Accurate Calculation of Far Field Diffraction Pattern   (Chuck Titus)
 

OUTLINE:


 
 

1.Diffraction Theory.
 

There are a number of different diffraction models.  These models can be divided into two classes, vector and scalar.  Scalar treatments include Huygens’ Principle[1,2,3], Rayleigh-Sommerfeld theory[1,2,3,4], the Kirchhoff formulation[1,2,3,4], and a model referred to as the angular spectrum of plane waves[3,5]. The well-known Fraunhofer and Fresnel approximations, as they appear in most introductory optics texts, are derived from the Kirchhoff model.  Vector models include a treatment by Stratton and Chu[6], another based on the surface equivalence principle[7], and a rigorous electromagnetic boundary value model which appears in Jackson’s text[7] (among other places).  Each of these theoretical models has strengths and weaknesses, and each can be satisfactorily employed for some range of problems.  The choice of an appropriate model is based on what is known about a specific problem.
 

Application of any diffraction model can be divided into two separate tasks[9].  First, one must obtain the fields exiting a diffracting object (i.e. the near fields, or the boundary field values), or a reasonable approximation thereof.  The second step involves propagating those fields to the desired observation point.  These are distinct and separate parts of the diffraction models listed above.  Most texts do not make this separation clear.  Instead, the boundary value assumptions and subsequent propagation into the far field are lumped together into one theoretical treatment.  If the resulting diffraction pattern is at all inaccurate, it is difficult to determine how much of that error is due to incorrect boundary fields and how much is the result of the propagation calculation.  Because of this, it is often difficult to know which model is appropriate for a particular problem.
 

We are not searching for an accurate complete diffraction model.  What is sought is only that portion of one of the diffraction theories which can accurately calculate far zone diffraction patterns given accurate near fields.  Our purpose here is not to provide rigorous proof of accuracy.  Instead, what follows is an attempt at sorting through the various theories in order to provide rationale for the choice of one such method.
 

It is often said that scalar diffraction models are not applicable for diffracting apertures possessing features with sizes on the order of a wavelength.  In addition, some models, notably the Fraunhofer and Fresnel approximations, are of questionable value for calculating nonparaxial diffracted fields.

In general, scalar theories assume the exact boundary field distribution (i.e. the near field) is not known, and make an intelligent guess about its values.  For instance, Rayleigh and Kirchhoff treatments of diffraction by a slit in an opaque screen often assume the fields inside the opening are the same as the undisturbed (i.e. incident) values, and are zero elsewhere on the screen.  That assumption produces boundary fields which are discontinuous at the edges of the slit.  Approximate boundary fields obtained in such a manner are not an exact representation of the true boundary fields.  The approximation neglects to apply Maxwell’s equations to obtain exact boundary conditions at the edges of the aperture.  As a result, subsequent calculation of the diffracted fields will contain some error.  The magnitude of error increases as the width of the slit is reduced to and below the wavelength of incident light.
 

In any case, scalar theory calculations based on simple assumptions about boundary fields generally depart somewhat from experimental data.  This is particularly true for cases when the diffracting aperture or features therein are not large compared to a wavelength[10,11].  Approximation of the boundary field distribution is at least partly to blame.
 

There have been attempts to obtain more accurate representations of boundary fields for some simple problems such as an infinite slit[12], an infinite half-plane[13], or a circular aperture[2].  However, those boundary field computations are difficult and not generally applicable to arbitrary diffracting apertures.
 

Vector models are also capable of obtaining more accurate near fields.  Such treatments apply electromagnetic boundary conditions, derived from Maxwell’s equations, to the diffracting aperture.  The accuracy of boundary fields obtained in such a manner depends on the ability to solve the boundary condition equations.  Again, for complicated diffracting objects, the calculation is often extremely difficult if not impossible.
 

So it is imperative that one obtain the most accurate possible representation of the near fields.  The FDTD method is capable of computing accurate near fields.  But if one can obtain accurate boundary fields by some method such as FDTD, which propagation method should be employed?  Are there restrictions which accompany any of these propagation methods?
 

First, we should make note of one feature of the FDTD computation method.  The core algorithm employed by the FDTD method is derived directly from the differential forms of Ampere’s and Faraday’s laws, also known as the two Maxwell curl equations.  The two divergence laws are not explicitly enforced in the FDTD method, but it has been shown to possess a divergence-free nature[14].  Consequentially, it can be said that fields computed by the FDTD method are solutions to Maxwell’s equations, to within any error incurred by the numerical nature of the computation.  FDTD can provide significantly  more accurate near fields than the approximations often encountered in scalar diffraction theory.
 

Given that the near fields produced by the FDTD computational method can be close approximations of Maxwell fields, all that should be required of a propagation model is that it retain the Maxwellian nature of those fields as they propagate away from the near zone.  Any such propagation model should produce an accurate diffraction pattern from accurate fields on the surface of a diffracting aperture.
 

With the exception of the original Huygens’ Principle, all of the diffraction theories listed above contain propagation models derived from Maxwell’s equations.  Propagation models in the scalar Rayleigh and Kirchhoff theories are based on the Helmholtz wave equation, which is obtained directly from Maxwell’s curl equations.  After application of Green’s Theorem, the propagation models are derived.  The scalar propagation models are then applied to each vector component of the boundary field.  If the vector near fields adhere to Maxwell’s equations, the component-by-component propagation of those fields employing a method derived from the wave equation should, in theory, produce Maxwell far fields.
 

It should be noted that the differences between these scalar theories result from different aspects of the boundary fields employed in their respective propagation models.  As a result, incorrectly specified boundary fields can lead to different diffraction patterns from these scalar theories.  Conversely, given the same accurate boundary values, these propagation models should produce the same diffraction pattern[9].
 

The preceding discussion provides reasonable expectation that we can safely use the propagation models contained in Rayleigh and Kirchhoff theories.  The major problem with those theories is the possibility of incorrectly specified boundary fields.  Use of FDTD-computed near fields significantly reduces that concern.
 

We have not discussed the angular spectrum representation (Dr. E. C. Gartland here in Kent State's Math and CS Dept. knows a lot about that) or any of the vector theories.  Each of these methods has its proponents.  We can say that there are doubts about the validity of the Stratton-Chu theory because its propagation model is not divergence-free.  The Surface Equivalence Principle is popular in the electrical engineering community, but is difficult to implement in its non-paraxial form.  In theory, it should produce Maxwell far fields from Maxwell near fields.  The angular spectrum of plane wave representation can be connected to the wave equation and, like the Rayleigh and Kirchhoff propagation models, is theoretically correct.


 
 

2.  Reduction of Two-D Kirchhoff Surface Integral to One-D Line Integral.
 

In theory, we could have employed either Rayleigh integral.  However, the Kirchhoff integral is a simple average of the two Rayleigh integrals.  One of the Rayleigh integrals makes use of near field magnitudes, and the other utilizes the derivatives of near fields.  It is possible that numerical errors may be more pronounced in one or the other Rayleigh integral, and that may change from one problem to another.  If so, an average of the two can reduce that concern.  For that reason, and for reasons of coding ease, we chose to implement the primitive form of the Kirchhoff propagation integral.

FIGURE 1.  Geometry of diffraction problem.  The diffracting object lies in the y’=0 plane  between x’=(-d) and x’=d.  Diffracted light is observed at (x,y,z).
 

The problem at hand is shown in Figure 1.  The drawing represents a planar cross-section from a three-dimensional diffraction geometry.  Points on the "exit" surface of the diffracting object are described by  (x',y',z').  The diffraction pattern is observed in the half-space y>0, at points (x,y,z).  For this general geometry, the primitive form of the Kirchhoff surface integral takes the form:
  

The Green’s function G(r,r’) is defined as: 

The location of any point on the surface of the layer is denoted r’=(x’,y’,z’), and the location of any observation point in the far zone is denoted r=(x,y,z).  The magnitude of the distance between any point on the aperture to any observation point is:

  
The near fields j(r') produced by FDTD propagation of light through the diffracting object are obtained at the object's "exit surface".  The gradient is evaluated with respect to changes in R.  The unit vector n is the normal to the aperture surface.

 

Now suppose the diffracting object and the fields emerging from it do not vary in the z direction, and that the exit surface of the diffracting object is flat, existing on the y'=0 plane.  Such cases may include flat liquid crystal layers of finite thickness whose director varies internally in the x and y directions but not in the z direction.  Then the above surface integral must be reduced to a line integral by carrying out the z' integration (over an infinite range).
 

Before we reduce the surface integral to a line integral, a few more aspects of this restricted problem:  Clearly the diffraction pattern produced by this one-dimensionally varying (along x') aperture will  not vary with z.  Because of the z-invariance of this problem, we can arbitrarily choose this problem to exist in the plane z=0.  Then the distance R becomes:
 

   
Also, such a diffracting object will be described by n=y everywhere on its exit surface.  The surface integral, over the aperture surface, is now evaluated over the semi-infinite "surface" invariant in z'. That is, it is to be evaluated for x’ over the range (-d,d) and z' over an infinite range.  Evaluating the normal component of the two gradient operations yields:
 

If the near fields emerging from the diffracting object (e.g. the flat liquid crystal layer) are non-zero only over a limited range from x'=-d to x'=d, the Kirchhoff integral may also be so restricted.  This may occur for a liquid crystal grating which has been illuminated by a gaussian (laser) beam, or a single transmitting pixel in an otherwise dark liquid crystal display.

Now we can reduce the surface integral to a line integral. Making use of the Hankel function of the first kind to evaluate the integrals along z’:

Which reduces the surface integral to a line integral.  After this  integration along z’, the distance R has become R||:
  
Since we are only concerned with diffracted light in the far zone, we can make use of the large-argument approximation of the Hankel function and evaluate the partial derivative inside the second integral term to give:
  
Note the occurrence of y in the numerator of the second integral term.  If the diffraction pattern were to be evaluated only over the paraxial region, y/R~1.  The above equation may finally be simplified, assuming kR||<<1, to give:
  
This integral could to be applied to all six components of the surface fields E(x’) and H(x’).  However, if the medium for y>0 is linear and isotropic, there are only two independent propagating modes.  Consequently, the above integral need only be evaluated for Ez(x’) and Hz(x’).  That will produce two far field distributions Ez(x,y) and Hz(x,y).  From the real part of those complex values the relative instantaneous intensity of the diffraction pattern is calculated:
  
where Z is the impedance of the half-space y>0.  In our implementation, the (x,y) dependence was converted to an angular distribution, for which q=arctan(x/y).  This concludes discussion of far field diffraction pattern computation.


 
 

3.  References (Bibliography).

1)  B. Baker, E.Copson; The Mathematical Theory of Huygens’ Principle; 2nd ed.; Oxford Univ. Press; London; 1950.

2)  C. Bouwkamp; Diffraction Theory; Rep. Prog. Phys.; 17; pp. 35-100; 1954.

3)  M. Born, E. Wolf; Principles of Optics; 6th Edition; Cambridge University Press; Cambridge; 1997.

4)  J. W. Goodman; Introduction to Fourier Optics; McGraw-Hill; New York; 1968.

5)  P. Clemmow; The Plane Wave Spectrum Representation of Electromagnetic Fields; Pergammon Press; London; 1966.

6)  J. A. Stratton; Electromagnetic Theory; 1st Edition; McGraw-Hill; New York; 1941.

7)  S. Schelkunoff; Some Equivalence Theorems of Electromagnetics and Their Application to Radiation Problems; Bell Sys. Tech. J.; 15; pp. 92-112; 1936.

8)  J. D. Jackson; Classical Electrodynamics; Wiley; New York; 1962.

9)  M. Kowarz; Diffraction Effects in the Near Field; PhD Thesis; University of Rochester; 1995.

10)  M. Totzeck; Validity of the Scalar Kirchhoff and Rayleigh-Sommerfeld Diffraction Theories in the Near Field of Small Phase Objects; J. Opt. Soc. Am. A; 8; pp. 27-32; 1991.

11)  M. Totzeck; Test of Various Diffraction Theories in the Near Field of Phase Objects; Ultramicroscopy; 57; pp. 160-164; 1995.

12)  P. M. Morse, P. J. Rubenstein; The Diffraction of Waves by Ribbons and Slits; Phys. Rev.; 54; pp. 895-898; 1938.

 13)  A. Sommerfeld, Lectures on Theoretical Physics; Volume VI; Academic Press; New York; 1954.

 14)  Taflove; Computational Electrodynamics, The Finite-Difference Time-Domain Method; Artech House; Boston; 1995.

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