1. 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
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Quantum Optics

2. 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.
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Quantum Optics

3. 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
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Quantum Optics

4. 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
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Quantum Optics

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

6. 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
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Quantum Optics

7. 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
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Quantum Optics

8. 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
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Quantum Optics

9. 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)
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Quantum Optics

10. 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
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Quantum Optics

11. 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)
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Quantum Optics

12. 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
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Quantum Optics

13. Homogeneous Broadening
Inhomogeneous Broadening
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Quantum Optics

14. 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
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Quantum Optics

15. 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
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Quantum Optics

16. 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
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Quantum Optics

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

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

19. 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
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Quantum Optics

20. 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
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Quantum Optics

21. 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.
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Quantum Optics

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

23. 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
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Quantum Optics