FRED uses a Gaussian beam decomposition algorithm to propagate coherent optical fields through the system geometry. A demonstration of this capability is shown in this webpage.

A generalized form of Gaussian Beam Decomposition (GBD) allows FRED to address a wide range of physical optics phenomena. Over the past quarter century, the GBD algorithm has been proven to accurately simulate both diffraction and interference effects with remarkable fidelity.  This webpage will demonstrate FRED's GBD capability to model partial coherence as observed with a Diffractometer.

The Diffractometer^[[1],[2]] is an apparatus helpful in demonstrating partial coherence. The experimental layout can be seen in the figure below. An extended, incoherent source s₀ is imaged by lens L₀ onto pinhole s₁. The light emerging from s₁ is collimated by lens L₁ and brought again to focus by Lens L₂ at plane F.  An opaque screen A containing two apertures P₁ and P₂ is placed between L₁ and L₂. The apertures P₁ and P₂ can be any size, shape and position.


Diffractometer (after B.J. Thompson and E Wolf)

 In our FRED model, the portion in the above figure surrounded by the red dashed line is replaced by a collection of randomly positioned point sources of differing wavelengths located within an area equal to the diameter of the pinhole area s₁. This collection of sources meets the definition of a quasi-monochromatic source given by Born & Wolf. In the plane F, each wavelength component of the source independently produces an interference pattern as a result of coherent summation. By design, FRED sums like wavelengths coherently and different wavelengths incoherently. Thus, the total irradiance pattern at F  becomes an incoherent summation of the various coherent components.

According to an important partial coherence theorem developed independently by P.H. van Cittert in 1934 and later by F. Zernike in 1938, the source collection at s₁ gives rise to a correlation between the field at any two points P₁ and P₂ on the screen A.  The van Cittert-Zernike theorem establishes the complex degree of partial coherence as

where   

and r is the radius of the pinhole at s₁, d is the center-to-center distance between P₁ and P₂, R is the focal length of L₁, r₁ and r₂ are the distances of P₁ and P₂ from the optical axis and l_m is the mean wavelength.
As a test of FRED’s capabilities, we have produced results in good agreement with Thompson and Wolf by calculating the irradiance pattern in the plane F  as the separation d between P₁ and P₂ is varied. The pertinent model parameters are f₁=f₂=R=1520mm; separation between L₁ and L₂: 14 mm; diameter of the pinhole area s₁: 90 mm; diameter of apertures P₁ and P₂: 1.4mm; mean wavelength l_m = 0.579 mm. The figures at right show the irradiance patter for d=6, 10 and 23 mm respectively and |m₁₂| = 0.593, .146 and 0.035.

For more information on FRED's coherent Gaussian Beam propagation read the Coherent Application note here and the Laser Application note here.

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(Above)  Partial Coherence where d = 6 mm. |m₁₂| = 0.593

(Above)  Partial Coherence where d = 10 mm. |m₁₂| = 0.146

(Above)  Partial Coherence where d = 23 mm. |m₁₂| = 0.035