Elli, Nergis, Daniel, Evan
Today we did a preliminary measurement of the Schnupp asymmetry to be 9.0cm +/- 1cm. We injescted a beam from the auxiliary laser on IOT2R which can be offset in frequency from the main laser. We measured the power of this beam at the AS port using a 1611 photodiode on ISCT6. With the michelson locked to a bright fringe, we measured the variation of auxiliary laser power with frequency at the AS port. Due to the Schnupp asymmetry, the power should change as the auxiliary laser offset from the carrier increases as cos(scnupp_assymetry*2*pi*f_offset/c).
By fitting a curve to our measured data, we can get an estimate of the Schnupp asymmetry. I did a fit using least squares fitting to a quadratic to get 9.0cm +/-1cm Schnupp asymmetry. The measured data and curve fitting codes are attached. Evan hopes that we can reduce our error estimet by doing the curve fitting more elegantly.
Summary
Attached are two plots showing the measured data, along with least-squares fits (using scipy.optimize.curve_fit). The horizontal axis is the detuning frequency of the aux laser from the PSL (and the sign is arbitrary). The vertical axis is rf power of the AS port beat note, as measured with an HP4396 (so these watts are electrical, not optical).
For both the upper and lower fringes, I've fit to a quadratic using the formula a(f−h)2 + k (the "vertex" form), where f is the detuning frequency. For the lower fringe, I find the minimum-power frequency occurs at f− = −840(3) MHz, and for the upper fringe I find f+ = 815(4) MHz. The uncertainties come entirely from the fitting routine; I have not included the measurement uncertainties (which we estimate to be 0.5 dBm or so). The Schnupp asymmetry is then found via the formula 2π(f+ − f−) LS / c = π; with this I find LS = 90.6(3) mm, where the errors come from the fit alone. I believe the difference between this uncertainty (<1%) and the uncertainty reported above (11%) is that the above analysis involves some covariances in the fitted parameters, and these must be accounted for when propagating the uncertainty forward to the Schnupp asymmetry.
More details
Instead of using the vertex form for the parabola, one can fit using the formula af2 + bf + c (the "standard" form). The minimum-power frequency is then f = −b/(2a). I ran a fit using this functional form, and for the upper fringe frequency I found a = 1.54e-4, b = -0.241, and c = 1.03e2. The nominal value of f+ is then 815 MHz. To find the uncertainty, I examined the covariance matrix of the fit parameters:
| Cov | a | b | c |
|---|---|---|---|
| a | 2.28e-10 | -3.74e-07 | 1.51e-04 |
| b | -3.74e-07 | 6.15e-04 | -0.250 |
| c | 1.51e-04 | -0.250 | 102 |
To find the variance var(f), we need the variances var(a) and var(b), as well as the covariance cov(a, b); the formula to find var(f) is then given in this Wikipedia article. Applying this formula gives an uncertainty [i.e., sqrt(var(f))] of 4 MHz, in agreement with the fit to the vertex form. If instead cov(a, b) is left out, I find that the reported uncertainty is much larger (114 MHz), and more in line with the larger uncertainty reported earlier.
The code I've used is attached.
