This page provides a simulation file containing several example structures created with the surface object, to demonstrate the types of structures that can be created with this object. Surface primitives can be used to define complex material volumes that exist above or below analytically defined surfaces. In 3D simulations, a surface (S) is defined as a function of variables u and v, i.e. S = S(u,v). The variables (u,v) can represent (x,y), (x,z) or (y,z) depending on the surface orientation. Similarly, in 2D simulations, a surface is defined as a function of u (S = S(u)) where u can represent x or y.

The simulation file usr_surface_example.fsp contains several examples of surface objects that can be created using conic, polynomial, and custom equation options. More examples of the surface objects can be found in the object library available in FDTD.

Note: Drawing resolution Initially the object will be drawn at a low resolution, as shown in the top half of this figure. You can increase the drawing resolution with the settings on the Graphical rendering tab. The bottom half of this figure shows the object displayed at the same resolution as the image that is was imported from. It is important to note that the graphical rendering does not affect the simulation. Irrespective of the drawing resolution, the meshed object used for the simulation is always as has been defined. |

The custom tab is only available for surface primitives in MODE and FDTD. The diagrams below show how the surface is used to create a volume of desired material.

The surface equation can contain up to three terms, conic, polynomial and custom which are added together to create the total surface. Not all terms have to be included.

$$ S = S_{conic} + S_{polynomial} + S_{custom} $$

**Conic term**:

\( S_{conic} = \frac{c r^2}{1 + \sqrt{1 - (\kappa + 1)c^2r^2} } \)

c = 1/R, c is the inverse of the radius of curvature of the surface at r = 0.

- \( \kappa \) is the conic constant. When \( \kappa = -1 \) we have a parabolic surface. When \( \kappa = 0\) we have a spherical surface.
- r2 = u2 + v2.

** Polynomial term**:

\( S_{polynomial} = \sum_{i,j=0}^5 M_{ij} u^i v^j \)

- There are 36 available coefficients

**Custom term**:

\( S_{custom} = f(u,v) \)

- You can choose any analytic function of (u,v).
- The syntax and functions available to specify the index are found in the Equation interpreter section.

For example, you could use

\( S_{custom} = \sin(u + v) \)

## Geometry tab

- X, Y, Z: The center position of the object
- X MIN, X MAX: X min, X max position
- Y MIN, Y MAX: Y min, Y max position
- Z MIN, Z MAX: Z min, Z max position
- X SPAN, Y SPAN, Z SPAN: X, Y, Z span of the object

## Surface tab

- ORIENTATION: The surface determines if S is a function of (x,y), (x,z) or (y,z)..
- ZERO PLANE: This determines if surface is measured from the lower edge of the rectangular volume or the upper edge. See diagrams above.

- MATERIAL POSITION: This determines if the material fills the regions above the surface or below the surface. See diagrams above.
- SET UNDEFINED TERMS TO: It is possible that the surface equation becomes undefined. For example, it could be sqrt(-1) for some values of (u,v). In this case, the surface function will become either zero or the maximum value allowed by the rectangular volume encompassing the surface object.
- U0, V0: The origin of the (u,v) coordinate system, If U0, V0 are zero, then (u,v) = (0,0) will correspond to the center of the surface object. Setting these values to non-zero will offset the origin by a desired amount.
- SURFACE UNITS: All quantities defining the surface must be measured in the same units. You can choose these units with this menu.
- CONIC: Check this option to include the conic term in the surface equation. If this is checked, then set

- RADIUS OF CURVATURE: curvature of the surface at the origin. This is equal to the inverse of the parameter c described in the definition of Sconic.
- CONIC CONSTANT: The k constant.

- CUSTOM: Check this option to include the custom term in the surface equation

- EQUATION: The equation as a function u and v.
- POLYNOMIAL: Check this option to include a polynomial term in the surface equation. If this option is checked, the M coefficients as explained above can be entered into a table.

To import a surface object, please see Import object surfaces

## Material tab

The material options are as follows:

- MATERIAL: This field can be set to any material included in the material database. It is possible to include new materials in the database, or edit the materials already included. See the material database section for more information.
- OVERRIDE MESH ORDER FROM MATERIAL DATABASE: Select to override the mesh order from the material database and manually set a mesh order. The mesh order is used by the simulation engine to select which material to use when two materials overlap. See the mesh order (optical) or mesh order (electrical) section for more details.
- MESH ORDER: Set the mesh order in this field if the OVERRIDE MESH ORDER FROM MATERIAL DATABASE option is selected. If the option is not selected, the field displays the material's default mesh order from the database. For example, a material of mesh order 1 will take precedence over a material of mesh order 2.

The following only applies to MODE and FDTD:

If <Object defined dielectric> is selected, then the INDEX property must be set.

- INDEX: The refractive index of the structure, when the material type is <Object defined dielectric>. The index must be greater than one.
- Anisotropic index: To specify an anisotropic refractive index, use a semicolon to separate the diagonal xx,yy,zz indices. Eg. 1;1.5;1
- Spatially varying index: It is possible to specify a spatially varying refractive index by entering an equation of the variables x,y,z in this field. Eg. 2+0.1*x will create an object where the refractive index increases in the X direction. The units of the spatial variables (x,y,z) must be set with the 'INDEX UNITS' property described below. The variables x,y,z will be zero in the center of the object. When using an equation in this field, consider using a mesh override region to control the simulation mesh size. For more information on entering equations, see the Equation interpreter section.

- INDEX UNITS: Only relevant when specifying a spatially varying equation in the INDEX properly described above. Specify the units (nm, um, m) of the x,y,z position variables.
- GRID ATTRIBUTE NAME: Enter the name of the grid attribute that applies to this object, see the grid attribute section

The following only applies to CHARGE, HEAT, FEEM, DGTD:

If the material chosen from the drop down menu is a binary alloy consisting of two semiconductors, then there will be an additional property, namely, the "composition fraction" to set as well.

- COMPOSITION FRACTION: This is x, the fraction of the semiconductor in the alloy. x can either take a fixed value or vary.

The user can see which semiconductor has fraction x and which has fraction (1-x) shown in a line above this drop down menu.

- FIXED: This means that fraction x will be a constant value between 0 and 1.
- LINEAR X/Y/Z: This means that the composition fraction x will vary as a function x or y or z. In this case, user can specify the min and max fraction values for the min and max spatial points and the fraction will be interpolated linearly in between. x,y,z here are those of the unrotated object. x,y,z are local to the object.
- EQUATION: The user can enter an equation for the fraction that varies with u,v and w. u is (x-x0), v is (y-y0) and w is (z-z0) where x0, y0 and z0 are the center coordinates of the object. This means that u,v and w are local to the object.

## Rotation tab

Rotate objects by setting the following variables:

- FIRST, SECOND, THIRD AXES: Select rotation axis. Up to three different rotations can be applied.
- ROTATION 1,2,3: The rotation of the object in a clockwise direction about each axis, measured in degrees.

## Graphical Rendering tab

The graphical rendering tab is used to change how objects are drawn in the layout editor. The options are:

- RENDER TYPE: The options for drawing the objects are detailed or wireframe. Detailed objects are shaded and their transparency can be set using OVERRIDE COLOR OPACITY FROM MATERIAL DATABASE.
- DETAIL: This is a slider which takes values between 0 and 1. By default it is set to 0.5. Higher detail shows more detail, but increases the time required to draw objects. This setting has no effect on the simulation.
- OVERRIDE COLOR OPACITY FROM MATERIAL DATABASE: When unselected the opacity is determined from the material database. When selected, you can specify a value for ALPHA between 0 (transparent) and 1 (opaque) for the object, depending on how transparent you want the object to be.