The Grating Type specifies the phase profile of the grating in OPD that determines the local grating frequency and orientation at any position on the grating surface. The following grating types are available to be specified: •Linear (Evenly spaced linear grating lines) •Two point exposure holographic optical element •Two source user-recorded holographic optical element
The Grating Type specification is found on the left hand side of the Grating tab in a surface dialog. Use the drop-list box to select the desired grating type specification.
Two point exposure holographic optical element The two-point exposure holographic optical element (HOE) uses the interference fringes formed by two idealized coherent sources to define the phase profile (OPD) of the grating, which in turn specifies the grating spacing and grating vector orientation at any position on the grating surface. The grating surface itself may be any arbitrary shape and is not restricted to planes. Given sources A and B in the two-point exposure model, where the sources can be any combination of point source or plane wave definitions, the grating lines and orientation at any position on the grating is determined by taking the gradient of the OPD for each of the sources and then adding them together. Defining a source as "virtual" simply negates the OPD gradient contribution of that source to the grating function such that the following table describes the summation of the A and B terms for the different combinations of real and virtual source definitions, where dA and dB are the gradients of the OPD contributions from the respective sources:
Additional x,y polynomial or radial polynomial phase departure terms can be optionally added to the two-source interference pattern in order to create linear or radial gratings with variable spacing. The phase departure terms have the following functional forms:
The local grating spacing is inversely related to the gradient of the phase (OPD) function. Taking the radial polynomial phase function, F(r), as an example, the local grating spacing is related to the phase function by:
d(r) = lref / (dF/dr) = lref / [ A1 + 2A2r + 3A3r2 + ... ]
where lref is the reference wavelength and d(r) is the grating spacing at the radial position, r.
Zemax Phase Term Conversion When a grating with phase departure terms are imported from Zemax, the coefficients are converted from radians to system units in the following way:
Fn = Zn (l / 2p) / rn
where Fn is the FRED phase function coefficient of the n'th order polynomial term, Zn is the Zemax coefficient of the n'th order polynomial term, l is the grating exposure wavelength and rn is the normalization radius to the power of the n'th order.
Example - Chirped grating with an XY Polynomial As an example, consider the case of a linearly chirped grating of width 50mm with reference wavelength 0.5um. There are 1250 lp/mm at position y = -25mm and 1500 lp/mm at position y = +25mm.
The polynomial equation for grating spacing is inversely related to the gradient of the phase function and is used to solve the set of equations below for coefficients, A11 and A22 :
d(-25) = 1/1250 = (5E-04)/[A11 +2A22*(-25)] d(25) = 1/1500 = (5E-04)/[A11 + 2A22*(25)]
Solving these two equation simultaneously yields:
A11 = 0.6875 A22 = 0.00125
In the Phase Departure subsection of the Grating dialog, choose XY polynomial and set the coefficients below.
X0Y0=0 X1Y0=0 X0Y1=0.6875 X2Y0=0 X1Y1=0 X0Y2=0.00125
NOTE: In the Phase Departure GUI, you must type a number in each cell in order to get the next term in the polynomial. Enter a value of 0.0 to skip a cell.
Example - Linear phase with an XY Polynomial The A11 coefficient of the XY Polynomial phase (OPL) function can be used to define OPL that is linear in Y for a collimated incident beam. If the refractive index of the transmitted wavefront is n' and the desired tilt of the wavefront is given by angle theta, then the value of the A11 coefficient can be set to n' sin(theta). To achieve this setting in the Grating dialog: •the diffraction order should be set to +1 or -1 in the efficiency table •the X0Y1 term in the XY Polynomial coefficients list should be set to n' * sin(theta) •the two-point construction parameters can be left at the default settings (no additional interference fringes defined)
Example - Focusing a collimated beam with a Radial Polynomial A collimated beam incident on a grating can be brought to focus using the Radial Polynomial phase departure. The A2 coefficient of the Radial Polynomial function represents the parabolic approximation of a spherical wavefront and can be used to define OPL(r) = -n'r2/(2d), where r is the radial position on the grating surface, n' is the refractive index of the transmitted wavefront, and d is the distance to focus from the grating. The diffraction order should be set to -1 in the efficiency table (or the sign of the A2 coefficient should be negative).
If, for example, n'=1.0 and the focus point is 25mm from the grating, then the A2 coefficient should be set to 0.02 if the diffraction order is set to -1.
Higher order terms in the spherical OPL expansion of our example, such as A4 = -r4/(8d3) = -8.0E-06 and A6 = r6/(16d5) = 6.4E-09, could be added to further improve the polynomial approximation of the converging spherical wave.
Example - Point imaging in first order with a Radial Polynomial A point source at a distance d from the grating gets imaged to a point a distance d' from the grating. The signs of d and d' are taken as negative when "left" of the grating and positive when "right" of the grating, in the usual Y-Z "layout" convention. The A2 coefficient of the Radial Polynomial function should be set to A2 = 0.5*(n'/d' - n/d) and the diffraction order set to -1 (or the sign of A2 should be negative). The refractive index of the object and image points are given by n and n', respectively. Since this represents the parabolic approximation to a spherical phase front, the image will be aberrated.
If, for example, n = n' = 1.0, d = -25 and d' = +50, the A2 coefficient should be set to 0.03.
Two-source user-recorded holographic optical element The "Two-source user-recorded holographic optical element" is a variation of the idealized "Two point exposure holographic optical element", except that the two beams used to record the hologram are generated by tracing real rays to the grating plane. By tracing real rays to the grating surface, which may have any arbitrary shape, the user is able to include a model of the recording optics in order to generate an accurate representation of the real phase profile at the hologram surface. Or, the user may generate custom raysets for the recording sources that will be used in the specification of the grating.
Consider the graphic below, which shows a simplistic example representing the intention of this feature. Nominally, the grating phase profile is intended to be defined by the interference of a plane wave (the red rays) and a spherical wave (the blue rays). To configure the model for recording the hologram, any relevant optics representing the recording system are made traceable in the model. In this case, recording source #1 is collimated by a thick singlet while recording source #2 has no intervening optics between itself and the grating plane. As can be seen in the image, the thick singlet provides a poorly collimated beam at the grating surface and this aberration is recorded into the grating phase profile.
After the grating phase is recorded, the recording optics are made not traceable and we playback the grating with a truly collimated plane wave. As can be seen in the image below, the grating forms a severely defocused beam at the detector plane as a result of the aberrations in the recording system.
The grating phase is determined using the following sequence: 1. Rays from the specified recording sources are traced to the grating surface in an auxiliary ray buffer 2. The OPL distribution for the rays from each recording source are individually fit to an XY polynomial of a user-specified order 3. During playback (i.e. a raytrace), the gradient of the OPL polynomials are evaluated for every ray intersection on the grating and summed together to determine the grating function at the ray's intersection point
Application of the "virtual" setting for a recording source has the effect of negating the OPL gradient contribution from that source when forming the grating function as can be seen in the table below, where dOPL1 and dOPL2 indicate the gradients of the polynomials from the respective sources.
In the case where the grating surface is non-planar, the OPL polynomials for each recording source are calculated for up to 4 planes along the z axis of the surface. Then, during raytracing, the final OPL polynomial evaluated is interpolated from the planes nearest to the ray's intersection point. This is visualized in the graphic below, where a hologram is recorded onto a hemispheric surface using the interference of a plane wave and a point source. Four planes are distributed along the length of the surface and the OPL polynomials for each recording source at each plane are determined. If, during raytracing, a ray intercepts the grating surface at a position between planes 2 and 3, the OPL polynomial coefficients are interpolated from the polynomials recorded at those two planes. For surfaces with aggressive curvatures, or grossly aberrated wavefronts, the numerics of fitting the OPL polynomials to the very high orders required will inevitably lead to a loss of precision in the representation. In such cases, the user must carefully evaluate the hologram playback to check for sufficient accuracy.
The intended workflow for the two-source user-recorded holographic optical element is the following: 1. Define the sources and optics that will be used for recording the hologram. This could be as simple as two sources (of any type) directly illuminating the grating surface, or as complicated as a fully specified optomechanical model of the recording layout. 2. Open the surface/edit dialog for the grating surface and go to the Grating tab. Select the grating type as, "Two source user-recorded holographic optical element". 3. For Source 1 and Source 2: a. Use the Node entry to designate the recording sources. Note that these source nodes do NOT need to be traceable on the tree for the recording procedure to work. If the "None" option is selected, FRED will trace any existing rays in the ray buffer. If using the "None" option, only one source should be recorded at a time. b. Set the Virtual flag as appropriate. Refer to the notes above for the mathematical effect of toggling the Real/Virtual state. Generally speaking, some trial and error is required to determine the proper Real/Virtual state for each source in order to achieve the desired grating effect. Note that this flag can be modified after the grating is recorded without needing to re-record the hologram. c. Specify the number of polynomial terms to be used in the fitting of the OPL for each recording source. The polynomial must have between 1 and 1024 terms. The coefficients of the polynomial fit will be displayed in the GUI after the recording process is completed, which allows the user to inspect the magnitude of the coefficients and determine whether more coefficients are needed (or, conversely, whether fewer coefficients could be used to the same effect). d. Right mouse click in the hologram section of the dialog and choose one of the available options; "Record Source 1", "Record Source 2", or "Record Both Sources". The source nodes designated in (a) do not need to be traceable on the object tree for recording to succeed, though any relevant optomechanical components necessary to the hologram recording optical path do need to be traceable at the time of recording. As mentioned above, the "Record Both Sources" option is not advised when the source nodes are set to "None". When using the "None" option for recording source designation (i.e. use rays that already exist in the ray buffer), the recording should be performed specifically for Source 1 or Source 2. 5. Select the desired Diffraction Efficiency specification on the right hand side of the surface edit/view dialog. 4. If necessary, make any components of the optomechanical system used for the hologram recording raytraces not traceable on the object tree.
After completing the steps above, the hologram is recorded and the grating surface is ready to be raytraced and analyzed.
•The recording sources can be coherent or incoherent, but randomized incoherent sources can be helpful for avoiding any aliasing in the OPL fitting that may result from the grid constructions of coherent sources. •During recording, overfill the grating surface with rays so that there are OPL data points available for fitting all the way to the outer aperture of the grating surface. •The recording sources are raytraced through all traceable geometry between the source and the grating surface. Be sure that only the geometry relevant for the recording paths are active at the time of recording, or that any geometry unrelated to hologram recording does not interfere with the recording paths. •When used with the Volume Hologram Efficiency specification, the grating surface should be immersed in the refractive index of the volume hologram emulsion. This ensures that during the recording process the recording rays are properly refracted into the volume hologram medium when their OPL data is recorded. •Polychromatic sources may be used for recording, but the wavelength of the last ray in the recording raytrace will be used as the "design wavelength". •When using incoherent, randomized raysets, the location of the central fringe may vary slightly due to the randomness in the average OPL of the recording rays. Increasing the number of rays in the recording sources will trend the central fringe to converge on its nominal location. This effect can be seen, for example, when comparing the same hologram definition constructed using the "Two-source user-recorded holographic optical element" and the idealized "Two-point exposure holographic optical element". For a discrete ray incident on each of these gratings under the same conditions, the OPL of the ray after diffraction from each grating may be different by a small amount that is attributable to the randomness of the rayset used during recording of the user-recorded hologram.
The "Two-source user-recorded holographic optical element" is accessible in the following way: 1. For a surface desired to have this grating attribute, double click on the surface node in the object tree to open its surface/edit dialog 2. On the Grating tab of the surface dialog, select the grating type, "Two-source user-recorded holographic optical element" from the drop-list
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