Autocorrelator

An autocorrelator is a device used to determine the autocorrelation function of an input signal . One of the most important implementations of such a device is the optical autocorrelator , which allows the duration of ultra-short light pulses to be determined. But there are also realizations in digital electronics that z. B. be used to measure the dynamics of diffusing particles in fluorescence correlation spectroscopy or dynamic light scattering .

general description

An autocorrelator calculates its autocorrelation function from an input signal I (t)

${\displaystyle G(\tau )=\langle I(t)\cdot I(t+\tau )\rangle :=\lim \limits _{T\rightarrow \infty }{\frac {1}{T}}\int \limits _{0}^{T}I(t)\cdot I(t+\tau )\;\mathrm {d} t}$

This provides information about the self-similarity of the signal I (t) with a delay τ and thus also allows signals to be analyzed that are hidden in the noise (e.g. in fluorescence correlation spectroscopy, as the autocorrelation function of white noise applies ), or are too fast for normal detection (see Optical Autocorrelator). ${\displaystyle g(\tau )\propto \delta (\tau )}$

Optical autocorrelator

background

A temporal resolution of a light pulse in the picosecond or femtosecond range is not possible with photodiodes , since the speed of a photodiode is limited by the recombination time of the electron-hole pairs, which is typically greater than 100 picoseconds. In order to temporally resolve a light pulse, you need reference processes that are shorter than the event to be measured. This is only possible with optical methods. In an autocorrelator, the impulse is measured “with itself” as a reference.

Structure and functionality

Schematic structure of an autocorrelator. BS: beam splitter, M1 and M2: mirror, M2 is mounted on a variable delay line, NC: non-linear crystal for generating the second harmonic (SHG) (e.g. BBO ), F: filter that only transmits the frequency-doubled light, D: Detector. red signal is input signal; blue signal corresponds to detected signal

The picture opposite shows a possible implementation of an autocorrelator. In principle, it represents a Michelson interferometer . The incident pulse is first split into two parts in a beam splitter. These pass through different paths independently of one another and are then brought together again in the beam splitter. The combined pulses hit a non-linear crystal (e.g. BBO ) in which the frequency-doubled (second harmonic) of the incident light is generated. The conversion efficiency, i.e. the intensity of the second harmonic, depends on the intensity of the light in the crystal. This in turn depends on the time lag between the two pulses. This offset is set by a mirror that is attached to a variable delay line (in the figure M2). By measuring the intensity of the frequency-doubled light as a function of the time offset, the autocorrelation of the incident pulse is measured. The duration can be determined from this, assuming the underlying pulse shape.

Mathematical description

A pulse with the temporal intensity distribution is first split into 2 pulses and then reunited. Since the impulses travel different paths, they have a time delay to each other: ${\displaystyle I(t)}$ ${\displaystyle \tau }$

${\displaystyle E_{\mathrm {ges} }(t)=E(t)+E(t-\tau )}$.

The following applies to the intensity of the frequency-doubled light in the non-linear crystal: ${\displaystyle I_{\mathrm {SH} }}$

${\displaystyle I_{\mathrm {SH} }(2\omega ,t,\tau )\propto \left[I_{1}(t)+I_{1}(t-\tau )\right]^{2}}$where is. ${\displaystyle I_{1}\propto \left|E_{1}\right|^{2}}$

The detector now measures the mean value of the 2nd harmonic over time , as its time constant is much greater than the pulse duration: ${\displaystyle S}$ ${\displaystyle T}$

${\displaystyle S(\tau )\propto {\frac {1}{T}}\int \limits _{0}^{T}I_{\mathrm {SH} }(t,\tau )\mathrm {d} t}$.

In an autocorrelator with a non-interfering beam path we get:

${\displaystyle S(\tau )\propto \,\langle I^{2}\rangle +\langle I^{2}\rangle +4\langle I(t)I(t-\tau )\rangle }$.

The angle brackets 〈·〉 mean the mean value over time. The last summand represents the autocorrelation function of the temporal intensity distribution of the pulse to be measured. Assuming the pulse shape, its duration can now be calculated.

If the pulse were not doubled in light frequency after the superposition, the detector would measure a signal that is independent of the time delay . So you don't get any information about the intensity of the pulse. In contrast to an autocorrelator, can the delay path be adjusted very finely, slowly and reproducibly, e.g. B. with an FTIR spectrometer , light frequency doubling is not necessary and the pulse duration can also be determined from the resulting interferogram . ${\displaystyle \tau }$

Electronic autocorrelator

Linear correlator

Schematic structure of a digital-electronic linear correlator

An electronic autocorrelator calculates its autocorrelation function from an analog or digital input signal I (t) . In many areas, the input signals are now digitized and then processed further with a so-called linear or multi-τ autocorrelator. The basic structure of a linear autocorrelator is shown on the right. The input signal is delayed by discrete steps ( ) and multiplied by the undelayed input signal. The result is summed up. To perform a discrete estimate of the autocorrelation function: ${\displaystyle \tau _{k}=k\cdot \tau _{\text{min}}=\tau _{\text{min}},2\tau _{\text{min}},3\tau _{\text{min}},...}$ ${\displaystyle k\in \mathbb {N} }$

${\displaystyle {\hat {G}}(\tau _{k})={\frac {1}{N}}\sum \limits _{n=1}^{N}I_{n}\cdot I_{n+k}}$

The continuous input signal is broken down into N discrete steps I _n . The delay stages um connected in series can be implemented with the aid of a shift register . Special microchips ( ASIC ) were developed for the first implementations and special applications (e.g. high speed) of this concept , later implementations use FPGAs and / or digital signal processors (DSPs). In the latter architectures, so-called multiply accumulate commands or blocks are available, with which the autocorrelation can be carried out very efficiently, since they realize exactly the single step shown on the right (multiply two numbers and then add them up). ${\displaystyle \tau _{\text{min}}}$

Multi-τ correlator

The so-called multi-τ correlator , which combines several linear stages, represents an extension of the linear correlator . The signal I _{n is} summed up over (typically 2) time periods between the stages . The next stage then correlates the signal . A semi-logarithmic distribution of the delays τ _k is thus achieved and a large delay range can be scanned with relatively little hardware expenditure. ${\displaystyle I_{n}'=I_{n}+I_{n-1}}$

literature

• Wolfgang Demtröder: Laserspektoskopie , 5th edition, Springer-Verlag, 2007

Individual evidence

1. ↑ M. Engels, B. Hoppe, H. Meuth, and R. Peters: A single chip 200 MHz digital correlation system for laser spectroscopy with 512 correlation channels . In: ISCAS'99. Proceedings of the 1999 IEEE International Symposium on Circuits and Systems, 1999 . tape 5 , 1999, p. 160-163 , doi : 10.1109 / ISCAS.1999.777535 .
2. ^ Steven B. Kaplan: Hybrid technology digital correlation spectrometers for astronomy and communications . In: Proceedings of the 22nd International Symposium on Superconductivity (ISS 2009) . tape 470 , no. 20 , 2009, p. 1538-1545 .
3. ↑ K. Schatzel: New concepts in correlator design . In: Inst. Phys. Conf. Ser. tape 77 , 1985, pp. 175-184 .