Description

The full diffraction efficiency table can interpolate power and phase (radians) efficiency values and is primarily designed to support the output from grating analysis programs such as GSolver.  This implementation of the diffraction efficiency table includes a dependence on wavelength and incident polar and azimuthal angles, distinguishes between diffraction orders in reflection and transmission, and can optionally include S and P polarization efficiencies and polarization cross-coupling.  No extrapolation is performed on the data when the incident ray's wavelength or angle lies outside the bounds of the sampled data; the efficiency data from the closest sample point will be applied.

After an efficiency file has been loaded into the dialog, summary information is provided back to the user regarding the number of entries and value ranges found for each parameter.  After applying the changes, the user can get an exact listing of the efficiency table data in the output window by right mouse clicking on the surface on the object tree and selecting "Detailed Report" from the menu options.

Because diffraction orders are specular operations, the surface coating definition will apply to the ray powers on top of the grating efficiency table value.  The user should create a fictitious sampled coating with R=1 and T=1 to be applied to the grating surface so that the ray power is completely determined by the full efficiency table.

The interpolation of the efficiency table occurs over wavelength, incident angle, and azimuthal angle (for each polarization component, if specified).  The eight nearest sample table values for each of wavelength, incident angle, and azimuthal angle define the corners of a hypercube in wavelength, incident angle and azimuthal angle space.  These eight corners are labeled E000, E100, E010, E001, E110, E101, E011 and E111.  The (wavelength, incident angle, azimuthal angle) point in the hypercube space is located inside the cube and the normalized coordinates of this point is denoted as (a,b,c).  For example, (0,0,0) is the corner with efficiency value E000, and (1,1,1) is the corner with efficiency value E111, etc.  A tri-linear interpolation in this space is performed and is mathematically equivalent to the following:

E(a,b,c) = (1-a)(1-b)(1-c)E000 + a(1-b)(1-c)E100 + (1-a)b(1-c)E010 + (1-a)(1-b)cE001 + ab(1-c)E110 + a(1-b)cE101 + (1-a)bcE011 + abcE111

From the scripting language, the GetInterpolatedDiffractEfficiency() function can be used to query a grating surface with the Full efficiency table specification for its interpolated efficiency values, which can be useful for debugging and verification.

Navigation

The Full efficiency table specification can be accessed from the Grating tab of a surface dialog.  The section on the right hand side of the dialog corresponds to the diffraction efficiency specification for the grating.  In the drop-list box, select "Full efficiency table".

Controls

Polarization dependent diffraction efficiency

The complex amplitudes of the diffracted ray's output S and P polarization states, As’ and Ap’, are given by the matrix equation below.  As and Ap are the complex amplitudes of the incident ray.  Each of the four components, SS, PP, SP and PS, have the form, Sqrt(power_efficiency)*exp( +i * phase_efficiency), with the phase efficiency given in radians.

The SP and PS coefficients represent the cross-coupling between the S and P states.  The S' polarization vector for the outgoing ray is perpendicular to the plane defined by the diffracted ray's direction vector and the surface normal, while the P' polarization vector is parallel to this plane.

The diffracted ray's flux is given by φ = |As’|*|As’| + |Ap’|*|Ap’|.

A procedure for verifying ray data

The complex nature of the Full efficiency table with dependencies on wavelength, polar and azimuthal angles, and polarization state can make it difficult to verify application of the efficiency data globally for an entire rayset.  However, the following procedure can be used to verify application of the efficiency data on a ray-by-ray basis for the purposes of debugging and validation.  If the user only wishes to check the interpolation of the grating data, then the GetInterpolatedDiffractEfficiency function can be used.

Step 1 - Idealized Behavior

The first step in this procedure is to establish the behavior of the diffracted rays in the idealized case for 100% efficiency (i.e. determination of As and Ap in the matrix construction above).

1. Configure the grating to use the Simple efficiency table option and configure the diffracted order to be evaluated with an efficiency of 1.0.

2. Perform a raytrace and then use the Tools > Reports > Ray Detail (for one ray) utility to retrieve the detailed ray data for the ray of interest.

3. From the detailed ray data in (2) above, retrieve the S and P direction vectors for the ray as well as the As and Ap complex amplitudes.

4. Calculate |As| and |Ap| as |A| = Sqr( Re{ }² + Im{ }² ) for each component.

5. Calculate the phase of each component, phi_s and phi_p, as Atan2(Re,Im) in radians.

Step 2 - Apply Diffraction Efficiency

The next step in this procedure is to apply the SS, PP, SP and PS efficiency components to the idealized ray data from Step 1.

1. From the efficiency file data, retrieve Axx * exp( i*phi ) for each component, where Axx = Sqrt(power_efficiency)  and phi = phase_efficiency.

2. Calculate the quantities:

As' = ( |Ass|*exp( i*phi_ss ) )( |As|*exp( i*phi_s ) ) + ( |Asp|*exp( i*phi_sp ) )( |Ap|*exp( i*phi_p ) )

Ap' = ( |Aps|*exp( i*phi_ps ) )( |As|*exp( i*phi_s ) ) + ( |App|*exp( i*phi_pp ) )( |Ap|*exp( i*phi_p ) )

which expand to,

As' = { |Ass||As| cos( phi_ss + phi_s ) + |Asp||Ap| cos( phi_sp + phi_p ) } + i { |Ass||As| sin( phi_ss + phi_s ) + |Asp||Ap| sin( phi_sp + phi_p ) }

Ap' = { |Aps||As| cos( phi_ps + phi_s ) + |App||Ap| cos( phi_pp + phi_p ) } + i { |Aps||As| sin( phi_ps + phi_s ) + |App||Ap| sin( phi_pp + phi_p ) }

3. Normalize As' and Ap' by the magnitude, Sqr( Re{As')² + Im{As'}² + Re{Ap'}² + Im{Ap'}² ) to get the final As' and Ap'

4. The diffracted ray's flux is given by |As'|*|As'| + |Ap'|*|Ap'|, assuming the incident ray's flux is 1.0

Phase Interpolation

Interpolation of phase components during raytracing requires special handling near sample points with 2PI phase jumps.  Consider two adjacent sample points, one defining a phase of +3 radians at a polar angle of 5 degrees and another defining a phase of -3 radians at a polar angle of 15 degrees.  Interpolation of the phase between 5 and 15 degrees should wind around the phase circle passing through the +/-PI phase discontinuity.  In a straight linear interpolation, however, the phase between 5 and 15 degrees would pass through zero phase going from +3 radians to -3 radians.  A signature of this zero-phase interpolation effect can be the presence of unexpected artifacts in an irradiance or field distribution that track +/-PI phase discontinuity boundaries in a phase efficiency map.

By default, if two adjacent data points have a phase difference within 0.5 radians of 2PI, FRED will automatically add +2PI or -2PI radians to one of the points so that the adjusted phase difference is less than 0.5 radians and the interpolated value does not pass through zero phase.

If a custom phase tolerance value (0.5 radians unless otherwise specified) is desired, the following header line can be optionally specified in the first row of the efficiency file:

phasetol n

where n is the phase tolerance value in radians (ex. phasetol 0.25).

Related Topics

Gratings Overview

Grating Types

Simple Efficiency Table

Volume Hologram Efficiency