Coherence, polarization and interference
Coherence expresses a very basic optical idea: the stochastic nature of real light beams. Real light
waves are unavoidably fluctuating. This is so because light is generated by electrons (free or
bounded) and the random dynamics of atoms and ions in gases and of electrons in solids translates
unavoidably into fluctuations of the light they generate.
This randomness has dramatic consequences in all optical phenomena, specially in polarization and
interference. Polarization is actually coherence at work between two orthogonal light vibrations
being reflected in the degree of polarization. Interference is coherence at work between
nonorthogonal light vibrations being measured by the visibility or the degree of coherence. In this
regard, the standard degrees of coherence, polarization and visibility are closely related since the
degree of polarization between two scalar field components E1 , E2 is the maximum degree of
coherence and the maximum visibility that can be reached between two waves obtained from E1 ,
E2 by energy conserving operations.
There are two main interpretations of coherence. The first interpretation identifies coherence as
electric field correlations. As a matter of fact, the standard degree of coherence is essentially the
degree of correlation  E1E2*  where the complex representation has been used and the acute
brackets denote ensemble averages (this is to repeat the experiment many times and average the
results). The second interpretation identifies coherence as the visibility of interference fringes
obtained where the two waves overlap.
Following the idea that polarization and interference are two particular manifestations of coherence
our aim has been to join both in a single formalism. This is as the same variable might be stored in
two different locations, interference and polarization, as mechanical energy of a particle can be split
into potential and kinetic energy. Thus, loosely speaking, interference can be converted into
polarization and vice versa.
There are two key points in our approach to these subjects; purity and distance. Throughout we
assess the amount of coherence, visibility and polarization of a given way as the distance to the light
wave fully devoid of the corresponding property. Using a simple and meaningful distance such as
the Hilbert-Schmidt we get that such distance becomes invariably equivalent to purity, in the same
sense employed in the quantum theory that is actually a measure of information. For a matrix M
 
purity is equivalent to the equality of trace functions tr M 2  trM 2 , or equivalently, that M has


just a non vanishing eigenvalue, M    , where  is a complex column vector, or loosely
speaking that the light is in a single field mode.
We have analyzed the interference-polarization conversion in the Young interferometer. We have
shown the existence of a periodic modulation of coherence in the interference plane fully analogous
to the usual fringe intensity distribution. Moreover, the amplitude of modulation of coherence
depends on the polarization state in the slits of the interferometer. Vice versa, the polarization state
in the interference plane depends on the coherence between the slits.
Modulation of coherence of vectorial electromagnetic waves in the Young interferometer
A. Luis, Opt. Lett. 33, 1497-1499 (2008)
Since interference and polarization can mutually exchanged being different sides of the same
phenomenon we have addressed the assessment of the amount of coherence of one or more than one
light waves. The key point is to include all the contributions that manifest in interference as well as
in polarization.
For definiteness let us consider two waves, each one with a vector complex amplitude with two
components. Thus the problem has four electric field components, say E1 , E2 , E3 , and E4 where
the two first components E1 , E2 , correspond to the first wave and the other two E3 , E4 to the
other wave.
If we understand coherence as correlations between electric fields all possible correlations (within
second order in the fields) are in the 16 correlation coefficients that make the 4  4 matrix
  E 2   E E*   E E*   E E*  
1
1 2
1 3
1 4 

2


*
*
 E2 E1   E2   E2 E3   E2 E4*  

M 
.
  E E*   E E*   E 2   E E*  
3
1
3
2
3
3
4


  E E*   E E*   E E*   E 2  
4 1
4 2
4 3
4


It is important to note that this matrix contains the polarization coherence (or coherence intra wave
we may say) in the submatrices
  E 2   E E*  
1
1 2 

2 

*
  E2 E1   E2  
  E 2   E E*  
3
3 4 .

2 

*
  E4 E3   E4  
The other matrix elements represent the interferometric coherence (or coherence inter waves we
may say).
We can assess the total coherence as the distance between the matrix M and the identity matrix I
that represents fully incoherent and fully unpolarized light (irrespectively of the pair of components
selected)
1

0
I 
0

0

0 0 0

1 0 0
.
0 1 0

0 0 1 
As a suitable and simple distance between matrices we may consider the Hilbert-Schmidt and
define a degree of coherence  as
 
2
 1
1   tr M 2 1
  tr 
M  I  
 ,
4   trM 2 4
 trM

where the factor tr M is included so that the result is independent of the light intensity. These
formulas properly reflect the two ideas of distance and purity.
The key point of this approach is that all components E1 , E2 , E3 , and E4 play exactly the same
role irrespectively of the wave they belong. Therefore the degree of coherence  contains all
coherence conveyed by the two waves, including interferometric as well as polarization coherences.
Moreover, the degree of coherence  also provides an upper bound to the visibility that can be
reached in the interference of any energy conserving combination of the original waves.
Degree of coherence for vectorial electromagnetic fields as the distance between correlation
matrices
A. Luis, J. Opt. Soc. Am. A 24, 1063-1068 (2007)
We have applied the same idea of the distance between a correlation matrix and the identity matrix,
or purity of the correlation matrix, to assess the degree of polarization of three-dimensional waves,
i. e., with three components. The result aggress wit previous approaches in terms of generalized
Stokes parameters.
Degree of polarization for three-dimensional fields as a distance between correlation matrices
A. Luis, Opt. Commun. 253, 10 (2005)
The case of a continuous distribution of two-dimensional waves has been also considered. In such a
case we have studied the global coherence including the polarization coherence of all waves and the
correlations between all them. We have found simple expressions by using the Wigner-function
formalism both in the scalar and vectorial cases. In both cases the global degree of coherence  can
be expressed as being proportional to the integration of the square of the corresponding Wigner
function W (z )
   dzW 2 ( z ) ,
where z represents all arguments of the Wigner function, that in the vectorial case includes also the
polarization degrees of freedom in addition to position and propagation direction. This formula
reproduces the idea of purity in the integration of the square and distance since the fully incoherent
and unpolarized light has a Wigner function that essentially tends to zero and  is the distance of
W (z ) to the zero function.
Overall degree of coherence for vectorial electromagnetic fields and the Wigner function
A. Luis, J. Opt. Soc. Am. A 24, 2070 (2007)
We have also addressed the problem of coherence between vectorial waves from the point of view
in fringe visibility. More specifically, we have examined the maximum visibility that can be
reached in the interference of two waves obtained from the original waves by any energy
conserving transformation. This leads to
  m
Vmax  M
 M  m
where M and m are the maximum and minimum eigenvalues of the correlation matrix M
introduced above. This formula for Vmax is the same formula for the degree of polarization of a
two-dimensional wave where the matrix M is 2  2 . Therefore, Vmax is a proper generalization of
the degree of polarization to multi-dimensional waves.
Maximum visibility in interferometers illuminated by vectorial waves
A. Luis, Opt. Lett. 32, 2191-2193 (2007)
We have studied the mutual relationships between coherence, visibility, polarization and phase
difference for an arbitrary number N of electromagnetic field components E1,  E N . In principle it
might be thought that the case of N components is merely an academic problem. However, this is a
very common situation in practice as revealed by the diffraction gratings and the coexistence of
many independent modes within a cavity. Following the identification of interference and
polarization discussed above we can refer naturally to the polarization properties of field of
arbitrary dimensions.
Within a second order theory all statistical information is provided by the N  N correlation matrix
with matrix elements M i, j  Ei E *j  . A key point of our work is the complete quantum-classical
equivalence if we identify the classical correlation matrix M with the density matrix  of a
quantum state in a Hilbert space of dimension N
1

M .
trM
This equivalence is interesting since it allows us to translate ideas an results from one theory to the
other, as it is the case of uncertainty relations, as we show below.
The interferometric visibility for N waves can be defined in very general terms as the distance
between the correlation matrix M and the diagonal matrix D with the intensities  E j
diagonal,
 E 2 

0

0
1


2


0
 E2  
0
.
D







2 
0
0
  EN  

2
 in the
This would be the matrix M if all relative phases between components were fully random so that
 Ei E*j  0 for i  j , this is when no interference whatsoever would be possible. The distance
between D and M is the amount of interference contained in M. A simple formula for this distance
is
2

2  tr  1 M  1 D   .
VM


trD  
 trM

The degree of polarization for an N-dimensional wave (or global degree of coherence) can be
assessed by a suitable generalization of the formula considered above for four components
 1
1
P 2  tr 
M
N
 trM

I

2
 
tr M 2
1

 ,
2
N
 trM 
where I the identity in dimension N. In quantum physics P is also the state purity and a measure of
the total information conveyed by the state.
We have found that these quantities are related by a kind of uncertainty relation for finitedimensional systems of the form
2  C 2  P2 ,
VM
where C represents the statistics of how the light intensity is distributed between the N components
2
 1
1  
C 2  tr 
D I 
N  
 trD

2  C 2  P 2 is a kind of quantum uncertainty relation for finite-dimensional
The relation VM
systems, as discussed in this web page in the notes about complementarity. This is also an example
of Pithagoras theorem as illustrated in the figure
2  C 2  P 2 means that for a given coherence P the maximum visibility
Classically the relation VM
VM  P is obtained when the intensity of the N waves are equal, this is C  0 . This is exactly the
same result that holds for two the interference of two waves where maximum visibility requires the
equality of the intensities of the interfering waves. In quantum mechanics this means that maximum
interference requires that each path for the interfering photons must be equally probable, this is null
information about the path followed by the photons.
Some other similar relations can be derived, such as
V2  C 2  P 2 ,
where V2 is the distance between the phase distribution I ( ) and the uniform distribution 1 /( 2 ) ,
2
1 

2
,
V   d  I ( ) 
2 


I ( )   exp i j  k    E j Ek*  .
j,k
In the classical domain I ( ) is the intensity distribution in the far plane produced by a grating with
N slits, being  the phase difference between consecutive slits. In this context V is the visibility of
the interference in the far plane measured as a suitable distance as detailed in other notes in this web
page. In the quantum domain I ( ) is the phase distribution in a finite-dimensional space, being V
a measure of phase uncertainty related to the Renyi entropy.
The chain of inequalities V  VM  P implied by the above relations has a clear interpretation.
The maximum visibility that can be achieved in an interferometric arrangement depends on the
number of degrees of freedom available. In the case of a diffraction grating in the figure we have in
(a) the amplitudes and phases of the slits are fixed, in (b) only the amplitudes are fixed and the
phases can be varied while in (c) all phases and amplitudes can be varied freely.
Quantum-classical correspondence for visibility, coherence, and relative phase for
multidimensional systems
A. Luis, Phys. Rev. A 78, 025802 (2008)
An Overview of Coherence and Polarization Properties for Multicomponent Electromagnetic
Waves
A. Luis, Advances in Information Optics and Photonics, International Commission for Optics, (Ed.
A. T. Friberg, R. Dändliker) Volume VI, c. 9, p. 171.
The standard second order degrees of coherence (both scalar and vectorial) may be insufficient to
analyze some relevant situations. This is the case of quantum metrology where the states reaching
maximum interferometric resolution may have vanishing degree of coherence. This vanishing
occurs because of symmetry properties of the field states instead of because of randomness.
In order to avoid this rather paradoxical situation (best interference with vanishing coherence) we
have developed an approach to coherence between two waves where this electromagnetic property
is described in terms of the statistics of the phase difference between the two waves. This provides a
simple generalization of the standard second order degree of coherence to all orders avoiding a very
complicate infinite hierarchy of correlation functions that become specially involved in the vectorial
case.
There are some reasons supporting the soundness of this approach. On the one hand, phase
fluctuations explain most examples of partial coherence, so that the identification of coherence with
phase difference might be regarded as a suitable approximation
E1 E2

E1
2
 e i ,
E2
2
  argE2   argE1 .
On the other hand this identification is exact from an ensemble picture of statistical optics where the
application of the standard formula to each sample realization (i. e., removing the averages) gives
exactly the exponential of the phase difference

E1E2
2
E1 E2
2
 ei .
Finally, coherence means essentially good interference and good interference means ability to
detect small phase shifts. This in turn depends on the phase difference statistics. The conclusion is
that coherence and phase-difference statistics must be closely related.
The statistics of the phase difference P ( ) can be derived from the probability distribution for the
complex amplitudes PE1, E2  by a suitable change of variables including the phase difference 
as one of the new coordinates, and then integrating over the other variables different from the phase
difference
PE1, E2   PE1 y1, yn , , E2  y1, yn ,  ,
P    dy1  dyn
 E1, E2 
PE1, E2  .
  y1, yn ,  
Then, the amount of coherence can be estimated by the distance D between the phase difference
distribution P ( ) and the uniform distribution 1 /( 2 ) representing completely incoherent light.
Larger distance means larger coherence. Since D has no upper bound we can define a suitably
normalized amount of coherence  between zero and one

1 

D  2  d  P  
2 
 
2

D
.
D 1
For thermal chaotic light with Guassian PE1, E2  we get that P ( ) depends exclusively on the
standard second order degree of coherence |  | . Moreover,  approximately coincides with |  | as
shown in the next figure.
Next we apply this to a quantum light field reaching maximum interferometric resolution, the so
called Heisenberg limit. For example this is the case of exactly n photons entering simultaneously
by the two input ports of a Mach-Zehnder interferometer. We are interested in the coherence and
phase difference properties of the internal beams after the beam splitter, since these are the
interfering waves.
The complex amplitude probability distribution can be given by the Q function which is obtained by
projection on the coherent states obtaining
Pn E1, E2  
E12  E22
2n
 n!2 22n
2
2
exp   E1  E2  .


This is an even function of its arguments so that the second order degree of coherence is exactly
zero.
Pn E1, E2   Pn  E1, E2   Pn E1, E2     E1E2  0
This holds because of symmetry but not because of a large amount of field fluctuations.
In this case the phase difference distribution P ( ) has two symmetric peaks. The width of the peaks
decreases when the number of photons n increases so that the distance D to the uniform distribution
increases when n increases. Similarly, the degree of coherence  tends to its maximum value
  1 as n increases, even if   0 . This is in complete agreement with their interferometric
usefulness. Interferometric resolution means ability to detect small phase shifts and this depends on
the width of the peaks and not on the number and location of the peaks.
In conclusion the phase difference provides a simple, physical and meaningful generalization of
coherence beyond second order including classic and quantum interferometric schemes consistently
in a single framework.
Ensemble approach to coherence between two scalar harmonic light vibrations and the phase
difference
A. Luis, Phys. Rev. A 79, 053855 (2009)
In a further work we have shown that the action of random unitary transformations introduces an
irreversible evolution in different quantities representing coherence for vectorial waves. In
particular, we have shown that the entropy, which is an important measure of disorder of the
electromagnetic fields, cannot decrease. Furthermore, different quantities recently introduced to
characterize coherence properties of partially polarized electromagnetic fields cannot increase with
the action of local random unitary transformations. Consequently, if the application of random
unitary modulations decreases one of these quantities, the decrease in this value cannot be
compensated without energy absorption, since neither deterministic nor random unitary
transformations can make this quantity increase.
Since partial coherence between two electromagnetic fields is related to the existence of
randomness between these fields, the analysis of random unitary transformations sheds light on
partial coherence properties of partially polarized lights, and we believe that such an analysis opens
many interesting perspectives.
Irreversible effects of random unitary transformations on coherence properties of partially
polarized electromagnetic fields
P. Réfrégier y A. Luis, J. Opt. Soc. Am. A 25, 2749-2757 (2008)
Inverted Huygens principle and generalized Stokes parameters
In the proper analysis of coherence for vectorial waves we have developed a version of the
geometrical optics that includes coherence properties. In standard geometrical optics each light ray
carries some intensity and polarization state represented by the Stokes parameters.
In a simple modification of this idea we can associate to each ray some generalized Stokes
parameters derived from the Wigner function formulation of wave optics so that all wave
phenomena can be exactly described within a geometrical picture, including all phase sensitive
cases such as interference of partially polarized waves.
This extended geometrical picture can be formulated as the Huygens formulation of wave optics but
replacing waves by rays. Further details about this can be found in other notes in this same web
page.
Scalar Wigner function for vectorial fields and spatial-angular Stokes parameters
A. Luis, Opt. Commun. 246, 437 (2005)
Properties of spatial-angular Stokes parameters
A. Luis, Opt. Commun. 251, 243 (2005)
Spatial–angular Mueller matrices
A. Luis, Opt. Commun. 263, 141–146 (2006)
Negativity, diffraction and interference for nongeometrical waves
A. Luis, Opt. Commun. 266, 426-432 (2006).
Ray picture of polarization and coherence in a Young interferometer
A. Luis, J. Opt. Soc. Am. A 23, 2855 (2006)
Complementary Huygens principle for geometrical and nongeometrical optics
A. Luis, Eur. J. Phys. 28 231-240 (2007)
Polarization ray picture of coherence for vectorial electromagnetic waves
A. Luis, Phys. Rev. A 76, 043827 (2007)
Polarization and coherence for vectorial electromagnetic waves and the ray picture of light
propagation
A. Luis, J. Eur. Opt. Soc. 2, 07030 (2007)