GRADIUM™
This surface type models lenses made from stock gradient index blanks available from LightPath Technologies, Inc. The surface shape is the same as the Standard surface. The blanks have an axial gradient index profile, which describes a reference index of refraction as a function of axial position within the glass blank. All that is required to define the lens is the starting position within the blank, the name of the stock blank profile, and of course the radii and thickness. The GRADIUM surface has a gradient index profile which is described by a polynomial of the form:
The z coordinate is the distance from the front vertex of the surface, zmax is the maximum z coordinate of the blank (also known as the boule thickness), and the value Δz is the "offset" distance along the profile. Unlike most other gradient glass models in OpticStudio, the GRADIUM surface uses only fixed, predefined axial profile coefficients. The only design parameter required is the offset value, Δz. The available profiles provided with OpticStudio are defined in the ASCII file PROFILE.GRD. Other profile files may be defined, if they end in the extension GRD and are placed in the <glass> folder (see "Folders").
To select a different profile file, see "GRADIUM Profile". For a list of the available profiles, see "Gradium™ Profile".
GRADIUM profile file format
The file format is a series of blocks of 13 lines of data defined as follows:
PROFILE_NAME GLASS_FAMILY MAX_Z DENSITY UNUSED n0 n1 ... n11
Each block of data starts with the name of the profile, which can be any valid ASCII name less than 20 characters. On the same line, the glass family name follows, which must be a gradient index material as defined in the SGRIN.DAT file, which is described in the section on the Gradient 5 surface type. The glass family name defines the reference wavelength which the profile describes. The last entry on the line is the maximum z coordinate of the blank. The 12 polynomial coefficients follow, from n0 to n11. The maximum allowed number of profiles is 100.
When OpticStudio performs ray tracing, the local z coordinate on the surface (which may be negative) is computed. The offset value is then added to determine where in the profile the coordinate lies.
Normally, the resulting value must always be positive and less than or equal to the maximum z, otherwise an error is generated (See the discussion of "Capping" below). The reference index is then evaluated. Once the reference index is computed, the index at the wavelength being traced is computed using the technique described in the section on the Gradient 5 surface.
The reference wavelength index at the front vertex is displayed as parameter 3. This value may be changed in the Lens Data Editor. When a new value is entered, OpticStudio computes the appropriate Δz to yield the specified reference index. However, Δz is the value that matters. The reference index is just displayed for convenience, and should not be made variable, or a multi-configuration operand. Note the reference index is the vertex index at the reference wavelength, which is the wavelength defined in the glass family definition file SGRIN.DAT. It is not necessarily the primary wavelength.
The GRADIUM surface model also supports 4 additional parameter terms that are intended for use in tolerancing: Decenter X, Decenter Y, Tilt X, and Tilt Y. These four terms model an axial gradient that is not perfectly centered and not perfectly parallel to the local Z axis. The tolerance terms modify the axial profile by redefining the axial coordinate z as follows:
Where
and tx, ty, tz are the coefficients of the unit vector which points along the axial gradient axis and dx and dy are the decenters in lens units of the start of the profile. If tx and ty are both zero, then the dx and dy values do not matter (since the gradient is only along the z axis) and the tz value is unity. The terms tx and ty determine the slope of the profile axis in x and y, which is intended to model the tolerance of axial alignment between the gradient axis and the mechanical axis of the lens. This expression is a linear approximation which is only valid for asymptotically small tx and ty.
The tolerance terms, dx , dy , tx , and ty , are ignored when performing paraxial ray tracing.
Usually, only the defined range of the profile is used. However, in some cases the profile may be extended in one or both directions to add additional glass to the ends of the profile, which allows the use of GRADIUM in thicker lenses. This technique is called "capping". By default, OpticStudio turns capping off, so that any ray trace which requires glass beyond the profile limits is flagged as an error. This automatically enforces boundary constraints during optimization. To remove this restriction, the capping flag can be set to be 1, 2, or 3. The default value of zero indicates the blank is bounded to the profile length at both ends. If the capping flag is 1, then only the left edge is bounded (the right edge is allowed to pass beyond the profile limit). If the capping flag is 2, then only the right edge is bounded. And if the capping is 3, then neither the left nor right edges of the blank are bounded, and both the thickness and the offset may take on any values. The additional material added to the beginning and end of the profile is assumed to be a homogeneous glass with the same index and dispersion as the respective end points of the profile. This assumption may be inaccurate if there is some slope to the profile at the defined endpoints, which is usually the case. Contact LightPath Technologies for more detailed information on designing with GRADIUM capping.
All GRADIUM profiles are defined in pairs; there is a "positive" and "negative" profile. These are identical profiles that are reversed in the axial direction. When modeling lenses in double pass, it is necessary to use one profile for the forward pass, and the other (reversed) profile for the backward pass.
The maximum step size Δt determines the trade-off between ray tracing speed and accuracy. For details see "Discussion on maximum step size for GRIN surfaces". Also see "Restrictions on surfaces following GRIN surfaces".
PARAMETER DEFINITIONS FOR GRADIUM SURFACES
| Parameter # | Definition |
| 0 | boule thickness |
| 1 | Δt |
| 2 | Δz |
| 3 | n_ref |
| 4 | dx |
| 5 | dy |
| 6 | tx |
| 7 | ty |
| 8 | Capping |
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