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1. Coherence X M1
2. Coherence X Vs Unite

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In physics, coherence length is the propagation distance over which a coherent wave (e.g. an electromagnetic wave) maintains a specified degree of coherence. Wave interference is strong when the paths taken by all of the interfering waves differ by less than the coherence length. A wave with a longer coherence length is closer to a perfect sinusoidal wave. Coherence length is important in holography and telecommunications engineering.

This article focuses on the coherence of classical electromagnetic fields. In quantum mechanics, there is a mathematically analogous concept of the quantum coherence length of a wave function.

Formulas[edit]

In radio-band systems, the coherence length is approximated by

${\displaystyle L=\frac{c}{n\mathrm{\Delta }f}=\frac{{\lambda }^2}{n\mathrm{\Delta }\lambda },}$

where ${\displaystyle c}$ is the speed of light in a vacuum, ${\displaystyle n}$ is the refractive index of the medium, and ${\displaystyle \mathrm{\Delta }f}$ is the bandwidth of the source or ${\displaystyle \lambda }$ is the signal wavelength and ${\displaystyle \mathrm{\Delta }\lambda }$ is the width of the range of wavelengths in the signal.

In optical communications, assuming that the source has a Gaussian emission spectrum, the coherence length ${\displaystyle L}$ is given by ^[1]

${\displaystyle L=\frac{1}{2}{L}_{FWHM}=\frac{2\ln 2}{\pi }\frac{{\lambda }^2}{n\mathrm{\Delta }\lambda },}$

where ${\displaystyle \lambda }$ is the central wavelength of the source, ${\displaystyle n}$ is the refractive index of the medium, and ${\displaystyle \mathrm{\Delta }\lambda }$ is the (FWHM) spectral width of the source. If the source has a Gaussian spectrum with FWHM spectral width ${\displaystyle \mathrm{\Delta }\lambda }$, then a path offset of ±${\displaystyle L}$ will reduce the fringe visibility to 50%.

Coherence length is usually applied to the optical regime.

The expression above is a frequently used approximation. Due to ambiguities in the definition of spectral width of a source, however, the following definition of coherence length has been suggested:

The coherence length can be measured using a Michelson interferometer and is the optical path length difference of a self-interfering laser beam which corresponds to a ${\displaystyle 1/e=37\mathrm{\%}}$ fringe visibility,^[2] where the fringe visibility is defined as

${\displaystyle V=\frac{I_{max}-I_{min}}{I_{max}+I_{min}},}$

where ${\displaystyle I}$ is the fringe intensity.

In long-distance transmission systems, the coherence length may be reduced by propagation factors such as dispersion, scattering, and diffraction.

Lasers[edit]

Multimode helium–neon lasers have a typical coherence length of 20 cm, while the coherence length of single-mode lasers can exceed 100 m. Semiconductor lasers reach some 100 m, but small, inexpensive semiconductor lasers have shorter lengths, with one source^[3] claiming 20 cm. Singlemode fiber lasers with linewidths of a few kHz can have coherence lengths exceeding 100 km. Similar coherence lengths can be reached with optical frequency combs due to the narrow linewidth of each tooth. Non-zero visibility is present only for short intervals of pulses repeated after cavity length distances up to this long coherence length.

Other light sources[edit]

Tolansky's 'An introduction to Interferometry' has a chapter on Sources which quotes a line width of around 0.052 Angstroms for each of the Sodium D lines in an uncooled low-pressure sodium lamp, corresponding to a coherence length of around 67 mm for each line by itself. Cooling the low pressure sodium discharge to liquid nitrogen temperatures increases the individual D line coherence length by a factor of 6. A very narrow-band interference filter would be required to isolate an individual D line.

See also[edit]

References[edit]

1. ^Akcay, C.; Parrein, P.; Rolland, J.P. (2002). 'Estimation of longitudinal resolution in optical coherence imaging'. Applied Optics. 41 (25): 5256–5262. doi:10.1364/ao.41.005256. PMID12211551. equation 8
2. ^Ackermann, Gerhard K. (2007). Holography: A Practical Approach. Wiley-VCH. ISBN978-3-527-40663-0.
3. ^'Sam's Laser FAQ - Diode Lasers'. www.repairfaq.org. Retrieved 2017-02-06.

• This article incorporates public domain material from the General Services Administration document: 'Federal Standard 1037C'. (in support of MIL-STD-188)

The currently implemented coherence metrics are described below including a description of thecontent type of the tcm that showed good performance in combination with a specific metric. For details on how to create tcm see the example section. For details on performance of metrics see the resources in the reference sectionthat served for definition of standard settings for individual metrics. Note that depending on the use case, still, different settings than the standard settings for creation of tcm may be reasonable. Note that for all currently implemented metrics the tcm is reduced to the top word space on basis of the terms in x.

Considering the use case of finding the optimum number of topics among several models with different metrics,calculating the mean score over all topics and normalizing this mean coherence scores from different metricsmight be considered for direct comparison.

Each metric usually opts for a different optimum number of topics. From initial experience it may be assumedthat logratio, pmi and nmpi usually opt for smaller numbers, whereas the other metrics rather tend to propose higher numbers.

Coherence X M1

Implemented metrics:

Coherence X Vs Unite

• 'mean_logratio' The logarithmic ratio is calculated as log(smooth + tcm[x,y]) - log(tcm[y,y]), where x and y are term index pairs from a 'preceding' term index combination. Given the indices c(1,2,3), combinations are list(c(2,1), c(3,1), c(3,2)).

  The tcm should represent the boolean term co-occurrence (internally the actual counts are used) in the original documents and, therefore, is an intrinsic metric in the standard use case.

  This metric is similar to the UMass metric, however, with a smaller smoothing constant by default and using the mean for aggregation instead of the sum.

• 'mean_pmi' The pointwise mutual information is calculated as log2((tcm[x,y]/n_doc_tcm) + smooth) - log2(tcm[x,x]/n_doc_tcm) - log2(tcm[y,y]/n_doc_tcm), where x and y are term index pairs from an arbitrary term index combination that subsets the lower or upper triangle of tcm, e.g. 'preceding'.

  The tcm should represent term co-occurrences within a boolean sliding window of size 10 (internally probabilities are used) in an external reference corpus and, therefore, is an extrinsic metric in the standard use case.

  This metric is similar to the UCI metric, however, with a smaller smoothing constant by default and using the mean for aggregation instead of the sum.

• 'mean_npmi' Similar (in terms of all parameter settings, etc.) to 'mean_pmi' metric but using the normalized pmi instead, which is calculated as (log2((tcm[x,y]/n_doc_tcm) + smooth) - log2(tcm[x,x]/n_doc_tcm) - log2(tcm[y,y]/n_doc_tcm)) / -log2((tcm[x,y]/n_doc_tcm) + smooth),

  This metric may perform better than the simpler pmi metric.

• 'mean_difference' The difference is calculated as tcm[x,y]/tcm[x,x] - (tcm[y,y]/n_tcm_windows), where x and y are term index pairs from a 'preceding' term index combination. Given the indices c(1,2,3), combinations are list(c(1,2), c(1,3), c(2,3)).

  The tcm should represent the boolean term co-occurrence (internally probabilities are used) in the original documents and, therefore, is an intrinsic metric in the standard use case.

• 'mean_npmi_cosim' First, the npmi of an individual top word with each of the top words is calculated as in 'mean_npmi'. This result in a vector of npmi values for each top word. On this basis, the cosine similarity between each pair of vectors is calculated.

  The tcm should represent term co-occurrences within a boolean sliding window of size 5 (internally probabilities are used) in an external reference corpus and, therefore, is an extrinsic metric in the standard use case.

• 'mean_npmi_cosim2' First, a vector of npmi values for each top word is calculated as in 'mean_npmi_cosim'. On this basis, the cosine similarity between each vector and the sum of all vectors is calculated (instead of the similarity between each pair).

  The tcm should represent term co-occurrences within a boolean sliding window of size 110 (internally probabilities are used) in an external reference corpus and, therefore, is an extrinsic metric in the standard use case.