Description

A diffractive optical element (DOE) or diffractive surface can be simulated in FRED by adding a holographic phase grating to a surface (HOE). The phase grating in FRED does not model the actual physical grating, which would be made with physical binary steps, blazed steps, or an amplitude grating.  Instead the phase grating in FRED adds the appropriate phase to the rays after they refract or reflect at the surface in question.

In FRED, the grating is constructed by interfering two point sources located at arbitrary distances from the surface. The interference pattern from the two points will generate a quadratic phase front.  Among other things, this quadratic phase front can be used to correct axial color in singlet lens, see the example below.

Aspheric phase departures from this quadratic phase front, in the form of either XY or radial polynomials, may be added to the grating phase on the surface. The aspheric phase departures may be used to correct higher order chromatic aberrations like spherochromatism (chromatic variation of spherical aberration).

The diffraction efficiency of the grating is based solely on user input. As a result, FRED permits the use of non-physical diffraction efficiencies (i.e., diffraction efficiency equal to 1 for multiple wavelengths and multiple orders). Use with caution. 

NOTE: The construction parameters and conventions used to define the HOE are analogous to those used in conventional lens design programs and can be entered or copied over accordingly.

Navigation

The grating control is found under the Grating tab in the Edit/View Surface dialog box.

Controls

The following steps illustrated in the diagrams below demonstrate how to add a diffraction grating to an arbitrary surface. This particular example uses the radial aspheric phase option, but an XY polynomial phase grating is created in a similar fashion.

Example

A DOE can be used to correct the axial chromatic aberration of a singlet, as shown in the following example. In the first case, the singlet is a BK-7 lens with an effective focal length of 100 mm. The front surface of the lens has a fourth order aspheric term to correct third order spherical aberration. The second surface of the lens is a simple sphere. A spot diagram for collimated input to the lens is shown at best geometric focus.

In the second case, a holographic phase grating is added to the second surface to bring the blue and the red to a common focus. The HOE is construction is shown in the final dialog above. Since both point sources are coincident and located at infinity (the default specification) the base holographic phase surface is a plane. Without further modification, this grating would have no effect on ray propagation. Correction is achieved by adding to this a radial phase departure with 2nd and 4th order terms. Note that only the 2nd order term is required to correct axial chromatic aberration.    

Create a source

Right mouse click on the Optical Sources folder and select, Create New Source Primitive > Plane Wave (coherent), and name the source, "multiwavelength source".  Per the image below, set the power of the source to 1.0 (parameter 0), make the x and y semi-apertures of the source 10 mm (parameters 1 and 2), and set the number of samples in x and y to 15 (parameters 4 and 5).

From the Wavelength Attributes options, choose "List" and then set the Source Draw Color to blue.  In the wavelengths list section, set three wavelengths as 0.4, 0.55 and 0.7 microns.  In this construction three plane waves, one at each wavelength, will be generated.

The dialog for the Plane Wave (coherent) type Source Primitive is shown below.  Press OK to create the source and add it to your document.

Aspheric lens without the diffractive

To build the first lens in this example, right mouse click on the Geometry folder and choose, "Create new subassembly".  Name the new subassembly node, "aspheric lens" and then hit OK.

Right mouse click on the new subassembly node in your tree and choose, "Create New Lens" from the menu.  Fill in the lens parameters as shown in the image below.

By default, the first surface of the lens is positioned at the local origin of it's parent node on the tree. In this construction, the parent node of the lens element is the "aspheric lens" subassembly node.  This relationship can be seen in the "Location of the Lens" section of the lens dialog above.

Expand the aspheric lens node on your tree so that you can see the individual surfaces.  Right mouse click on the first surface of the lens node and select, Edit/View Surface" from the list of options.  In the resulting dialog, change the surface type from Conicoid to Standard Asphere using the Type drop down list. Depending on your Preferences, you may see the following dialog

Click on the Yes button to continue with the modification.

The coefficient for the R² term is zero. The coefficient for the R⁴ term is –5.0214e-07, as shown below.

Press Apply to accept the changes and leave the dialog open, or OK to accept the changes and close the dialog.

Go back to the "aspheric lens" subassembly folder on the tree, right mouse click and select: Create Element Primitive > Plane.  The image surface is a plane 4 mm in diameter and located 102.1 mm from the front surface of the lens, as shown below in the creation dialog.  Hit OK to add the detector plane to your model.

Expand out the tree so that you can see the surface node underneath the "det" element primitive node you just created.  Right mouse click on the surface and select the option, "Auto Create and Attach an Analysis Surface".  This will add a new analysis surface to the Analysis Surface(s) folder of your object tree.

Trace and render the rays, as shown below. Note the blue (short wavelength) comes to a focus before the longer green and red wavelengths. This is axial chromatic aberration, which is defined as the variation in focal length with color. It arises from the spectral dispersion inherent to optical materials. The aspheric front surface has removed the spherical aberration, so each color comes to a sharp focus.

The Position Spot Diagram shows the blur sizes at each of the three wavelengths.

Aspheric lens with a diffractive surface to correct axial chromatic aberration

In this section, we will create a new version of the geometry so that we can modify the lens prescription to use a holographic grating in order to correct for the axial color.

Right mouse click on the "aspheric lens" subassembly and select, "Copy".  Now, right mouse click on the Geometry folder and select, "Paste".  Select the new subassembly node to highlight it on the object tree and then press "F2" on your keyboard, which allows you to rename the subassembly node to "diffractive lens".

Open the dialog for the lens node of your "diffractive lens" subassembly and edit the name, description, and radius of the back surface.  Hit OK on the dialog to accept the changes.

Expand the tree view so that you can see "Surface 1" of the "DOE lens" element.  Double click on the "surface 1" node to open its dialog and then change the value of the r^4 aspheric term to -4.34355E-07 and then hit OK on the dialog.

Now double click on "Surface 2" of the "DOE lens" element to open its dialog and then switch to the Grating tab.

Use the drop-list selection box on the Grating section to choose the grating type as, ‘Two point exposure holographic optical element’.  Next, edit the point source locations and wavelength. In this example, we are accepting the default point source locations. That is, both point sources are located at infinity because we have specified propagation direction instead of a physical location. The former results in a traveling plane wave, the latter in a convergent or divergent spherical wavefront. In practice, either point source positions or directions are acceptable and they can be used in combination with one or the other. The direction of propagation is determined by the Real or Virtual setting. Selecting a Real point source tells FRED that the propagation between the source and the hologram plane has a positive path length. Conversely, the Virtual setting tells FRED that the propagation from the point source to the hologram plane represents a negative optical path length. As these settings are a matter of convention, you may need to experiment with the inputs, changing the Real and Virtual buttons and/or the sign of the diffraction order to achieve the desired grating effect. The X, Y, and Z locations refer to the physical location of the point(s) in the case of a position specification, or, when normalized, the direction cosine of the plane wave in the case of a direction specification. Point source locations and directions are always measured in the local coordinate system of the surface. By accepting the defaults, we are interfering two coincident plane waves, traveling in the same direction and having the same optical path length. This results in a plane of constant phase at the hologram. Without the addition of additional phase departure terms, the grating will have no effect. The exposure wavelength determines the grating frequency or, equivalently, the fringe spacing of the interference pattern formed by the two sources. In this example, change the exposure wavelength from the default setting of 0.589 microns to 0.55 microns. The exposure wavelength can take any value, independent of the source wavelengths defined in the model.

To add aspheric phase terms, left click on the pull down button in the column next to Phase Departure and select Radial polynomial.  A Radial polynomial adds an aspheric phase departure that goes as |R|ⁿ , where R is the radial distance from the local origin. Initially, only R0 is shown. Enter a coefficient value or 0 to add a row for R1. Additional rows are added in a similar fashion. For this example, enter the following coefficients: R0 = 0, R1 = 0, R2 = 0.00031065, for R3 = 0, and R4 = –2.0685e-08, as shown in the figure below.

On the right hand side of the dialog, in the Diffraction Efficiency section, use the drop-down menu to select, "Simple efficiency table" as the diffraction efficiency specification type.  Left click in the column header where the diffraction order is specified to make the up/down spinner control appear, which allows you to change the diffraction order. Click on the up or down arrow to add or subtract from the order. For this example, use –1 as the diffraction order.

Right mouse click in the diffraction efficiency table area to bring up a context menu that allows you to append, insert or delete additional wavelength rows or diffraction order columns from the spreadsheet.

There is no limit to the number of allowable orders or wavelengths. In this example, add the system wavelengths: 0.4 microns, 0.55 microns, and 0.7 microns. The diffraction efficiency for each wavelength is unity, which means that 100% of the light incident on the grating at each of the 3 wavelengths is diffracted into the –1 order. It is not necessary to enter all of the system wavelengths uniquely, since FRED will interpolate through the table. Similarly, the diffraction efficiency can be defined at wavelengths independent of those contained in the source or sources. FRED linearly interpolates the diffraction efficiency between adjacent (lower and higher) wavelengths already defined inside a range. Alternatively, FRED assigns the diffraction efficiency the same value as that of the nearest wavelength if only a single wavelength is entered or the new wavelength is defined outside an existing range (i.e. FRED does not extrapolate the data). Note that FRED allows you to enter a diffraction efficiency of 1 for multiple diffraction orders at the same wavelength. This potentially violates conservation of energy, as FRED does not re-normalize, but rather accepts the user- entered values as absolute.

The completed dialog is shown in the following figure.

Press OK or Apply to accept the changes.

Finally, expand out the object tree so that you can see the surface underneath the "det" element primitive in the "diffractive lens" subassembly.  As was done previously, right mouse click on the surface and choose, "Auto Create and Attach an Analysis Surface".

Right mouse click on the "aspheric lens" subassembly node and select, "Not Traceable" to remove it from the raytrace.

Perform a raytrace so that the source propagates through the diffractive lens subassembly. The DOE has been designed to correct the red and blue wavelengths. These now share a common focus. This focus is not coincident with the green focus because of a residual chromatic error called secondary spectrum, which cannot be corrected by a diffractive surface. For this example, the image surface is coincident with the paraxial focus for green light. In practice, however, the image plane can be shifted towards the lens slightly so that all three colors have the same blur size.

Perform another Positions Spot Diagram analysis using the analysis surface for the diffractive lens detector surface.  The spot distribution for the blue and red wavelengths now match, while the residual secondary spectrum error gives a smaller spot size for the 0.55 micron wavelength.