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Derivation of the coherence length of light waves

Here, we will derive the relationship between the coherence length of the light wave and the light source spectrum width. The coherence length is a factor that determines the depth resolution in OCT (Optical Coherence Tomography). However, it is important to note that in OCT, the depth resolution is half the value of the coherence length because the light reflected from the measurement object is detected.

This shows the coherence length. The coherence length is the optical path length difference that can maintain the interference fringes, and is defined as the half-width. The shorter the wavelength width of the light source (the higher the monochromaticity), the longer the coherence length.

Step 1: Derive the spectral width in k-space

The spectral characteristics of a light wave are expressed as a normal distribution in k-space (wave number space) as follows:

The wavenumber distribution is shown as a normal distribution.

where I0 is the total intensity of the light wave, kp is the wavenumber at which S(k) peaks, and w is the standard deviation of the spectrum in k-space.

The diagram of S(k) is as follows:

The wavenumber distribution is shown as a normal distribution.

Here, the following definitions are made:

・The wave number at which S(k) is half the peak is kH.

・The half-width of the spectrum in k-space is Δk.

In this case,

Derivation of coherence length_051.png

Therefore,

is the full width at half maximum (FWHM) of the wavenumber.

Step 2: Derive the spectral width in λ space

In ①, k=2π / λ, k p = 2π / λ p Then, when converted to λ space (wavelength space),

S(λ) can be illustrated as follows:

This is a graph showing the wavenumber distribution as a normal distribution, converted into wavelength.

Here, the following definitions are made:

・λ H± is the wavelength when S(λ) is half the peak

・Δλ is the full width at half maximum of S(λ)

In this case,

Derivation of coherence length_053.png

Therefore,

This is the result of calculating the wavelength half width (FWHM).

Step 3: Derive the coherence length in real space

Let E(x) be the distribution of light waves in real space (x space).

Furthermore, the position where the amplitude reaches its peak is denoted by x p , and the phase at the position x p is denoted by −δ.

In this case,

This is the equation that shows the spatial distribution of light waves.

Here, the following is self-evident.

Therefore,

Derivation of coherence length_030.png

Here, the following is set:

Derivation of coherence length_031.png

From the interchange of integrals and derivatives theorem, we can say the following.

Derivation of coherence length_032.png

Solving this equation gives us the following:

Derivation of coherence length_042.png
コヒーレンス長の導出_055.png

Where:

Derivation of coherence length_035.png

Therefore,

Derivation of coherence length_036.png

Therefore,

Derivation of coherence length_037.png

E(x) can be illustrated as follows:

This shows the spatial distribution of light waves as seen in OCT etc.

Here, the following definitions are made:

・E A± (x) is the amplitude of E(x)

・x H± is the position when E A± (x) is halfway to the peak

・Δx is the full width at half maximum of E A± (x)

 

In this case,

Derivation of coherence length_048.png

Therefore,

Step 4: Derive the relationship between Δx and Δλ

From ② and ③, we finally obtain the following relationship.

Coherence length calculation formula (relationship between spatial resolution and wavelength width)
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