# Knowledge Base

Standard mathematical and matrix functions are listed in this page. Users have also the option to define their own custom functions using user defined functions

Trigonometric and complex

 Command Description sin Trigonometric sin function. cos Trigonometric cos function. tan Trigonometric tan function. asin Inverse trigonometric sin function. acos Inverse trigonometric cos function. atan Inverse trigonometric tan function. atan2 Same as atan, but returns angle in correct quadrant. real Returns the real part of variable imag Returns the imaginary part of variable conj Complex conjugate abs Absolute value angle Phase of a complex number. unwrap Removes phase difference of more than 2π.

Logarithmic, exponential and power

 Command Description log The natural logarithm. Input can be complex or negative. log10 The log, base 10. Input can be complex or negative. sqrt The square root. exp The exponential.

Matrix functions

 Command Description size Returns the dimensions of a matrix. length Returns the total number of elements in a matrix. pinch Remove singleton dimensions from a matrix. sum The sum of a matrix. prod The product of elements in a matrix. max The max value in a matrix. min The min value in a matrix. amax The maximum value in a specified dimension of a matrix. amin The minimum value in a specified dimension of a matrix. dot The dot product of two vectors. cross The cross product of two vectors. flip Flip a matrix in one dimension. interp Linear interpolation function. spline Cubic spline interpolation. polyfit Polynomial fit. normpdf Evaluate the normal (Gaussian) probability density function (PDF) pearson4pdf Evaluate the Pearson IV probability density function (PDF) fitnormpdf Fit to the normal (Gaussian) probability density function (PDF) fitpearson4pdf Fit to the Pearson IV probability density function (PDF) integrate Integrate a matrix. integrate2 Integrate a matrix, ignore singleton dimensions. find Find values that satisfy a condition in a matrix. findpeaks Find peaks in a matrix. findresonances Find the frequency, decay constant and Q-factor of resonances extracted from the time trace of a signal. transpose Transpose a matrix. ctranspose Transpose a matrix, and do complex conjugate. mult Perform matrix multiplication of two or more matrices. reshape Reshape the matrix to have different dimensions conserving the overall product of the dimensions. eig Calculate the eigenvalues and/or eigenvectors of a matrix. permute Rearrange the dimensions of a matrix. inv Calculate the inverse of a matrix. mean Return the mean value of a matrix. var Returns the variance. std Returns the standard deviance. mapfind Returns a string value associated to specified point, given a file containing a map of values to a string. svd Returns a 3-cell array for the decomposition. chol Returns the lower triangular matrix. norm Returns the matrix y to the L2-norm. cov Calculates the covariance matrix. corrcoef Calculates the correlation matrix. scorrcoef Generates a spatial correlation matrix. corrtransf Calculates the transformation matrix. chpts Samples function on a Chebyshev grid. chebin Returns Chebyshev interpolation of a sampled function. chebpol Chebpol is similar to chebin command, but it offers additional control over the interpolation process as it allows to specify the polynomial order. chebpol1 Returns the first derivative of Chebyshev polynomials of a function sampled on the Chebyshev grid. sort Sorts a matrix in ascending or descending order. sortmap Sorts matrices in more complex ways than simply ascending or descending order of the array. conv2 Convolves two 2-dimensional arrays.

Manipulating variables

String functions

 Command Description num2str Convert number to a string. str2num Convert a string into a floating point number. eval Execute string containing Lumerical scripting language. feval Run a Lumerical script file. length Returns the total length of the string. substring Returns a substring of a string, as a specified position and length. findstring Returns the position of a substring in a string. replace Replaces a part of a string with another, at a specified position. replacestring Replaces all instances of a substring with another string. splitstring Split a single long string into a cell (string) array based on a delimiting character. upper Convert a string to upper case. lower Convert a string to lower case. toscript Returns a string containing the equivalent script of a generate variable.

Frequency and time-domain

 Command Description fft Fast Fourier transform. fftw Returns the angular frequency vector. fftk Returns the spatial wavevector kx. invfft Inverse fft. czt Chirped z-transform. sroughness Returns a matrix containing a rough surface characterized by an RMS amplitude.

Line and polygon functions

 Command Description polyarea Returns the area of a polygon. centroid Returns the center of mass of a polygon. polyintersect Determines if two polygons intersect. inpoly Determines if a series of points are inside our outside a polygon. polygrow Grows or shrinks a polygon by a specified amount. polyand Combines two polygons into one with an and operation. polyor Combines two polygons into one with an or operation. polydiff Combines two polygons into one by taking the difference. polyxor Combines two polygons into one with a xor operation. lineintersect Returns the intersection of line segments. linecross Determines if line segments cross each other.

Colorimetry

 Command Description colormatchfunction Returns a set of color matching functions. colormatch Calculates the X, Y, Z tristimulus values for a set of color matching functions. colormatchxy Calculates the x, y chromaticity values for a set of color matching functions. colormatchuv Calculates the u, v chromaticity values for a set of color matching functions.

Multilayer stack calculations

 Command Description stackrt Calculates the reflection and transmission of a plane wave through a multi-layer stack using the analytic transfer matrix method. stackfield Calculates the fields within a multilayer stack illuminated from below by a plane wave using the analytic transfer matrix method. stackdipole Analytically calculates the dipole emission for a multilayer stack.

Multi-quantum well calculations

 Command Description mqwgain Calculates gain and spontaneous emission in multiple quantum well structures buildmqwmaterial Returns material properties of the type and format required by mqwgain

Ion implant doping calculations

 Command Description implantrange Calculates the range of the doping profile from ion implant. implantstraggle Calculates the straggle of the doping profile from ion implant. implantskewness Calculates the skewness of the doping profile from ion implant. implantkurtosis Calculates the kurtosis of the doping profile from ion implant. implantlateralscatter Calculates the lateral scatter of the doping profile from ion implant.

Miscellaneous

 Command Description ceil Round up. floor Round down. mod Modulus after division. round Rounds to the nearest integer. rand Returns a uniformly distributed random number between 0 and 1. lognrnd Returns a lognormal distributed random number. randn Returns a normally distributed random number. randreset Resets the random number seed. finite Determines if a number is finite or NaN. solar Returns the solar power spectrum all Returns 1 if all of the specified matrix entries are nonzero and returns 0 otherwise. any Returns 1 if any of the specified matrix entries are nonzero and returns 0 otherwise. interptri Interpolates a data set from a triangular to a rectilinear grid. interptet Interpolates a data set in 3D from a tetrahedral to a rectangular grid. quadtri Returns approximated integration (first order quadrature) of data on a 2D finite element mesh. quadtet Returns approximated integration of data on a 3D finite element mesh. precision Returns truncated value to a user specified precision. erf Returns the error function. erfc Returns the complementary error function. erfinv Returns the inverse error function. erfcinv Returns the inverse complementary error function. unique Returns an array containing all unique values in a given matrix. uniquevertices Given a matrix of vertices, returns a matrix of unique vertices with differences in values larger than a specified tolerance. icht Takes the Chebyshev interpolation coefficients and returns the corresponding function samples dcht Returns the Chebyshev interpolation coefficients besselj Bessel function of the first kind bessely Bessel function of the second kind besseli Modified Bessel function of the first kind besselk Modified Bessel function of the second kind mie3d Analytically calculates scattering, absorption, and extinction coefficients of a spherical particle. mie3ds12 Analytically calculates scattered farfield functions of a spherical particle.

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