Fluctuation properties of chaotic light The uncertainty principle applies to the lineshape broadening (consequence of the finite time during which the atom is unperturbed while it is emitting/absorbing)   property of the Fourier trasform Spectral linewidth broadening of the light source Fluctuations of the electric field (around mean value) The spectral linewidth broadening of the light source produces fluctuations of the electric field and of the beam intensity around the mean value on a time-scale inversely proportional to the frequency breadth of the light. The frequency spread and the temporal fluctuations are manifestations of the same physical properties of the radiating atoms. Both complementary aspects are needed to interpret the optical experiments Complementary aspects contained in the optical experiments F. De Matteis Quantum Optics

Fluctuation properties of chaotic light Two type of light source. Chaothic sources like gas discharge lamps, thermal cavities, conventional filament lamps, light emitting diodes Coherent sources like laser A classical description of the light beam is proper to describe the properties of the first type of sources. A quantistic description does not add any further kind of information. The second type needs a quantistic description of the radiative field to clearly discern the peculiarities of the coherent source against the background of the classic theory. F. De Matteis Quantum Optics

Spectrum of a fluctuating light beam Fourier component of electric field Frequency spectrum Some kinds of optical interference experiments perform exactly such an integral. But it does not integrate an infinite time First-order electric-field correlation function Describes the way in which the value of the electric field at time t affects the probability of the possible values at a later time t+t F. De Matteis Quantum Optics

First-order correlation The nature of the fluctuations determines the shape of this function Stationary fluctuations  determined by a physical phenomenon which does not vary with time Correlation does not depend on initial time provided that T is long compared to the characteristic time scale of the fluctuations Experimentally one determines the correlation function by means of a time average, but it can be calculated by means of a statistical average over all values of the field at time t and t+t The time averaging samples all the E-field value allowed by the statistical properties of the source F. De Matteis Quantum Optics

First-order correlation Where we use the definition of the function d(x) Normalized spectral distribution function Degree of first-order temporal coherence F. De Matteis Quantum Optics

Collision-broadened Light Source Source in which the predominant broadening mechanism is by collision (elastic, phase shift but no atomic state transitions) Field amplitude of the wave emitted by an atom suffering elastic collisions Field amplitude of the wave emitted by N atoms suffering elastic collisions Carrier wave at frequency w0 with a random modulation of amplitude and phase The spectral decomposition shows the lineshape of collision broadening Inclusion of other line-broadening source does not change substantially the description Coherence length Coherence time F. De Matteis Quantum Optics

First-order Coherence – Frequency Spectrum Correlation Function of single atom is proportional to the probability that no collision occurs during time t . < >=0 many collisions < >=1 no collision In presence of both radiative and collision broadening the parameter g will be the sum of two (g’) Degree of first-order coherence plays an important role in the description of interference experiments F. De Matteis Quantum Optics

Young Interference Field fluctuations, resulting from the cahotic nature of the source, affect the visibility of the interference fringes. The ui are purely imaginary and inversely proportional to si (Huygens principle) The field intensity is recorded by means of instruments with recording times longer than the coherence time of cahotic light The fringes arise from the term that involves the correlation of the fields at the two slits. There is a generalization to take into account the different spatial position F. De Matteis Quantum Optics

Young Interference Let the direction of propagation be taken as z-axis. Fringe Visibility. The chaotic nature of the source affects the fringe If w0>>g’ many fringes are generated by the cos term About 104 maxima are visible around the centrum. Other factors actually limit the fringe visibility (finite extension of the source) F. De Matteis Quantum Optics

Degree of first-order coherence Light at two point in space and time COHERENT if, even in principle, can produce interference effects. The potential magnitude of interference effects is governed by the first-order coherence of the light beam employed. Degree of first-order coherence Incoherent Partially coherent First-order coherent Treated exemples all involve plane parallel light beams  single spatial coordinate z Module |t| is introduced to include positive and negative delays F. De Matteis Quantum Optics

Degree of first-order coherence - Classic wave of stable amplitude and phase (laser beam in single-mode) - Single mode of a cavity (all cavity mode are removed except one). Many oscillators each with a randomly distributed but fixed phase. First-order coherent beam in any point of the cavity. - Two classic wave - Stable amplitude, fixed phase but different frequency First-order coherent beam in any point In general coherent if: Only a single cavity mode is excited Field can be specified precisely with no statistical features - Two classic wave - different mode frequency each with fluctuating amplitude and random phase (equal average intensity) F. De Matteis Quantum Optics

Degree of first-order coherence Inhomogeneous broadening (Doppler) Every atom emits a different frequency with a random fixed phase The cross-terms i≠j average to zero The sum is converted to a Gaussian distribution Combining homogeneous and inhomogeneous broadening F. De Matteis Quantum Optics

Homogeneous Broadening Inhomogeneous Broadening F. De Matteis Quantum Optics

Intensity fluctuations of chaotic light Fluctuations of the electric field Intensity fluctuations Let’s suppose we can perform instantaneous measurements of intensity Ī(t) (average over a cycle) Chaotic monochromatic light I(t) Ī Long Time Average Intensity Mean Square Intensity Root Mean Square Deviation N ways to choose i N-1 ways to choose j Root-mean-square deviation F. De Matteis Quantum Optics

Intensity fluctuations of chaotic light Random walk problem Gaussian distribution of |E(t)| Ī Most probable value of |E(t)| and/or I(t) is zero Average value is different of zero F. De Matteis Quantum Optics

Degree of second-order coherence Degree of second-order temporal coherence Hence the degree of second-order temporal coherence at zero delay satisfies the inequality F. De Matteis Quantum Optics

Degree of second-order coherence For finite time delays it holds simply Hence the degree of second-order temporal coherence soddisfies the inequality F. De Matteis Quantum Optics

Degree of second-order coherence for chaotic light Coherence time tc for chaotic light from N independent emitters Assuming N>>1 F. De Matteis Quantum Optics

Degree of second-order coherence for chaotic light Collision-broadened light source (homogeneous broadening g’) Doppler-broadened light source (inhomogeneous broadening d) In general for chaotic light F. De Matteis Quantum Optics

Degree of second-order coherence for chaotic light Generalized to include the spatial dependence Light at (r1,t1) and (r2,t2) points is called second-order coherent Chaotic light is never second-order coherent Classical stable wave is always second-order coherent F. De Matteis Quantum Optics

Experiment of Hanbury Brown Twiss The first experiment showing the degree of second-order coherence Any detector measure the fluctuation of the beam intensity (AC coupling) For chaotic light the measured signal is 1 for t<<tc and fall to 0 for t>>tc It has been verified that for coherent light we get 0 for any delay value Homogeneous broadening. F. De Matteis Quantum Optics

Experiment of Hanbury Brown Twiss Homogeneous Broadening. Detectors have a response time tr not zero Valid in general for any type of line-broadening F. De Matteis Quantum Optics

Spatial Coherence The effects of intensity interference depend on the second-order temporal coherence. But there are effects from spatial coherence which are associated to finite dimensions of the source and detector. In the Young interference it causes the decay of the fringe visibility away from the axis of the slits. There is a phase difference of 180° between the rays from the two ends of the source in a position of the detector a/2 away from the axis when The a dimension yields the size at which it begins to suffer cancellations between the contributions from different parts of the source The spatial coherence controls the phase variation in a direction perpendicular to the beam and it is determined by the closeness of the light beam to the condition of plane parallelism. The temporal coherence controls the fluctuation properties and the correlation of the fields and intensity in a direction parallel to the beam and it is determined by emission properties of the atoms of the source F. De Matteis Quantum Optics