Computationally-assisted multi-heterodyne spectroscopy
According to one aspect, a multi-heterodyne system is disclosed, which comprises a first laser source for generating multi-mode radiation having a frequency spectrum characterized by a first plurality of phase coherent frequencies, and a second laser source for generating multi-mode radiation having a frequency spectrum characterized by a second plurality of phase coherent frequencies. The system further comprises at least one detector for detecting a combination of the multi-mode radiation generated by the first and second laser sources so as to provide a multi-heterodyne signal having a frequency spectrum characterized by a plurality of beat frequencies, each beat frequency corresponding to a pairwise difference in the first and second plurality of phase coherent frequencies. The system further comprises an analyzer for receiving said multi-heterodyne signal and configured to employ a predictive model of the multi-heterodyne signal to provide estimates of any of phase error and timing error associated with the beat frequencies.
1. A multi-heterodyne system, comprising:
a first laser source for generating multi-mode radiation having a frequency spectrum characterized by a first plurality of phase coherent frequencies,
a second laser source for generating multi-mode radiation having a frequency spectrum characterized by a second plurality of phase coherent frequencies,
at least one detector for detecting a combination of said multi-mode radiation generated by said first and second laser sources so as to provide a multi-heterodyne signal having a frequency spectrum characterized by a plurality of beat frequencies each beat frequency corresponding to a pairwise difference in said first and second plurality of phase coherent frequencies,
an analyzer for receiving said multi-heterodyne signal and configured to employ a predictive model of said multi-heterodyne signal to provide estimates of any of phase error and timing error associated with said beat frequencies.
2. The system of claim 1 , wherein said analyzer corrects any of said phase error and timing error of said detected multi-heterodyne signal based on said estimates so as to generate a corrected multi-heterodyne signal.
3. The multi-heterodyne system of claim 2 , wherein said predictive model of said multi-heterodyne signal (y(t)) is defined as:
y
(
t
)
=
∑
n
A
n
e
i
2
π
∫
f
n
d
t
=
∑
n
r
n
e
i
φ
n
e
i
2
π
∫
f
n
dt
wherein,
A n denotes a complex amplitude associated with n th beat frequency characterized by a real amplitude r n and a phase φ n ,
ƒ n denotes the frequency of the n th beat frequency.
4. The multi-heterodyne system of claim 3 , wherein said analyzer minimizes an error function defined as:
J
(
x
)
=
∑
k
y
k
-
h
(
x
k
)
R
-
1
2
+
x
k
-
f
(
x
k
-
1
)
Q
-
1
2
wherein,
x k denotes a state of the system at time k,
y k denotes measurement of the multi-heterodyne signal at time k,
h(x k ) denotes the measurement function h evaluated at state x k and defined as follows:
h(x k )=Σ n A nk e i2πϕ nk =Σ n r nk e iφ nk e iϕ nk , wherein A nk , r nk , φ nk , ϕ nk denote, respectively, A n , r n , φ n and ϕ n evaluated at time k,
ƒ(x k ) denotes the time evolution function ƒ evaluated at state x k such that:
r n(k+1) =r nk
φ n(k+1) =φ nk
ϕ n(k+1) =ϕ nk +2πΔ t ƒ nk
wherein r n(k+1) , φ n(k+1) , ϕ n(k+1) , denote, respectively, r n , φ n , and ϕ n evaluated at time k+1,
R is said measurement noise covariance, and
Q is said process noise covariance.
5. The multi-heterodyne system of claim 1 , wherein said analyzer is further configured to minimize an error function associated with a difference between said detected and said predicted multi-heterodyne signal to provide said estimated phase and timing errors.
6. The multi-heterodyne system of claim 5 , wherein said error function comprises any of an extended Kalman filter, an unscented Kalman filter, and a particle filter.
7. The multi-heterodyne system of claim 5 , wherein said predictive model of said multi-heterodyne signal (y(t)) is defined as:
y
(
t
)
=
∑
n
A
n
e
i
(
φ
0
+
n
Δφ
)
=
∑
n
r
n
e
i
φ
n
e
i
(
φ
0
+
n
Δφ
)
wherein,
A n denotes a complex amplitude associated with n th beat frequency characterized by a real amplitude r n and a phase φ n ,
φ 0 denotes frequency offset phase between the multimode radiation from said first and second lasers and is defined as follows:
f
0
=
1
2
π
d
φ
0
dt
,
where f 0 denotes a time-dependent frequency offset between two lowest frequencies of said first and second plurality of frequencies,
Δφ denotes repetition rate phase and is defined as follows:
Δ
f
=
1
2
π
d
Δφ
dt
,
where Δf denotes said repetition rate of said beat frequencies.
8. The multi-heterodyne system of claim 7 , wherein said error function is defined as:
J
(
x
)
=
∑
k
y
k
-
h
(
x
k
)
R
-
1
2
+
x
k
-
f
(
x
k
-
1
)
Q
-
1
2
wherein,
x k denotes a state of the system at time k,
y k denotes measurement of the multi-heterodyne signal at time k,
h(x k ) denotes measurement function h(x) evaluated at state x k as follows:
h(x k )=Σ n A nk e i(φ 0k +nΔφ k ) =Σ n r nk e iφ nk e i(φ 0k +nΔφ k ) , wherein A nk , r nk , φ nk , φ 0k , Δφ k denote, respectively, A n , r n , φ n , φ 0 , and Δφ evaluated at time k,
ƒ(x k ) denotes time evolution function ƒ(x) evaluated at state x k such that:
r n(k+1) =r nk
φ n(k+1) =φ nk
φ 0(k+1) =φ 0k +2πΔ t ƒ 0k
Δφ k+1 =Δφ k +2πΔ t Δƒ k
wherein r n(k+1) , φ n(k+1) , φ 0(k+1) , and Δφ k+1 denote, respectively, r n , φ n , φ 0 , and Δφ evaluated at time k+1,
R is said measurement noise covariance,
Q is said process noise covariance.
9. The multi-heterodyne system of claim 1 , wherein said frequency spectrum of any of said first and second pluralities of phase coherent frequencies spans a range of at least about 1 octave.
10. The multi-heterodyne system of claim 1 , further comprising an optical combiner for receiving the radiation from said first and second lasers and generating a combined radiation beam directed to said at least one detector.
11. The multi-heterodyne system of claim 1 , wherein at least one of said first and second lasers generates continuous-wave (CW) radiation.
12. The multi-heterodyne system of claim 1 , wherein at least one of said first and second lasers generates pulsed radiation.
13. The multi-heterodyne system of claim 12 , wherein at least one of said first and second lasers generates chirped pulsed radiation.
14. The multi-heterodyne system of claim 1 , wherein at least one of said first and second lasers comprises a quantum cascade laser.
15. The multi-heterodyne system of claim 1 , wherein at least one of said first and second laser sources comprises an infrared laser source.
16. The multi-heterodyne system of claim 1 , wherein at least one of said first and second laser sources comprise a terahertz laser source.
17. The multi-heterodyne system of claim 1 , wherein at least one of said first and second laser sources comprise a laser diode.
18. The multi-heterodyne system of claim 1 , wherein the multimode radiation generated by each of said first and second laser sources comprises a frequency comb.
19. The multi-heterodyne system of claim 1 , wherein said at least one detector comprises two detectors, each of said detector receiving a combination of the multi-mode radiation generated by said first and second laser sources to generate a multi-heterodyne signal, and wherein said analyzer operates on the multi-heterodyne signal associated with one of said detectors to generate said estimates of any of phase error and timing error and applies said estimates to multi-heterodyne signal generated by the other detector to generate a corrected multi-heterodyne signal.
20. The multi-heterodyne system of claim 1 , wherein at least one of said laser sources comprises a micro-ring resonator.
21. The multi-heterodyne system of claim 20 , wherein said micro-ring resonator generates a frequency comb.
22. A method for processing a multi-heterodyne signal comprising:
generating from a first laser source multi-mode radiation having a frequency spectrum characterized by a first plurality of phase coherent frequencies,
generating from a second laser source multi-mode radiation having a frequency spectrum characterized by a second plurality of phase coherent frequencies,
detecting a combination of said multi-mode radiation generated by said first and second laser sources so as to provide a multi-heterodyne signal having a frequency spectrum characterized by a plurality of beat frequencies, each beat frequency corresponding to a pairwise difference between said first and second plurality of phase coherent frequencies, and
employing a predictive model of said multi-heterodyne signal to provide estimates of any of phase error and timing error associated with said beat frequencies.
23. The method of claim 22 , further comprising correcting any of said phase error and timing error of said detected multi-heterodyne signal based on said estimates so as to generate a corrected multi-heterodyne signal.
24. The method of claim 22 , further comprising minimizing an error function associated with a difference between said detected and said predicted multi-heterodyne signal to provide said estimated phase and timing errors.
25. The method of claim 24 , further comprising using any of an extended Kalman filter, an unscented Kalman filter, and a particle filter to minimize the error function.
26. The method of claim 22 , wherein said predictive model of said multi-heterodyne signal is defined as:
y
(
t
)
=
∑
n
A
n
e
i
(
φ
0
+
n
Δφ
)
=
∑
n
r
n
e
i
φ
n
e
i
(
φ
0
+
n
Δφ
)
wherein,
A n denotes a complex amplitude associated with n th beat frequency characterized by a real amplitude r n and a phase φ n ,
φ 0 denotes frequency offset phase and is defined as follows:
f
0
=
1
2
π
d
φ
0
dt
,
where f 0 denotes a time-dependent frequency offset between two lowest frequencies of said first and second plurality of frequencies,
Δφ denotes repetition rate phase and is defined as follows:
Δ
f
1
2
π
d
Δφ
dt
,
where λf denotes said repetition rate of said beat frequencies.
27. The method of claim 22 , further comprising combining the multi-mode radiation generated by said first and second lasers to generate a combined beam for detection by said detector.
28. The method of claim 22 , wherein at least one of said first and second lasers generates continuous-wave (CW) radiation.
29. The method of claim 22 , wherein at least one of said first and second laser sources generates pulsed radiation.
30. The method of claim 22 , wherein at least one of said first and second laser sources comprises a quantum cascade laser.
31. The method of claim 22 , wherein at least one of said first and second laser sources comprises a infrared laser source.
32. The method of claim 22 , wherein at least one of said first and second laser sources comprises a terahertz laser source.
33. A multi-heterodyne spectrometer comprising:
a first laser source for generating multi-mode radiation having a frequency spectrum characterized by a first plurality of phase coherent frequencies,
a second laser source for generating multi-mode radiation having a frequency spectrum characterized by a second plurality of phase coherent frequencies,
a sample holder arranged such that the multi-mode radiation generated by at least one of said first and second laser sources passes through said sample holder so as to interact with a sample contained therein,
at least one detector for detecting a combination of the multimode radiation generated by said first and second lasers, wherein the combination includes at least one multimode radiation having passed through the sample holder, so as to generate a multi-heterodyne signal having a frequency spectrum characterized by a plurality of beat frequencies, each beat frequency corresponding to a pairwise difference between said first and second plurality of phase coherent frequencies, and
an analyzer for receiving said multi-heterodyne signal and configured to employ a predictive model of said multi-heterodyne signal to provide estimates of any of phase error and timing error associated with said beat frequencies.
34. The multi-heterodyne spectrometer of claim 33 , wherein said analyzer corrects any of said phase error and timing error of said detected multi-heterodyne signal based on said estimates so as to generate a corrected multi-heterodyne signal.
35. The multi-heterodyne spectrometer of claim 34 , wherein said analyzer determines at least one property of said sample based on an analysis of said corrected multi-heterodyne signal.