Interesting! Using optics to "polarise" light usually entails the rejection of all light except that polarised in a certain direction. Depending on the source, it may be that this reduced radiation field is more coherent than the complete field. If the polarisation is induced in a resonator, then the light buildup in only in the selected polarisation, which should have more coherence in the polarised direction, than if it was not polarised.

Although coherence and polarisation are not necessarily directly linked, the effects of polarising a beam internal or external to the source can effect the overall coherence, I would think!

In the TE mode, electric field vector is perpendicular to the propagation vector and in TM mode magnetic field vector is perpendicular to the propagation vector. If a laser beam is either unpolarized or in TE mode passing through the  wire-grid polarizer and after that it passes through the linear polarizer. So if we rotate the linear polarizer then in tensity of the light will be changes according to the Malus' law. If the laser light is in the TM mode then  wire-grid polarizer is not able to block any component of the electric field (magnetic field) beam and hence there is no change in the output.

Interesting! Using optics to "polarise" light usually entails the rejection of all light except that polarised in a certain direction. Depending on the source, it may be that this reduced radiation field is more coherent than the complete field. If the polarisation is induced in a resonator, then the light buildup in only in the selected polarisation, which should have more coherence in the polarised direction, than if it was not polarised.

Although coherence and polarisation are not necessarily directly linked, the effects of polarising a beam internal or external to the source can effect the overall coherence, I would think!

This is actually a rather complicated question. In practice, at the classical level, the two have to be linked. Imagine the situation that you have a light source that has a very narrow spectral bandwidth. Such a light source should have a very long coherence length, because bandwidth and coherence time and Fourier transform pairs of each other. Hence, a spectral measurement would tell you that these two parameters are orthogonal to each other. But, we can also measure the coherence length through autocorrelation.

So, if we set up a michelson interferometer and start scanning one of the mirrors, then the output port should start steadily varying from bright to dark as the interference moves from constructive to destructive. But, if the light source is unpolarized, then the electric field orientation at time time t0 and time t1 is uncorrelated, so the interference between the two fields is imperfect and the fringes rapidly wash out. So, we would measure a very short coherence time. The two measurements (spectral and temporal do not agree)

Or, if we were to think of this a different way: if we take our very narrow band but unpolarized light source and pass it through a polarizer, then the measured light field intensity will have strong amplitude fluctuations on the time scale that the electric field orientation varies. These amplitude variations would then show up in the spectrum as a spectrally broadened light source and the coherence length (or time) would be measured to be shorter. Thus, a spectral measurement of any two orthogonal polarization measurements should give a coherence length that agrees with our direct measurement in an interferometer

You can also think of this from the point of view of individual photons. The natural description of photons is to be circular polarized. In a coherent light field, all the photons must add up in phase. And, for a given polarization state, the phase and number of LCP and RCP photons is fixed. But, in an unpolarized light field, the number of LCP and RCP photons must fluctuate randomly. Hence, the phase of each photon is may be correlated, but the number of photons in each field is still uncorrelated.  

Polaarization can be considered the coherence in the transverse plane. For unpolarized light, the x and y (or any pair, right or left, etc) are uncorrelated and can't form interference fringes. As the coherence between x and y increases, the degree of polarization increases. For polarized light, linear, elliptical, circular, (DOP=1) the components are in-phase or coherent.

Dear Russell

The interference pattern is formed when there is  a degree of temporal and spatial coherence and the polarization of light  is not necessary to create interference fringes!

That is why, I would claim the coherence length should be identical if we use unpolarized or polarized light in a Michelson interferometer.

Moreover, the Newton rings is due to the interference of sun light as an unpolarized  and incoherent (very low degree of coherence) source!! Eventhough the interference may occur within its coherence length.

Correct. In these cases the x-component is interfering with the x-component and the y-component is interfering with the y-component.

When we split the polarization components, as with a polarizing beam splitter, rotate one component, and recombine the beams and try to form interference fringes, this is where the coherence of the tranverse components is observed.

Doesn't the degree of polarization have a role in the visibility of fringes (though will not directly effect the coherence length)?

The degree of coherence affects the visibility of the fringes: 

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Polarizations of two interfering beam do play a very  important role. If the two beams are in orthogonally polarized states, no interference takes place and difference in Polarization State reduces the visibility of interference fringes. Two beams will be in orthogonally polarization states if average over a complete cycle of scalar product of their electric fields is zero, e.g., two orthogonal linearly polarization or two circular polarizations.

Thus, absolutely essential conditions for two beams of same mean frequency  to interfere are

Further desirable conditions for getting high visibility are

For obtaining appreciable fringe width, a further desirable condition is

To understand the importance of polarization in interference one may consider a simple situation where two light beams having perfect coherence and the same intensity, say I, interfere but one beam has a linear polarization while the other is circularly polarized. Circularly polarized beam of intensity I can be considered to be a superposition of two linearly polarized beams, each of intensity I/2, one having linear polarization identical to  the other interfering beam and the other having the orthogonal polarization. Beams of intensities I and I/2 having same polarizations and perfect coherence  will generate interference with maximum and minimum intensities ((3/2) +  21/2)I and   ((3/2) + 21/2)I and the orthogonally polarized beam shall give uniform  background intensity I/2. Thus the resultant maximum and minimum intensities will be (2+21/2)I and (2-21/2)I, and  visibility will be equal to 1/21/2. The visibility here is less than one in spite of the same intensities of the two interfering beams and perfect coherence.

Polarization would affect the interference result but does not alter the degree of coherence of the given EM waves.

Prof. Russell A Chipman points out correctly that the concept of coherence is a wider one and contains in a sense the concept of polarization. Classic books by Born and Wolf and by Mandel and Wolf describe at length the concepts of coherence including coherence matrix and coherence tensors but then only for lowest order correlations. The lowest order correlation functions (coherence functions) describe the Stokes parameters and polarization property in linear optics, where observations are confined to Intensity of polarized components only. However, if one tries to measure higher order correlations, e.g., the square of the intensity of polarized components (may be by letting the light pass through a polarizer and then through a non-centrosymmetric crystal, which generates second harmonic, before detecting the intensity of this second harmonic which is proportional to the square of the incident intensity), Stokes parameters are found insufficient to describe polarization properties and light beams having the same values of the Stokes parameters may give different measurement results.

Also, it must be emphasized that if orthogonally polarized light beams do not show interference it is not because of any lack of coherence. In fact if we extract the same polarized component of the two orthogonally polarized beams beams the interference may be seen. In Babinet Compensator we see the interference of ordinary and extra-ordinary beams which are orthogonally polarized by extracting linearly polarized components along a direction which bisects the directions of the two orthogonal linear polarizations.