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Degree of polarization (DOP) is a quantity used to describe the portion of an electromagneticwave which ispolarized. A perfectly polarized wave has a DOP of 100%, whereas an

unpolarized wave has a DOP of 0%. A wave which is partially polarized, and therefore can berepresented by a superposition of a polarized and unpolarized component, will have a DOP

somewhere in between 0 and 100%, calculated as the fraction of the polarized component power

with respect to the total power.

DOP can be used to map the strain field in materials when considering the DOP of the

photoluminescence. The polarization of the photoluminescence is related to the strain in amaterial by way of the given material'sphotoelasticity tensor.

DOP is also visualized using the Poincar sphere representation of a polarized beam. In this

representation, DOP is equal to the length of the vectormeasured from the center of the sphere.

Degree of coherence

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In optics, correlation functions are used to characterize the statistical and coherence properties of

an electromagnetic field. The degree of coherence is the normalized correlation of electricfields. In its simplest form, termed g

(1), it is useful for quantifying the coherence between two

electric fields, as measured in a Michelson or other linear optical interferometer. The correlationbetween pairs of fields, g

(2), typically is used to find the statistical character of intensity

fluctuations. It is also used to differentiate between states of light that require a quantummechanical description and those for which classical fields are sufficient. Analogous

considerations apply to any Bose field in subatomic physics, in particular to mesons (cf. BoseEinstein correlations)

Contents

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y 1 Degree of first-order coherenceo 1.1 Examples of g(1)

y 2 Degree of second-order coherenceo 2.1 Examples of g(2)

y 3 Degree of nth-order coherenceo 3.1 Examples of g(n)

y 4 Generalization to quantum fieldso 4.1 Examples of nonclassical states

y 5 Photon bunchingy 6 See alsoy 7 References

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y 8 Suggested reading

Degree of first-order coherence

Figure 1: This is a plot of g(1)

as a function of the delay normalized to the coherence length /c.

The blue curve is for a coherent state (an ideal laser or a single frequency). The red curve is forLorentzian chaotic light (e.g. collision broadened). The green curve is for Gaussian chaotic light

(e.g. Doppler broadened).

Where denotes an ensemble (statistical) average. For non-stationary states, such as pulses, the

ensemble is made up of many pulses. When one deals with stationary states, where the statisticalproperties do not change with time, one can replace the ensemble average with a time average. If

we restrict ourselves to plane parallel waves then . In this case, the result for stationary

states will not depend on t1, but on the time delay = t1 t2 (or if

).

This allows us to write a simplified form

where we now average over t.

In optical interferometers such as the Michelson interferometer, Mach-Zehnder interferometer, orSagnac interferometer, one splits an electric field into two components, time delays one

component, and then recombines them. The intensity of resulting field is measured as a functionof the time delay. The visibility of the resulting interference pattern is given by | g(1)() | . More

generally, when combining two space-time points from a field

visibility=

The visibility ranges from zero, for incoherent electric fields, to one, for coherent electric fields.

Anything in between is described as partially coherent.

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Generally, g(1)

(0) = 1 and g(1)

() = g(1)

( )*

.

Examples ofg(1)

For light of a single frequency (e.g. laser light):

For Lorentzian chaotic light (e.g. collision broadened):

For Gaussian chaotic light (e.g. Doppler broadened):

Here, 0 is the central frequency of the light and c is the coherence time of the light.

Degree of second-order coherence

Figure 2: This is a plot ofg(2)

as a function of the delay normalized to the coherence length /c.The blue curve is for a coherent state (an ideal laser or a single frequency). The red curve is for

Lorentzian chaotic light (e.g. collision broadened). The green curve is for Gaussian chaotic light(e.g. Doppler broadened). The chaotic light is super-Poissonian and bunched.

Note that this is not a generalization of the first-order coherence

If the electric fields are considered classical, we can reorder them to express g(2) in terms ofintensities. A plane parallel wave in a stationary state will have

The above expression is even, g(2)() = g(2)( ) For classical fields, one can apply Cauchy-Schwarz inequality to the intensities in the above expression (since they are real numbers) to

show that and that . Nevertheless the second-ordercoherence for an average over fringe of complementary interferometeroutputs of a coherent state

is only 0.5 (even though g(2) = 1 for each phase). And g(2) (calculated from averages) can be

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reduced down to zero with a proper discriminating triggerlevel applied to the signal (within therange of coherence).

Examples ofg(2)

Chaotic light of all kinds: g(2)

() = 1 + | g(1)

() |2

. Note the Hanbury-Brown and Twiss effect usesthis fact to find | g(1)() | from a measurement ofg(2)().

Light of a single frequency: g(2)

() = 1

Also, please seephoton antibunching for another use ofg(2)

where g(2)

(0) = 0 for a single photon

source because

where n is the photon number observable.[1]

Degree of nth-order coherence

A generalization of the first-order coherence

A generalization of the second-order coherence

or in intensities

Examples ofg(n)

Light of a single frequency:

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Using the first definition: Chaotic light of all kinds:

Using the second definition: Chaotic light of all kinds: Chaotic light of all kinds:

g(n)

(0) = n!

Generalization to quantum fields

Figure 3: This is a plot of g(2)

as a function of the delay normalized to the coherence length /c.

A value of g(2)

below the dashed black line can only occur in a quantum mechanical model oflight. The red curve shows the g

(2)of the antibunched and sub-Poissonian light emitted from a

single atom driven by a laser beam.

The predictions ofg(n)

forn > 1 change when the classical fields (complex numbers orc-

numbers) are replaced with quantum fields (operators orq-numbers). In general, quantum fieldsdo not necessarily commute, with the consequence that their order in the above expressions can

not be simply interchanged.

With

we get

Examples of nonclassical states

Photon bunching

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Figure 4: This is a plot of g(2)

as a function of the delay normalized to the coherence length /c.

This is an example of a g(2)

that indicates antibunched light but not sub-Poissonian light.

Figure 5: Photon detections as a function of time for a) antibunching (e.g. light emitted from asingle atom), b) random (e.g. a coherent state, laser beam), and c) bunching (chaotic light). c is

the coherence time (the time scale of photon or intensity fluctuations).

Light is said to be bunched if and antibunched if .