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Abstract

Coherence function is a complex quantity and follows wave equation similar to the optical field. Utilizing this analogy, wepropose a new method to craft the correlation structure of the light. This is realized by considering the interference of coherence waves due specially positioned incoherent sources in the Fourier space. Basic principle of the proposed approach is described and simulation results for two different cases are presented.
Crafting correlation structure by interference
Tushar Sarkar, Manisha Dixit, Rakesh Kumar Singh
Department of Physics, Indian Institute of Technology (Banaras Hindu University), Varanasi, Uttar Pradesh, India
Author e-mail address:
tusharsarkar.sarkar@gmail.com
,
Abstract: Coherence function is a complex quantity and follows wave equations similar to the optical field.
Utilizing this analogy, we proposed a new method to craft the correlation structure of the light.
This is realized by considering the interference of coherence waves due to specially positioned
incoherent sources in the Fourier space. Basic principle of the proposed approach is described and
simulation results for two different cases are presented.
Keywords: Coherence, Light shaping, Diffractive Optics
1. Introduction
Light fields are inherently of a statistical nature and correlations play a fundamental role in describing the
statistical properties of the light. The cross correlation between the light fields at different space-time points, known
as complex coherence, plays a significant role in characterizing the light fields [1]. The theory of partial coherence is
generally described in terms of space-time correlation functions. The commonly used correlation function of this
type is the mutual coherence function. The complex coherence functions follow two-four dimensional wave
equations. One of the significant results in the statistical optics is the van Cittert-Zernike theorem (vZT) which
relates the spatial incoherent structure with the spatial coherence in the far field.
Another important result is the relation between fourth order and second order correlations. One significant outcome
of this relation is the Hanbury Brown-Twiss (HBT) intensity interferometer. The HBT approach provides modulus
square of the complex coherence function with auto-covariance of the intensity at two points. This relation can be
utilized to design and develop new interferometric effects with higher order correlation. Learning from the analogy
between the optical field and the coherence, higher order interferometers can be designed. For instance, intensity
interferometer can be used to examine the coherence wave interference [2]. In this paper, we use an intensity
interferometer and coherence waves interference to craft the correlation structure. Our work is inspired from a
reverse engineering applied in non-diffraction and beam shaping in the coherent optics. Purpose is to explore
similarity between ‘coherence’ and other waves on the basis of interference phenomena.
2. Principle
For a quasi-monochromatic light source, the second order correlation is
(1)
The angular bracket < > is ensemble average, * stands for complex conjugate and V is analytic signal of the
fluctuating field at position u
. Consider an instantaneous field in Fig. 1. The coherence at the back focal plane is
connected with source as
(2)
Equation (2) is derived on the basis of incoherent sources. Position vector at the source is , wavelength , f is the
focal length of the lens and is light intensity from nth incoherent source in the mask. For Gaussian random fields,
Equation (2) is derived on the basis of incoherent sources. Position vector at the source is , wavelength , f is
the focal length of the lens and is light intensity from nth incoherent source in the mask. For
Gaussian random fields,
Fig. 1: A schematic diagram to craft the correlation structure by incoherent source masks
(3)
Therefore, change in shape, size and position of the incoherent sources effect composition of the coherence function
and hence intensity correlation. Analogy with this behavior can be found at other places like interference of waves
from independent harmonic oscillators, and modes etc.
3. Results and discussion
We consider two different masks to craft the coherence functions. In the first case, we consider a set of four circular
filters in a mask as in Fig. 2(a). These filters in the mask are considered to be incoherent and their coherence
functions are shown in Fig. 2 (b)- (e). These coherence functions are complex quantities and their phases are
controlled by positions and separation of filters in the mask according to the vZT. The resultant coherence function
due to combination of filters is shown in Fig. 2(f). Fig. 2(f) can be obtained using intensity correlation, i.e. left side
of Eq. (4) or using interference of the coherence waves as in right side of Eq. (4). Both give similar results.
Fig.2: Crafter coherence functions for combination of (a) mask; coherence function by individual mask (b) 1 (c) 2 (d) 3 (e) 4 (f) resultant
coherence by all mask; (g) another mask, coherence functions from individual masks (h)- (k), and (g) resultant coherence
4. References
[1] L. Mandel L, and E. Wolf , Optical coherence and quantum optics
(Cambridge University Press, 1995).
[2] J. Goodman, Statistical Optics (John Wiley & Sons Inc, New York, 2000).
[2] R. K. Singh, S. Vyas, and Y. Miyamoto, “Lensless Fourier transform holography for coherence waves”J. Opt.19 (2017) 115705 /1-5.
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Article
Coherence functions behave like a wave and follow two-four dimensional wave equations. Utilizing this feature, we propose a lensless Fourier transform holography for coherence waves. Experimental demonstration is carried out by synthesizing the coherence function of a stationary random field as a superposition of two independent coherence functions in the Fresnel domain. Using the connection between two point intensity correlation and coherence function for the Gaussian random field, the application of this technique is demonstrated in imaging through a random scatterer in a lensless geometry from a single measurement of the speckle field.