Spectral Uncertainties

Spectral uncertainties in measuring ˜s^TC(λ₀) arise from the following con-tributions being discussed in this subsection:

• Deviations from the center wavelengthλ0

• Current generation from undesired spectral components λ_bl

• Limited bandwidth of the quasi-monochromatic radiation λbw

A deviation of the actual wavelength λfrom the intended wavelength λ₀ causes discrepancies in j^TC and j^RC, as the differential spectral re-sponsivity of test and reference cell are generally wavelength-dependent.

For correcting an erroneous wavelength deviation,j^TC andj^RCmight be expanded with a Taylor series up to its first order around the intended center wavelengthλ₀ Thus, the corrected current response is achieved from j^TC,RC(λ₀) = fλ0j^TC,RC(λ) with the correction term

f_λ0^TC,RC= 1−∂j^TC,RC(λ) Apart from any known deviation in center wavelength that can be corrected with f_λ0 (given that the deviations are not to strong), each measurement suffers from intrinsic uncertainties of the center wavelength

∆λ0. Typically two different contributions to ∆λ0 are considered that are, on the one hand, resulting from the (wavelength-dependent) repeata-bility of the applied components ∆λ₀₁ and, on the other hand, from a wavelength-independent offset ∆λ02[116].

The wavelength-dependence of ∆λ01in the previously presented Laser-DSR setup is predominantly caused by the variation of the output sources

(either THG or SHG are used directly or a prism or grating monochro-mator is applied, see Section 6.2.3). For THG and SHG ∆λ01 directly depends on the laser repeatability that is estimated to be approximately

±1 nm (corresponding to plus-minus one wavelength increment).¹¹ Con-sequently, ∆λ01≈ ±0.33 nm for THG and ∆λ01≈ ±0.5 nm for SHG. The wavelength repeatability of the grating monochromator is specified with

±0.04 nm for λ≤1200 nm and ±0.08 nm forλ > 1200 nm. The prism monochromator repeatability is determined by the dispersion characteris-tics of the prism and the repeatability of the motorized translation axis. In the prism monochromator configuration presented in Section 6.2.3 ∆λ01

varies from approximately 0.01 nm at 520 nm to 0.13 nm at 1040 nm.

The wavelength-independent offset is essentially given by the uncer-tainty in wavelength calibration of the setup and estimated to be ∆λ02≈

±1 nm in this work. In contrast to the repeatability uncertainty ∆λ01, the offset uncertainty ∆λ₀₂accounts for both, test and reference cell measure-ment in an identical manner given that the measuremeasure-ment system remains unchanged in between the two measurements. In order to account for this correlation, the correction terms for test and reference cell (see Eq. (6.5)) are combined to

The spectrally resolved standard measurement uncertainties¹²uλ0 result-ing from ∆λ₀ = ∆λ₀₁(λ) + ∆λ₀₂ are exemplified in Table 6.1 in Sec-tion 6.4.3 assuming rectangular probability density funcSec-tions (PDFs) for both ∆λ01 and ∆λ02. In Table 6.2 in Section 6.4.3 their relative contri-bution to the combined standard measurement uncertainty is given.

11This estimation is based on manufacturer’s data.

12Please note that general uncertainties are denoted by capitalUand thatstandard measurement uncertainties are denoted by lowercaseuthroughout this work.

In addition to the uncertainty in center wavelength, spectral compo-nents other than the desired quasi-monochromatic radiation¹³ and inci-dent onto the reference or test cell contribute to Q^TC,RC via j^TC,RC = j_mono^TC,RC+j^TC,RC_bl , with”mono”denoting the desired quasi-monochromatic and ”bl” the undesired current response contribution (”bl” symbolizes that these spectral components are supposed to beblocked). Expressing the total spectral irradianceE_λ^tot(λ) as the sum of quasi-monochromatic and undesired radiation, E_λ^tot(λ) = E_λ^mono(λ) +E^bl_λ (λ), yields the cor-rection term

f_λbl^TC,RC= 1−

R s˜^TC,RC(λ)E^bl_λ (λ) dλ

Rs˜^TC,RC(λ)E_λ^tot(λ) dλ. (6.7) As monochromators are used over wide spectral ranges in the new Laser-DSR setup, thereby blocking the undesired radiation, Eq. (6.7) only needs to be considered for THG and SHG radiation. However, the dichroic mirrors and filters used in the THG and SHG beam path (see discussion in Section 6.2.3) eliminate the undesired wavelengths in such a manner, that a correction using f_λbl^TC,RC is not reasonable. Instead, the still remaining undesired spectral contributions are taken into account as measurement uncertainties¹⁴ The insignificance ofUλbl becomes apparent in Table 6.1 in Section 6.4.3 where the standard measurement uncertainties u_λbl ≈ 0 in the entire spectral range.

Finally, a further wavelength-related measurement uncertainty arises from the bandwidth of the monochromatic radiation that deviates from the idealized case of true monochromaticity. If the second-order derivative

13These spectral components might e.g. result from remaining fundamental radia-tion after nonlinear conversion processes.

14In Eq. (6.8) a monochromatic approximation is applied that is reasonable when considering the quasi-monochromatic nature of the laser radiation in both desired and undesired spectral components. The sum over i represents that more than a single undesired quasi-monochromatic wavelength component might be present as e.g. in case of THG, when remaining SHG and fundamental wavelength components might exist simultaneously.

of ˜s^TC,RCafterλis non-vanishing (non-zero), the impact of bandwidth of the monochromatic radiation is comparable to a deviation in the center wavelength as discussed above.¹⁵ This is expressed as a second-order Taylor expansion with vanishing first order

j^TC,RC(λ)≈j^TC,RC(λ0) +∂²j^TC,RC(λ)

Instead of applying a correction using Eq. (6.9), the impact of the radi-ation bandwidthλbw might be considered as a measurement uncertainty that is subject to an additional uncertainty arising from potential varia-tionsδλ_bw induced by the laser or the monochromators. Thus, the entire bandwidth-related uncertainty reads ∆λbw=λbw±δλbw.

Similar to the previously introduced uncertainty due to a wavelength offset ∆λ₀₂, the impact ofλ_bw acts identically on both the reference cell and the test cell measurement. Thus, it becomes significant if the second-order derivatives of test and reference cell SR differ. This correlation is expressed by the correction term

fλbw=

In Table 6.1 in Section 6.4.3 the standard measurement uncertainties uλbwassigned to the spectral bandwidths are exemplified for the combina-tion of a GaAs test and a Si reference cell and the spectral bandwidths of the presented setup. In a rather conservative estimation the uncertainty of the setup bandwidths is assumed to be δλbw = ±2 nm. In spectral ranges using the prism monochromator a rectangular PDF is assumed for δλ_bwresulting from a convolution of the rectangular slit function with the

15This becomes clear when considering e.g. a linearly decaying differential spectral responsivity ˜s(λ) around a center wavelengthλ0. As long as∂˜s(λ)/∂λis a constant (which is the case for a linear decay as in this example and is equivalent to a vanishing second order derivative), the measurement outcome does not change with spectral widthλbwof the quasi-monochromatic radiation.

much smaller Gaussian beam width in the slit plane. A triangular PDF is assumed for the grating monochromator that is obtained from a convo-lution of entrance and exit slit and, finally, a normal PDF is assumed for THG and SHG radiation that is not passing any monochromator.