty
\omega2 d\omega = 2 \pi Z \infty
\omega2 d\omega = 4 \pi Z \infty
\omega2 d\omega = 2 \pi Z \infty −\infty
\omega2
(2.165)
Chapter 2. Classical Coherence where we have used Eq. (2.98) to evaluate the correlation functions. Then the variance becomes
D
−1 \pi2 Z \infty −\infty d\omega Z \infty −\infty dt′ Z \infty −\infty
\omega′2
= D
−2 \pi Z \infty −\infty
\omega2
= D
−4 \pi Z \infty
\omega2
(2.166) Finally, using Eq. (2.101) for the infinite-time variance \langle [∆\phi(\tau)]2\rangle , we obtain
\pi Z \infty
\omega2 1 −sinc2 \omegaT (2.167) as its generalization for finite observation times. Now we must be a bit more careful in interpreting the observation time. In an observation time Tobs, the field is measured over this time interval and then used to construct the correlation function (and thus the spectrum). To construct the correlation function at delay \tau, only a time of Tobs −|\tau| is actually useable in the correlation-function time average, and delays |\tau| > Tobs are nonsensical. Thus, we should take T = Tobs −\tau in Eq. (2.167), with the rsult
\pi Z \infty
\omega2 1 −sinc2
. (variance of phase fluctuations related to frequency-noise spectrum) (2.168) Note that this is the same as Eq. (2.101), except for the replacement
1 −sinc2
, (replacement to account for observation time) (2.169) where the sinc2 part came from the square of the finite-time-mean fluctuation. This correction factor scales as \omega2 as \omega −\rightarrow 0, and so also serves to cut off the dc divergence due to the 1/f noise spectrum. The normalized, one-sided spectrum of the laser, including the observation time Tobs, is thus given by Eq. (2.107) as
2\pi Z Tobs −Tobs
1 −|\tau| Tobs exp −2 \pi Z \infty
\omega′2 1 −sinc2
d\omega′ , (spectrum of the signal, including observation time) (2.170) where ∆= \omega −\omega0, and we are assuming a narrow spectral line compared to the central frequency \omega0, and
factor of 1 −|\tau|/Tobs in the integral here, which bears some explanation. When we construct the temporal correlation function (2.114) for the intensity power spectrum, this is a finite-time average over the observation time Tobs, as in Eq. (2.16). But the intensity signal I(t) itself is windowed within the observation time, so the overlap I(t)I(t+\tau) is only nonzero over the time interval Tobs−|\tau|. Thus, the correlation function itself should be modified for the finite observation time by multiplying by (Tobs −|\tau|)/Tobs, with the 1/Tobs coming from the time average, and the correlation function is zero if |\tau| > Tobs. In terms of the spectrum, the observation time imposes a rectangular temporal window, which is equivalent to convolution with sinc (\omegaTobs/2) in the spectrum. However, we have just argued that in the correlation function, the windowing function is the triangular pulse (Tobs −|\tau|)/Tobs, which is essentially the self-convolution of the rectangular pulse (since the pulse is symmetric under time reversal). The effect of this in the power spectrum is a convolution with the Fourier transform of the triangle pulse, which is sinc2(\omegaTobs/2). For very short observation times, the correlation function is constant over the observation time, and thus the spectrum is just sinc (\omegaTobs/2). Asymptotically, then for small Tobs, we expect the spectrum to have an angular FWHM of 4\alpha/Tobs, where \alpha \approx 1.39156 is the positive solution of sinc x = 1/2.
2.8 Optical Linewidth Measurements¶
time:
2\pi Z Tobs −Tobs
1 −|\tau| Tobs exp −2\gamma \pi Z \infty
\omega′2 1 −sinc2
= 1 2\pi Z Tobs −Tobs
1 −|\tau| Tobs \times exp " − \gamma 12 (Tobs −|\tau|)2 h (2|\tau| −Tobs)3 −|2|\tau| −Tobs|3 + 2(Tobs −|\tau|)3i# . (2.171) Notice that the exponent reduces to exp(−\gamma\tau/2) as Tobs −\rightarrow \infty, as it should. The width of this spectrum is plotted here. goTobs 10-2 10-1 (full width at half maximum)/g 10-1 (321/2ologo2)o/goTobs Note that for \gammaTobs < 1, the Fourier-broadened asymptotic result for small observation times is a good approximation. For large observation times, the width converges to \gamma as appropriate for the long-time Lorentzian shape of the line. For intermediate observation times, there is a more complicated step-like dependence on the width, due to the fringes of varying width in the spectrum from the convolution with sinc2(\omegaTobs/2). Note that the fringes cause some apparent narrowing of the spectrum for intermediate observation times, compared to the Lorentzian result.
2\pi Z Tobs −Tobs
1 −|\tau| Tobs exp −2k \pi Z \infty
\omega′3 1 −sinc2
= 1 2\pi Z Tobs −Tobs
1 −|\tau| Tobs exp −2k \pi −7\tau 2 48 −\tau 2 \tau 2
log |\tau| + (2\tau −Tobs)4 48(Tobs −|\tau|)2 log 2|\tau| −Tobs
−(Tobs −|\tau|)2 log (Tobs −|\tau|) + T 4 obs 48(Tobs −|\tau|)2 log Tobs = 1 2\pi Z Tobs −Tobs
1 −|\tau| Tobs
\times 2|\tau| −Tobs
obs . (2.172) The width of this spectrum is plotted below as Tobs varies:
Chapter 2. Classical Coherence k1/2oTobs 10-2 10-1 (full width at half maximum)/k1/2 10-1 (321/2ologo2)o/k1/2oTobs The behavior here is similar to the white-noise case, but as Tobs increases, the spectral width continues to in- crease. Again, for small Tobs, the width matches the Fourier-broadened asymptotic result. For large Tobs, the heuristic arguments (for a different model of the cutoff) lead to the asymptotic scaling8 of (log \betak1/2Tobs)1/2 (for some constant \beta), which is very slow divergence in the line width. We have essentially used the convolution theorem in deriving the finite-observation-time spectrum, where the windowing function fT (t)/T appeared as its Fourier transform as a high-pass filter in the frequency- noise spectrum, as well as in the form of its self-convolution as a windowing function for the correlation function. Of course, any other windowing function f(t) may be used here, so long as it represents a unit-area pulse, and then, for example, the square of its Fourier transform will appear in place of sinc2. Since f(t) is normalized, the correction factor will always vanish at \omega = 0, taking care of the divergence due to the 1/f noise. A non-square windowing function could better model the observation time inherent in the scanning of a spectrum analyzer, for example, where f(t) would be a scaled version of the response function of the final low-pass filter. For example, for a Gaussian window ∝exp[−(4 log 2)t2/T 2 obs] with a full width at half maximum of Tobs, Eq. (2.170) is modified to read
2\pi Z \infty −\infty
−(2 log 2)t2 T 2 obs \times exp −2 \pi Z \infty
\omega′2 1 −exp −T 2 obs\omega′2 16 log 2 d\omega′ . (spectrum of the signal, including Gaussian-window observation time) (2.173) The windowing function exp[−(4 log 2)t2/2T 2 obs] that appears here has exp[−T 2 obs\omega2/8 log 2] as its Fourier transform. If we compare this to the spectral resolution function exp[−(4 log 2) \omega2/\delta\omega2] of the spectrum analyzer, where \delta\omega is the (full width at half maximum) resolution bandwidth, then the resolution bandwidth is given in terms of the observation time by \delta\omega = \sqrt
this as a Fourier transform,
2\pi Z \infty −\infty
−(2 log 2)t2 T 2 obs \times exp −\gamma |\tau| erfc
Tobs − Tobs
obs −1 , (spectrum of the signal, Gaussian-window observation time, white noise) (2.174) where \gamma is as usual the long-time Lorenztian width. The behavior of this width is similar to the rectangular- window observation case, except that the function is smoother (there are no fringes associated with the 8Gianni Di Domenico, Stéphane Schilt, and Pierre Thomann, ‘‘Simple approach to the relation between laser frequency noise and laser line shape,’’ Applied Optics 49, 4801 (2010) (doi: 10.1364/AO.49.004801).
2.8 Optical Linewidth Measurements Gaussian convolution) and has a less pronounced dip. The short-time asymptotic form also has a slightly different coefficient: 4 \sqrt 2 log 2/Tobs. goTobs 10-2 10-1 (full width at half maximum)/g 10-1 (321/2ologo2)o/goTobs For 1/f noise, the expression is
2\pi Z \infty −\infty
−(2 log 2)\tau 2 T 2 obs \times exp −k\tau 2 3\piT 2 obs (log 2)\tau 2 2F2 1, 1; 5 2, 3; −(4 log 2)\tau 2 T 2 obs −3T 2 obs
T 2 obs , (spectrum of the signal, Gaussian-window observation time, 1/f noise) (2.175) where here \gamma is Euler’s constant, and 2F2(a1, a2; b1, b2; z) is a generalized hypergeometric function. The be- havior is similar to the rectangular-window observation case, but again smoother and with a less pronounced minimum. k1/2oTobs 10-2 10-1 (full width at half maximum)/k1/2 10-1 (321/2ologo2)o/k1/2oTobs The asymptotic scaling for small Tobs is the same as for the white-noise case, and the scaling for large Tobs appears to match the rectangular-window observation case.
2.9 Exercises¶
Chapter 2. Classical Coherence 2.9 Exercises Problem 2.1 In the classical model of spontaneous emission, an atom impulsively excited at t = 0 gives rise to radiation in the far-field of
r
(2.176)
second-order degrees of temporal coherence for this field, and then show that the radiated spectrum s(\omega) is a Lorentzian lineshape with a full width at half maximum of \gamma. Problem 2.2 In molecular spectroscopy, Fourier-transform infrared (FTIR) spectroscopy is an important technique. The basic idea is to use a Michelson interferometer to measure the correlation function g(1)(\tau) of some input (infrared) field on a detector, which is then digitized. The experimenter takes the Fourier transform on the computer to give the spectrum. This in principle gives the same spectral information as a grating spectrometer, which uses a diffraction grating and an aperture to limit the light so that only a certain range of frequencies hits the detector at any time; scanning the grating position gives the direct spectrum. (a) Give a (qualitative) argument to justify the following statement: in the infrared, thermal detector noise is significant, so for a given measurement time the FTIR method gives an improved signal/noise ratio compared to the grating method. Assume the same detector is used in both setups. (b) Give a different (qualitative) argument to justify the following statement: for small detector noise, a grating spectrometer system is superior to an FTIR-type system if it is important to have a large dynamic range in the measured spectrum. Problem 2.3 Consider the Young double-slit experiment, where two slits are illuminated with classical, coherent light. The setup produces interference fringes on a distant screen due to the variation in path-length difference to the two slits. The fringe visibility V for a single detector is the one we defined in class, and V = 1 for coherent light. We can define a two-detector fringe visibility for simultaneous detection by
max −G(2) min G(2) max + G(2) min , (2.177) where
(2.178) is the unnormalized intensity correlation function for simultaneous detection at two points x1 and x2 on the screen (x is the direction across the fringes). (a) What is V (2) for this double-slit experiment? (b) Suppose a ‘‘phase scrambler’’ is placed in front of one slit to randomize the phase of its transmitted wave. How are V and V (2) changed? Problem 2.4 Given a signal with time dependence of the form exp[−i\phi(t)], a (one-sided) phase-fluctuation spectrum defined by
Z \infty −\infty
(2.179)
2.9 Exercises and a (one-sided) frequency-fluctuation spectrum defined by
Z \infty −\infty D
E
(2.180) show that the spectra are related by
(2.181) Qualitatively, what conditions must be satisfied for this relation to hold? Problem 2.5 Given a Gaussian random variable X with zero mean and variance \sigma2, show that
e−iX
(2.182) Problem 2.6 Suppose we define the exponent A of Eq. (2.134) by A :=
. (2.183) Show that A = 2
- 2
−
−
. (2.184) Do not make any assumptions about whether ∆\phi(\tau) is correlated at different times t, or about the probability density of ∆\phi(\tau), other than the fact that the variances above are well-defined and that
Problem 2.7 Fill in the steps between Eq. (2.135), Ganalyzer(\tau)∝\eta2
−
+
/2, (2.185) and Eq. (2.136), Ganalyzer(\tau)∝\eta2
cos \omegam\tau exp −8 \pi Z \infty
\omega′2 d\omega′ . (2.186)