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.
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:
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:
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,
Therefore,
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:
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,
Therefore,
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,
Here, the following is obvious (from the properties of odd functions).
Therefore,
Here, the following is set:
From the interchange of integrals and derivatives theorem [1], we can say the following.
Solving this equation gives us the following:
Where:
Therefore,
Therefore,
E(x) can be illustrated as follows:
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,
Therefore,
Step 4: Derive the relationship between Δx and Δλ
From ② and ③, we finally obtain the following relationship.
The relationship between Δx and Δλ can be illustrated as follows:
Application: Relationship between OCT depth resolution and coherence length
As shown in the figure below, when observing reflected light from an observation surface in a material with a refractive index n, consider the relationship between the depth resolution Δd and the coherence length Δx.
The coherence length Δx calculated so far is the distance in air, so the distance in a material must be considered in terms of the air-equivalent length. Also, when observing reflected light, the optical path length is twice the actual distance, since it is the round-trip distance. Taking this into consideration, the relationship between coherence length and depth resolution is as follows:
