Varying optical coefficients to generate spectra for polishing control

A method of generating a library of reference spectra, comprising:

• storing an optical model for a layer stack having a plurality of layers, wherein the layer stack comprises a stack of P+1 layers, and wherein the stack includes a first layer, and with layer 0 being a bottom layer and layer P being an outermost layer;
• receiving user input identifying a set of one or more refractive index functions and a set of one or more extinction coefficient functions for the first layer from the plurality of layers, wherein the set of one or more refractive index functions includes a plurality of different refractive index functions or the set of one or more extinction coefficient functions includes a plurality of different extinction coefficient functions;
• and
• for each combination of a refractive index function from the set of refractive index functions and an extinction coefficient function from the set of extinction coefficient functions, calculating a reference spectrum using the optical model based on the refractive index function, the extinction coefficient function and a first thickness of the first layer, to generate a plurality of reference spectra,
• wherein calculating the reference spectrum comprises calculating a stack reflectance RSTACK comprising
• R
• STACK
• ⁢
• ⁢
• 2
• =
• E
• P
• -
• H
• P
• μ
• P
• E
• P
• +
• H
• P
• μ
• P
• where for each layer j>0, Ej and Hj are calculated as
• [
• E
• j
• H
• j
• ]
• =
• [
• cos
• ⁢
• ⁢
• g
• j
• i
• u
• j
• ⁢
• sin
• ⁢
• ⁢
• g
• j
• i
• ⁢
• ⁢
• μ
• j
• ⁢
• sin
• ⁢
• ⁢
• g
• j
• cos
• ⁢
• ⁢
• g
• j
• ]
• ⁡
• [
• E
• j
• -
• 1
• H
• j
• -
• 1
• ]
• where E0 is 1 and H0 is μ0, and where for each layer j≧0, μj=(nj−ikj)·cos φj and gj=2π(nj−ikj)·tj·cos φj/λ, where nj is an index of refraction of layer j, kj is an extinction coefficient of layer j, tj is a thickness of layer j, φj is an incidence angle of light to layer j, and λ is a wavelength of the light.

A method of generating a library of reference spectra, comprising:

• storing an optical model for a layer stack having a plurality of layers;
• receiving user input identifying a set of one or more refractive index functions and a set of one or more extinction coefficient functions for a first layer from the plurality of layers, wherein the set of one or more refractive index functions includes a plurality of different refractive index functions or the set of one or more extinction coefficient functions includes a plurality of different extinction coefficient functions;
• and
• for each combination of a refractive index function from the set of refractive index functions and an extinction coefficient function from the set of extinction coefficient functions, calculating a reference spectrum using the optical model based on the refractive index function, the extinction coefficient function and a first thickness of the first layer, to generate a plurality of reference spectra,
• wherein calculating the reference spectrum comprises calculating a first spectrum RSTACK1 using the optical model and combining the first spectrum RSTACK1 with a second spectrum RSTACK2, and wherein calculating the reference spectrum RLIBRARY comprises calculating
• R
• LIBRARY
• =
• 1
• R
• REFERENCE
• ⁡
• [
• X
• *
• R
• STACK
• ⁢
• ⁢
• 1
• +
• (
• 1
• -
• X
• )
• *
• R
• STACK
• ⁢
• ⁢
• 2
• ]
• where RSTACK1 is the first spectrum, RSTACK2 is the second spectrum, RREFERENCE is a spectrum of a bottom layer of the first stack and the second stack, and X is a value between 0 and 1.