The accurate determination of spectral linewidths is essential to much of this research. The typical testing arrangement in Fig. 2.4 employs an external-cavity tunable diode laser to sweep across the desired wavelengths and a computer to synchronize the laser’s reference output (or start trigger) with the analog transmission/reflection signal acquired by the photodetector. The method and accuracy of this synchronization depends on the laser model and is particular to each unit. For example, the piezo-tuning mode for the New Focus Velocity lasers applies a sawtooth waveform to a piezoelectric element attached to the tuning mirror; the computer also receives this waveform and applies a set calibration to its amplitude to determine the wavelength. Unfortunately, the wavelength span can vary up to 10% from day to day for a given voltage, and the forward and backward scans are hysteretic (see Ref. [60] for more information). Another common problem in older Velocity lasers is that the position sensor on their tuning arm (used for coarse motor scans) can begin to malfunction and produce an offset between the laser’s true wavelength and the wavelength indicated by the laser head—this offset can be removed during post-processing using the OSA. In general, the repeatability issues with the Velocity lasers do not impede daily testing, but precise measurements over several days (e.g., Ref. [56]) require calibration.
In contrast, the discontinued Vidia lasers only send a single pulse at the beginning of the scan (after their acceleration phase), and the computer simply assumes the laser is sweeping at the set rate after that time. To date, there has been no need to repeatedly calibrate narrow Vidia sweeps.
To calibrate a laser’s sweep range, we use a fiber-based Mach-Zehnder interferome-
ter consisting of two 50:50 splitters connected by two SMF-28e patch cords with different lengths. The entire assembly is placed inside a insulated cooler with fiber thru-ports to prevent thermal drift. The interference fringe spacing is given by δfs = ¯λ2/¯neffL, where ¯λ is the average wavelength between the two fringes, ¯nis the fiber’s effective index at ¯λ, and L is the path length difference between the two arms. For L ≈ 1 m, the reference spac- ing and wavelength are δo = 1.57±0.03 pm at λo = 1460.5 nm. The fringe spacing can be measured by performing laser scans over a small range without the customary acceleration phase, finding the start and stop wavelengths on the OSA, and dividing by the number of fringes. The resulting transmission interferogram consists of a chirped signal as a function of wavelength:
T(λ) =Tosin 2πλ2o
δo 1
λ− 1 λ1
+Tdc, (A.1)
where{To, Tdc}are amplitudes to account for incomplete interference contrast andλ1 is the wavelength of a fringe maximum [to set the phase ofT(λ)]. AtN fringes from the maximum at λ1, the wavelength (λ) and the uncertainty (σ∆) in the separation (∆ =λ−λ1) are
λ= λ2o
λ2o/λ1−N δo and σ∆=
N λ2o (λ2o/λ1−N δo)2
σδ, (A.2)
where +N corresponds to a red shift (λ > λ1), σδ is the uncertainty in the measured δo fringe spacing, and σ∆ assumes δo is the only significant source of error. For practical measurements, the fractional uncertainty in the calibrated linewidth (σ∆/∆) is the same as the fractional uncertainty in the fringe spacing (σδ/δo).
The calibration procedure for spectral measurements depends on the desired accuracy.
For general exploratory testing, the piezo-tuning ranges should be calibrated when the laser arrives from the factory and then periodically verified; the DC motor accuracy can be assessed by comparing the wavelength set point and the laser output on the OSA. Much of the early data in §3.2 and §2.2 was taken by checking the piezo calibration daily or even more frequently. A more accurate procedure is to acquire a real-time calibration signal on a second detector, as in Fig. 2.4. The most expedient use of this data is to find the true wavelength span of the data by counting fringes and then scale any raw linewidths by the ratio of the true and raw spans. This method was used throughout §4.2, but it assumes the scan rate is constant. The most accurate method is to simply count fringes starting at the beginning of the scan and to ignore the wavelength data from the laser. Time used for
further improvements would likely be better spent building another MZI with smaller δo and σδ.
The pure fringe counting method has been implemented to calibrate large wavelength ranges while studying dispersion-engineered microdisks for degenerate four-wave mixing.
Designed and fabricated by Q. Lin, the disks consist of a Si core and SiO2 cladding to control the GVD, similar to the waveguides in Ref. [117]. The disks are repeated oxidized to tune the ZDWL for the TMp=1 modes toward 1550 nm in order to achieve momentum and energy conservation for four-wave mixing between three consecutive modes. Because future experiments call for a coarse wavelength division multiplexer (20 nm channel spac- ing), the FSRs separating the signal and idler modes from the pump mode must be ap- proximately equal to the channel spacing (2530 GHz) with mismatched comparable to the modes’ linewidths (∼0.6 GHz). For comparing two wavelength spans, the measurement is less sensitive to uncertainty inδo, and the uncertainty in the difference (∆1−∆2) between the two spans is
σ(∆1−∆2) =
N1λ2o
(λ2o/λ1−δoN1)2 − N2λ2o (λ2o/λ2−δoN2)2
σδ. (A.3)
Using the MZI and sampling every ∼0.1 pm, Fig. A.1 shows a sample calibrated data set for a device with signal and idler modes well matched about the pump mode at 1528.8 nm.
The difference between the calibrated and raw wavelengths is at most 0.25 nm, and the scan rate is clearly not linear. The pump-signal and pump-idler separation is∼2676.7 GHz with a difference of 0.2±2.8 GHz, which is a relative error of 0.1%. To achieve the necessary ac- curacy of.0.02%, the MZI fringe spacing must be increased toδo≈7 pm if the uncertainty in the spacing cannot be reduced (σδ=±0.03 pm).
In the current implementation, the MZI reference signal is loaded into an (N×1) matrix and analyzed as a function of the matrix index. First, the mean is subtracted, and a zero- finding routine determines the “fractional” indices where the fringes cross the axis. These positions in the matrix are separated in wavelength by half the fringe spacing, and they pro- vide a piecewise function for interpolating the indices of the original vector onto calibrated wavelengths. Since the reference and transmission signals are sampled simultaneously, the same calibrated wavelength vector applies to the transmission data. The dependence of the fringe spacing on wavelength is included by quadratically scalingδfsat each “zero” crossing
1510 1520 1530 1540 1550 1560 1570
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
Wavelength (nm)
Detector Signal (V)
Δλ = 10 pm
−0.05 0.00 0.05 0.10 0.15 0.20 0.25
1510 1520 1530 1540 1550 1560 1570
Wavelength (nm) λcalib - λdata (nm)
(a)
(b)
2676.80 GHz 2676.60 GHz 2678.06 GHz
TM trans.
TE trans.
MZI ref.
Figure A.1: (a) Raw TE- and TM-polarized transmission spectra for a Si-SiO2 microdisk after calibrating the wavelength. The MZI reference signal was acquired concurrently with the TM spectrum. Nearly equal FSRs between the highlighted TMp=1 modes shows the ZDWL is near 1530 nm. The inset contains a 10-pm section of the calibration signal showing measured data points (∗) and the interpolated “zero” crossings (◦). (b) Difference between the calibrated and raw wavelengths for a long Vidia laser scan.
to δ′fs =δo(λ′/λo)2 where the reference splitting again is δo = 1.57 pm atλo = 1460.5 nm.
It assumes the starting wavelength is accurate; the lasers’ ±0.05 nm repeatability is suffi- cient for these purposes. This method also ignores dispersion in the optical fiber because it introduces a correction an order of magnitude smaller than the present uncertainty in δfs,o. For completeness, the dispersion can be calculated using parameters obtained from the manufacturer (Corning for SMF-28e) and is presented below for 1200 nm ≤λ≤1625 nm:
D ≡ −2πc λ2
∂2k
∂ω2
= πcSo 2
1 ω −ω3
ωo4
, (A.4a)
ng = π2c3So ω2
2ωo2 + 1 2ω2
+C1, (A.4b)
neff = π2c3So ω2
6ωo4 − 1 2ω2
+C1+C2
ω , (A.4c)
where ωo/2π = 228.3 THz (λo = 1313 nm) is the zero-dispersion frequency, the slope of D(ω = ωo) is So = 0.086 ps/(nm2·km), C1 = 1.45647 is an integration constant to give ng = 1.4682 at 1550 nm and 1.4676 at 1310 nm (specified by Corning), and the second integration constant C2 = −4.6547×1012rad/s gives neff = 1.4462 at 1550 nm (via FEM using 0.36% core-cladding contrast, a core diameter of 8.2µm, and a cladding index of refraction for pure silica n = 1.4440). With the correction to neff, the fringe spacing at 1568 nm is 1.8128 pm, and it decreases by less than 2 fm when dispersion is neglected (δfs = 1.8109 pm). The (λ′/λo)2 scaling has been verified to within the accuracy of the measurements for wavelengths from 1460 to 1625 nm.