Calibration of frequency monitors having dual etalon signals in quadrature
Improved calibration of a dual-etalon frequency monitor having x-y outputs is provided. An ellipse is fit to the (x,y) points from a set of calibration data. For each (x,y) point, an angle θ is determined. A linear fit of frequency to θ is provided. Differences between this linear fit and the determined values of θ are accounted for by including a spline fit to this difference in the calibration.
1. A method of calibrating a frequency monitor having dual etalon signals in quadrature, the method comprising:
a) providing calibration data in the form of triples (x i , y i , ν i ) indexed by i, wherein for each triple, x i is a first etalon signal at a known frequency ν i and y i is a second etalon signal at said known frequency ν i , and wherein said first and second etalon signals are substantially in quadrature;
b) fitting an ellipse to the set of points (x i , y i ), wherein said ellipse is parametrically defined according to
x=x 0 +A cos θ
y=y 0 +B sin(θ+ε);
c) calculating an angle parameter θ i of said ellipse for each of said points (x i , y i );
d) fitting a line to the set of points (θ i , ν i ) to provide a linear angle-frequency fit, wherein said line is parametrically defined according to ν=sθ+ν 0 ;
e) providing said parameters s and ν 0 and said parameters A, B, x 0 , y 0 , and ε as ν(θ) calibration outputs;
f) calibrating said frequency monitor having dual etalon signals in quadrature using said ν(θ) calibration outputs.
2. The method of claim 1 , further comprising:
g) determining a difference e i of said linear angle-frequency fit at each of said triples according to ε=ν i −(sθ i +ν 0 );
h) fitting a spline to the set of points (θ i , e i ) to provide a spline difference fit e(θ);
i) providing said spline difference fit e(θ) as part of said ν(θ) calibration outputs.
3. The method of claim 1 , wherein said calculating an angle parameter θ i of said ellipse for each of said points (x i , y i ) comprises setting θ i =a tan 2(A(y i −y 0 )−B(x i −x 0 )sin ε, B(x i −x 0 )cos ε).
4. The method of claim 1 , wherein said frequencies ν i of said calibration data are equally spaced.
5. The method of claim 4 , wherein said frequencies ν i of said calibration data are determined by a reference optical cavity.
6. The method of claim 1 , wherein said fitting an ellipse comprises:
fitting a quadratic form a 0 x 2 +a 1 xy+a 2 y 2 +a 3 x+a 4 y+a 5 =0 to said set of points (x i , y i ) to provide values for fitting parameters a 0 , a 1 , a 2 , a 3 , a 4 , and a 5 ;
calculating values for x 0 , y 0 , A, B, and ε, from said fitting parameters a 0 , a 1 , a 2 , a 3 , a 4 , and a 5 .
7. The method of claim 2 , wherein said fitting a spline comprises fitting cubic B-splines defined on a uniformly spaced set of knots.
8. A method of measuring an operating frequency of an optical source in a system including a frequency monitor having dual etalon signals in quadrature, the method comprising:
calibrating said frequency monitor according to the method of claim 1 ;
measuring source operating data (x op , y op ), wherein x op and y op are respectively said first and second etalon signals provided by said frequency monitor responsive to radiation from said optical source having said operating frequency;
calculating an angle parameter θ op of said optical source according to a θ op =a tan 2(A(y op −y 0 )−B(x op −x 0 )sin ε, B(x op −x 0 )cos ε);
determining said operating frequency from said ν(θ) calibration outputs and from said angle parameter θ op ;
providing said determined operating frequency as an output.
9. The method of claim 8 , wherein said determining said operating frequency from said ν(θ) calibration outputs and from said angle parameter θ op comprises:
providing a secondary measurement of said operating frequency;
defining a set of angles θ n =θ op +2πn, where n is any integer that results in θ n being within a predetermined calibration range;
calculating a set of frequencies ν n =θ(θ n ) from said ν(θ) calibration outputs and from said set of angles θ n ;
selecting one of said set of frequencies ν n as said determined operating frequency according to said secondary measurement.
10. A method of setting an optical source to operate at or near a predetermined frequency in a system including a frequency monitor having dual etalon signals in quadrature, the method comprising:
calibrating said frequency monitor according to the method of claim 2 ;
determining an angle θ in corresponding to said predetermined frequency ν in from said ν(θ) calibration outputs;
controlling the output frequency of said optical source such that w 1 (x−x 0 )+w 2 (y−y 0 )=0, wherein x and y are said first and second etalon signals respectively, and wherein
w
1
=
-
sin
(
θ
in
+
ɛ
)
A
cos
ɛ
and
w
2
=
cos
θ
in
B
cos
ɛ
.
11. The method of claim 10 , further comprising:
providing coarse frequency control of said optical source sufficient to distinguish said predetermined frequency ν in from any frequency ν n =ν(θ in +2πn), where n can be any non-zero integer;
wherein said frequencies ν n are determined from said corresponding angles θ in +2πn according to said ν(θ) calibration outputs.
12. A method of calibration update in a system having a frequency monitor having dual etalon signals in quadrature, the method comprising:
setting an optical source to operate at or near each of a set of predetermined frequencies ν j according to the method of claim 10 ;
measuring an output frequency error Δν j =ν j,out −ν j at each of said set of predetermined frequencies, where ν j,out is optical source frequency when said source is set to operate at ν j , wherein ν j,out is measured independently of said ν(θ) calibration outputs;
revising said spline difference fit e(θ) to decrease said output frequency error, by making incremental changes to spline fit coefficients.
13. The method of claim 12 , wherein said revising said spline difference fit is repeated one or more times.
14. A method of calibrating a wavelength monitor having dual etalon signals in quadrature, the method comprising:
a) providing calibration data in the form of triples (x i , y i , λ i ) indexed by i, wherein for each triple, x i is a first etalon signal at a known wavelength λ i and y i is a second etalon signal at said known wavelength λ i , and wherein said first and second etalon signals are substantially in quadrature;
b) fitting an ellipse to the set of points (x i , y i ), wherein said ellipse is parametrically defined according to
x=x 0 +A cos θ
y=y 0 +B sin(θ+ε);
c) calculating an angle parameter θ i of said ellipse for each of said points (x i , y i );
d) fitting a line to the set of points (θ i , λ i ) to provide a linear angle-wavelength fit, wherein said line is parametrically defined according to λ=sθ+λ 0 ;
e) providing said parameters s and λ 0 and said parameters A, B, x 0 , y 0 , and ε as λ(θ) calibration outputs;
f) calibrating said wavelength monitor having dual etalon signals in quadrature using said λ(θ) calibration outputs.
15. The method of claim 14 , further comprising:
g) determining a difference e i of said linear angle-wavelength fit at each of said triples according to e i =λ i −(sθ i +λ 0 );
h) fitting a spline to the set of points (θ i , e i ) to provide a spline difference fit e(θ);
i) providing said spline difference fit e(θ) as part of said λ(θ) calibration outputs.