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## 6.10 Incident Beams

Different types of incident beams can be sourced into the simulation grid using the TFSF group.

Global variable: group TFSF

The TFSF group contains definitions for various types of incident electromagnetic beams required for scattering problems. Angora uses the total-field/scattered-field (TF/SF) technique to source incident beams into the simulation grid (see TafloveHagness). In this technique, a rectangular surface surrounding the scatterer is designated the total-field/scattered-field boundary (or the TF/SF box in short); and the electromagnetic field on this surface is supplemented by certain terms proportional to the incident electromagnetic field. These additional terms create the incident field inside the surface (suggesting the term injection), while maintaining a very small electromagnetic field outside the surface. The region outside the TF/SF box only harbors the scattered field created by the scatterers inside the TF/SF box. The field inside the box is the total field, which is a sum of the incident field and the scattered field. Since the boundary divides the grid into total-field and scattered-field regions, the term "TF/SF boundary" is justified.

TF/SF incident beam injection is supported for infinite planar layered media. Infinite planar layers are created using the MaterialSlabs variable (see section Planar Layers). Currently, only layers with different permittivities and electrical conductivities are supported. Permeability variations across layers will also be supported in the future. Angora also supports evanescent plane waves resulting from plane waves passing from a high-permittivity layer to a low-permittivity one at a low grazing angle. Angora supports evanescent waves only for narrowband plane waves, which have appreciable frequency components only in a small band around a center frequency. A modulated Gaussian waveform with a large can be used as a narrowband waveform in cases where evanescent waves might be present (see Modulated-Gaussian Waveforms).

Different types of incident beams are defined in their respective lists inside the TFSF group. These are explained in the following subsections.

 TFSF: { PlaneWaves: ( { … … } ); FocusedLaserBeams: ( { … … } ); }; 

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### 6.10.1 Plane Waves

A plane wave is one of the simplest solutions of Maxwell’s equations; with the electric field

where is the unit vector in the direction of propagation, is the electric-field unit vector, is the electric field amplitude, and is the distance vector. The velocity of propagation is determined by the material properties in the direction from which the plane wave is incident. The time waveform is arbitrary. Inserting the above expression into Maxwell’s equations, it is found that the electric-field unit vector is perpendicular to the direction of propagation, as well as the magnetic-field unit vector.

In a discrete FDTD grid, a plane wave propagates at a slightly lower velocity than in continuum. Furthermore, there is an intrinsic grid velocity anisotropy that results from the inherent rotational asymmetry of the rectangular FDTD grid. These are partially alleviated in Angora by the use of the matched numerical dispersion (MND) technique (see TafloveHagness).

Sub-variable of TFSF: list PlaneWaves

Plane waves are defined inside a PlaneWaves list inside the TFSF group:

 TFSF: { PlaneWaves: ( { theta = 40.0; phi = 90.0; psi = 90.0; waveform_tag = "waveform1"; pw_extra_amplitude = 1.0; tfsf_back_margin_x_in_cells = 6; tfsf_front_margin_x_in_cells = 6; tfsf_left_margin_y_in_cells = 6; tfsf_right_margin_y_in_cells = 6; tfsf_lower_margin_z_in_cells = 6; tfsf_upper_margin_z_in_cells = 6; pw_origin_x = 0.0; pw_origin_y = 0.0; pw_origin_z = 0.0; display_warnings = true; min_cells_per_lambda = 15.0; }, { … … } ); }; 
Sub-variable of PlaneWaves: floating-point theta (units: degrees)
Sub-variable of PlaneWaves: floating-point phi (units: degrees)

The incidence angles of the plane waves are defined in terms of the traditional spherical-coordinate variables . The first of these angles is the zenith angle, while the second is the azimuth angle. In this figure, the definitions of these angles are shown schematically. The theta variable specifies the zenith angle in degrees. Although this angle is traditionally defined between 0 and 180deg, theta can be assigned any negative or positive value. The phi variable specifies the azimuth angle in degrees. Although this angle is traditionally defined between 0 and 360deg, phi can be assigned any negative or positive value.

Note that the incidence angles specify the direction from which the plane wave is incident; not the direction in which it propagates.

Sub-variable of PlaneWaves: floating-point psi (units: degrees)

This variable is used to specify the polarization of the electric field of the incident plane wave. Maxwell’s equations dictate that the electric field is perpendicular to the incidence vector In order to define the orientation of the electric vector unambiguously, a local coordinate system is defined, such that and The unit vectors are perpendicular to each other, and lie in the plane perpendicular to the incidence vector The polarization angle of the electric-field unit vector is defined as the left-handed (clockwise) rotation angle around the axis defined by the incidence vector The variable psi sets this angle in degrees.

Figure 6.7: Graphical description of the incidence and polarization angles associated with a plane wave.

Sub-variable of PlaneWaves: string waveform_tag

This string variable specifies the electric-field waveform in the above equation. The waveform is interpreted in (Volts/m) units. This should match a previously-defined tag in a Waveforms definition (see section Waveforms).

Sub-variable of PlaneWaves: floating-point pw_extra_amplitude (units: V/m, default: 1.0)

This variable sets the electric field amplitude in the above equation.

Sub-variable of PlaneWaves: floating-point tfsf_back_margin_x (units: m)
Sub-variable of PlaneWaves: floating-point tfsf_front_margin_x (units: m)
Sub-variable of PlaneWaves: floating-point tfsf_left_margin_y (units: m)
Sub-variable of PlaneWaves: floating-point tfsf_right_margin_y (units: m)
Sub-variable of PlaneWaves: floating-point tfsf_lower_margin_z (units: m)
Sub-variable of PlaneWaves: floating-point tfsf_upper_margin_z (units: m)
Sub-variable of PlaneWaves: integer tfsf_back_margin_x_in_cells (default: 6)
Sub-variable of PlaneWaves: integer tfsf_front_margin_x_in_cells (default: 6)
Sub-variable of PlaneWaves: integer tfsf_left_margin_y_in_cells (default: 6)
Sub-variable of PlaneWaves: integer tfsf_right_margin_y_in_cells (default: 6)
Sub-variable of PlaneWaves: integer tfsf_lower_margin_z_in_cells (default: 6)
Sub-variable of PlaneWaves: integer tfsf_upper_margin_z_in_cells (default: 6)

By default, the total-field/scattered-field (TF/SF) surface is placed 6 grid cells away from the PML boundary (see Perfectly-Matched Layer (PML)). You can specify different margins to reduce the computational burden associated with the TF/SF operation. This burden is directly proportional to the area of the TF/SF surface. The margins can be specified in meters or in grid cells. For the latter, the _in_cells suffix should be appended to the variable name. If given in meters, the margins are rounded to the nearest multiple of the spatial step size.

Sub-variable of PlaneWaves: floating-point pw_origin_x (units: m)
Sub-variable of PlaneWaves: floating-point pw_origin_y (units: m)
Sub-variable of PlaneWaves: floating-point pw_origin_z (units: m)
Sub-variable of PlaneWaves: floating-point pw_origin_x_in_cells (default: 0)
Sub-variable of PlaneWaves: floating-point pw_origin_y_in_cells (default: 0)
Sub-variable of PlaneWaves: floating-point pw_origin_z_in_cells (default: 0)

These variables set the coordinates of the point relative to which the distance is defined in the above equation. The coordinates are with respect to the grid origin (see section Coordinate Origin). The units are either in meters or grid cells. For the latter, the _in_cells suffix should be appended to the variable name.

Sub-variable of PlaneWaves: boolean display_warnings (default: true)
Sub-variable of PlaneWaves: floating-point min_cells_per_lambda (default: 15)

The boolean variable display_warnings enables or disables the printing of warning messages. Currently, a warning is displayed only when there are not enough grid cells per "minimum" wavelength in the excitation waveform. This minimum wavelength is defined to be the one at which the spectrum of the waveform falls to -40dB below its maximum. The number of required grid cells per the minimum wavelength is determined by the min_cells_per_lambda variable.

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### 6.10.2 Focused Laser Beams

Angora can synthesize focused laser beams created by an aplanatic optical system (i.e., free of spherical aberration and coma) illuminated by a normally-incident paraxial Hermite-Gaussian laser mode. The electromagnetic formulation of the focused beam is based on the classic work of Richards and Wolf (see Richards59). This formulation is interpreted as a sum of plane waves, and approximated by a finite sum in the FDTD implementation (see Capoglu08). It is assumed that the Hermite-Gaussian laser mode filling the entrance pupil of the optical system has a beam width much larger than the wavelength, and is therefore in the paraxial regime. In this regime, the wavefronts are almost planar, perpendicular to the optical axis, and the electric field of the beam has a negligible longitudinal component.

The incidence geometry for the laser mode illuminating the entrance pupil of the system is shown on the upper right in this figure. A local coordinate system is defined on the plane of the entrance pupil, such that and The unit vectors are perpendicular to each other, and lie in the plane perpendicular to the incidence vector The symmetry axes of the Hermite-Gaussian beam are rotated at an angle of with respect to the axis, in a clock-wise (left-handed) sense with respect to

On the plane of the entrance pupil, which is assumed to coincide with the waist of the Hermite-Gaussian beam, the electric field is given by

where is the beam half width (or beam waist radius), is the electric-field unit vector, is the electric field amplitude, and are the (physicists’) Hermite polynomials of order m. The first few Hermite polynomials are

The time waveform is arbitrary. Since we assume that the width of the beam at the entrance pupil is much larger than the wavelengths contained in the waveform , it is reasonable to employ the paraxial approximation, in which the rays propagate parallel to the optical axis and the electric-field unit vector is perpendicular to the direction of propagation

The focusing optical system between the entrance pupil and the exit pupil is assumed aplanatic. This means that, in addition to focusing on-axis points stigmatically (without spherical aberration), the system also correctly focuses off-axis points up to the first order in off-axis distance. The latter condition corresponds to the absence of circular coma. The sine condition, first derived by Ernst Abbe in 1881, is an expression of this in mathematical terms. The sine condition reads

where is the radius of the entrance pupil, and is the back focal length of the focusing system. The sine condition is also shown geometrically in this figure.

Figure 6.8: Geometry of the focused laser beam.

Sub-variable of TFSF: list FocusedLaserBeams

Focused laser beams are defined inside a FocusedLaserBeams list inside the TFSF group:

 TFSF: { FocusedLaserBeams: ( { theta = 40.0; phi = 90.0; psi = 90.0; alpha = 0; x_order = 1; y_order = 1; waveform_tag = "waveform1"; flb_extra_amplitude = 1.0; ap_half_angle = 23.5782; back_focal_length = 0.1; filling_factor = 1; object_space_refr_index = 1.0; tfsf_back_margin_x_in_cells = 6; tfsf_front_margin_x_in_cells = 6; tfsf_left_margin_y_in_cells = 6; tfsf_right_margin_y_in_cells = 6; tfsf_lower_margin_z_in_cells = 6; tfsf_upper_margin_z_in_cells = 6; flb_origin_x = 0.0; flb_origin_y = 0.0; flb_origin_z = 0.0; display_warnings = true; min_cells_per_lambda = 15.0; }, { … … } ); }; 
Sub-variable of FocusedLaserBeams: floating-point theta (units: degrees)
Sub-variable of FocusedLaserBeams: floating-point phi (units: degrees)

The incidence angles for the focused laser beam are defined in reference to the paraxial beam that hits the entrance pupil (see figure). The incidence angles are defined in terms of the traditional spherical-coordinate variables . The first of these angles is the zenith angle, while the second is the azimuth angle. The theta variable specifies the zenith angle in degrees. Although this angle is traditionally defined between 0 and 180deg, theta can be assigned any negative or positive value. The phi variable specifies the azimuth angle in degrees. Although this angle is traditionally defined between 0 and 360deg, phi can be assigned any negative or positive value.

Note that the incidence angles specify the direction from which the paraxial beam is incident; not the direction in which it propagates.

Sub-variable of FocusedLaserBeams: floating-point psi (units: degrees)

This variable is used to specify the polarization of the transverse electric field of the paraxial beam on the entrance pupil. The electric field is perpendicular to the incidence vector The polarization angle is defined as the left-handed (clockwise) rotation angle with respect to the incidence vector between the symmetry axis and the electric-field unit vector (see figure). The variable psi sets this angle in degrees.

Sub-variable of FocusedLaserBeams: floating-point alpha (units: degrees) (default: 0)

This angle specifies the rotation of the symmetry axes of the Hermite-Gaussian beam with respect to the local coordinate axes . The rotation is clock-wise (left-handed) with respect to the incidence vector (see figure).

Sub-variable of FocusedLaserBeams: integer x_order
Sub-variable of FocusedLaserBeams: integer y_order

These positive integers specify the orders of the Hermite polynomials in the definition of the Hermite-Gaussian beam in the above equation. x_order and y_order correspond to m and n, respectively.

Sub-variable of FocusedLaserBeams: string waveform_tag

This string variable specifies the electric-field waveform in the above equation, except a time advance equal to the time of propagation from the entrance pupil to the focal point F. In other words, the actual waveform on the entrance pupil is advanced in time by with respect to the waveform represented by waveform_tag. This is needed because we simulate the fields around the focus F; not the entrance pupil. The waveform is interpreted in (Volts/m) units. The string waveform_tag should match a previously-defined string tag in a Waveforms definition (see section Waveforms).

Sub-variable of FocusedLaserBeams: floating-point flb_extra_amplitude (units: V/m, default: 1.0)

This variable sets the electric field amplitude in the above equation.

Sub-variable of FocusedLaserBeams: floating-point ap_half_angle (units: degrees)

This variable sets the half-angle of the illumination cone in degrees (see figure).

If a planar layered medium is present in the grid (see Planar Layers), the incidence cone bounded by the angle should lie entirely within the upper or lower half space. In other words, the incidence cone cannot cut across the grazing direction . Angora will throw an error if this is found to be the case.

Sub-variable of FocusedLaserBeams: floating-point back_focal_length (units: m)
Sub-variable of FocusedLaserBeams: floating-point back_focal_length_in_cells

This variable specifies the back focal length of the optical system. The units are either in meters or grid cells. For the latter, the _in_cells suffix should be appended to the variable name.

Sub-variable of FocusedLaserBeams: floating-point filling_factor

This dimensionless parameter represents the ratio between the half-width of the incident paraxial beam and the radius of the entrance pupil. The larger this ratio, the more overfilled the pupil, and vice versa. This parameter is defined as

Sub-variable of FocusedLaserBeams: floating-point object_space_refr_index (default: 1.0)

This variable specifies the refractive index of the object space of the optical system; i.e., the space from which the paraxial beam is incident. Note that the FDTD simulation grid is in the image space of the optical system; therefore the refractive index of the image space is determined by the material filling the FDTD grid. If you want to simulate an oil immersion scenario, for example, you should fill the FDTD simulation space with the material representing the immersion oil. The object-side refractive index object_space_refr_index is seldom different than 1.0, which is the default value.

Sub-variable of FocusedLaserBeams: floating-point tfsf_back_margin_x (units: m)
Sub-variable of FocusedLaserBeams: floating-point tfsf_front_margin_x (units: m)
Sub-variable of FocusedLaserBeams: floating-point tfsf_left_margin_y (units: m)
Sub-variable of FocusedLaserBeams: floating-point tfsf_right_margin_y (units: m)
Sub-variable of FocusedLaserBeams: floating-point tfsf_lower_margin_z (units: m)
Sub-variable of FocusedLaserBeams: floating-point tfsf_upper_margin_z (units: m)
Sub-variable of FocusedLaserBeams: integer tfsf_back_margin_x_in_cells (default: 6)
Sub-variable of FocusedLaserBeams: integer tfsf_front_margin_x_in_cells (default: 6)
Sub-variable of FocusedLaserBeams: integer tfsf_left_margin_y_in_cells (default: 6)
Sub-variable of FocusedLaserBeams: integer tfsf_right_margin_y_in_cells (default: 6)
Sub-variable of FocusedLaserBeams: integer tfsf_lower_margin_z_in_cells (default: 6)
Sub-variable of FocusedLaserBeams: integer tfsf_upper_margin_z_in_cells (default: 6)

By default, the total-field/scattered-field (TF/SF) surface is placed 6 grid cells away from the PML boundary (see Perfectly-Matched Layer (PML)). You can specify different margins to reduce the computational burden associated with the TF/SF operation. This burden is directly proportional to the area of the TF/SF surface. Because the focused beam is actually a collection of many (often hundreds) of plane waves, the reduction of the surface area of the TF/SF box greatly helps the simulation performance.

The margins can be specified in meters or in grid cells. For the latter, the _in_cells suffix should be appended to the variable name. If given in meters, the margins are rounded to the nearest multiple of the spatial step size.

Sub-variable of FocusedLaserBeams: floating-point flb_origin_x (units: m)
Sub-variable of FocusedLaserBeams: floating-point flb_origin_y (units: m)
Sub-variable of FocusedLaserBeams: floating-point flb_origin_z (units: m)
Sub-variable of FocusedLaserBeams: floating-point flb_origin_x_in_cells (default: 0)
Sub-variable of FocusedLaserBeams: floating-point flb_origin_y_in_cells (default: 0)
Sub-variable of FocusedLaserBeams: floating-point flb_origin_z_in_cells (default: 0)

These variables set the coordinates of the back focal point of the focusing lens. (see figure). The coordinates are with respect to the grid origin (see section Coordinate Origin). The units are either in meters or grid cells. For the latter, the _in_cells suffix should be appended to the variable name.

Sub-variable of FocusedLaserBeams: boolean display_warnings (default: true)
Sub-variable of FocusedLaserBeams: floating-point min_cells_per_lambda (default: 15)

The boolean variable display_warnings enables or disables the printing of warning messages. Currently, a warning is displayed only when there are not enough grid cells per "minimum" wavelength in the excitation waveform. This minimum wavelength is defined to be the one at which the spectrum of the waveform falls to -40dB below its maximum. The number of required grid cells per the minimum wavelength is determined by the min_cells_per_lambda variable.

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