Method for measuring characteristics of sample
A method for measuring characteristics of a sample is provided. The method includes the following steps: obtaining an interference spectrum of the sample; transforming the interference spectrum into a temporal interference signal via a Fourier transform, in which the temporal interference signal includes a plurality of coherence wave packets; separating the wave packets; transforming the wave packets into a plurality of interface interference signals via an inverse Fourier transform; and fitting a plurality of factors of the interface interference signals into a model for obtaining the refractive indexes, the extinction coefficients, and a thickness of the sample.
1. A method for measuring characteristics of a sample, the method comprising:
obtaining an interference spectrum of the sample, wherein the interference spectrum is measured by an interferometer, the interferometer includes a broadband light source, the light source emits a broadband spectrum with a shape similar to a shape of a Gaussian function;
transforming the interference spectrum into a temporal interference signal via a Fourier transform, wherein the temporal interference signal includes a plurality of coherence wave packets;
separating the wave packets;
respectively transforming the separated wave packets into a plurality of interface interference signals via an inverse Fourier transform; and
fitting a plurality of factors of the interface interference signals into a model for obtaining refractive indexes for different wavelengths, extinction coefficients for different wavelengths, and a thickness of the sample.
2. The method of claim 1 , wherein the interference spectrum is measured by a Michelson interferometer.
3. The method of claim 1 , wherein a frequency bandwidth of the broadband light source is approximately 1×10 14 Hz to 1×10 15 Hz.
4. The method of claim 1 , further comprising:
filtering out parts of the temporal interference signal.
5. The method of claim 1 , wherein a Hilbert transform is applied to separate an envelope amplitude and a carrier phase.
6. The method of claim 1 , wherein the number of the wave packets is two, and the interface interference signals include a front interface interference signal and a rear interface interference signal.
7. The method of claim 6 , wherein the front interface interference signal is formed by the interference of a reference light and a light reflected by a front interface of the sample, and the rear interface interference signal is formed by the interference of the reference light and a light reflected by a rear interface of the sample.
8. The method of claim 6 , wherein the model includes the following equations:
A
=
G
R
front
B
=
G
t
front
t
front
′
r
rear
exp
(
-
4
π
kfl
/
c
)
C
=
∠
(
t
front
t
front
′
r
rear
r
front
)
+
4
π
nfl
/
c
wherein A is an amplitude spectrum of the front interface interference signal, B is an amplitude spectrum of the rear interface interference signal, and C is phase differences between the two interface interference signals, G is an interferometer response coefficient, t front is a complex transmission coefficient when a light enters the sample, t front ′ is a complex transmission coefficient when a light exits the sample, r front is a complex reflection coefficient of the front interface of the sample, r rear is a complex reflection coefficient of the rear interface of the sample, n is the refractive index of the sample, k is the extinction coefficient of the sample, l is the thickness of the sample, f is a frequency of a light, and A, B, and C are represented as functions of n, k and l.
9. The method of claim 8 , wherein G is determined by the following equation:
G=2ηa s a r √{square root over (I s I r )}e iφ
wherein η is an interference efficiency, a s is an attenuation factor of a sample light, a r is an attenuation factor of a reference light, I s is an incident intensity of the sample light, I r is an incident intensity of the reference light, and φ is a phase related to an optical path difference of the reference light and the sample light.
10. The method of claim 8 , wherein the Gauss-Newton's algorithm is adopted for the model fitting.
11. The method of claim 1 , wherein the thickness of the sample has a minimum λ 0 2 /(2nΔλ), wherein λ 0 is a center wavelength of a light source, Δλ is a wavelength bandwidth of the light source, and n is the refractive index of the sample.
12. A method for measuring characteristics of a sample, the method comprising:
emitting a sample light to a position of the sample for obtaining an interference spectrum of the position of the sample; and
analyzing the interference spectrum by performing the method of claim 1 for obtaining a thickness of the position of the sample.