Numerical dispersion compensation for Partial Coherence Interferometry and Optical Coherence Tomography

A. F. Fercher; C. K. Hitzenberger; M. Sticker; R. Zawadzki; B. Karamata; T. Lasser

doi:10.1364/OE.9.000610

Optics Express
Vol. 9,
Issue 12,
pp. 610-615
(2001)
•https://doi.org/10.1364/OE.9.000610

Numerical dispersion compensation for Partial Coherence Interferometry and Optical Coherence Tomography

A. F. Fercher, C. K. Hitzenberger, M. Sticker, R. Zawadzki, B. Karamata, and T. Lasser

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A. F. Fercher, C. K. Hitzenberger, M. Sticker, R. Zawadzki, B. Karamata, and T. Lasser, "Numerical dispersion compensation for Partial Coherence Interferometry and Optical Coherence Tomography," Opt. Express 9, 610-615 (2001)

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Abstract

Dispersive samples introduce a wavelength dependent phase distortion to the probe beam. This leads to a noticeable loss of depth resolution in high resolution OCT using broadband light sources. The standard technique to avoid this consequence is to balance the dispersion of the sample by arranging a dispersive material in the reference arm. However, the impact of dispersion is depth dependent. A corresponding depth dependent dispersion balancing technique is diffcult to implement. Here we present a numerical dispersion compensation technique for Partial Coherence Interferometry (PCI) and Optical Coherence Tomography (OCT) based on numerical correlation of the depth scan signal with a depth variant kernel. It can be used a posteriori and provides depth dependent dispersion compensation. Examples of dispersion compensated depth scan signals obtained from microscope cover glasses are presented.

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Fig. 1.

Fig. 1. Resolution improvement obtained by dispersion compensation using depth-variant numerical correlation. Light of a filtered Hg high pressure lamp (λ ₀≅710 nm; Δλ≅210 nm) is incident from left (thick arrow). The object has been synthesized using experimentally obtained non-dispersed front and dispersed back scattered depth scan signals of a single microscope cover slide. Shown are the (direct) auto- and cross-correlation depth scan signals (red) of the two front and the two back interfaces and the corresponding numerically dispersioncompensated correlation signals (black).

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l_{c} = \frac{4 \ln 2}{π} \frac{λ_{0}^{2}}{Δ λ}

n_{s} = c \frac{d k}{d ω} = n - λ \frac{d n}{d λ} .

\frac{d n_{g}}{d λ} = - λ \frac{d^{2} n}{d λ^{2}} = - λ \frac{2 π v^{3}}{c} \frac{d^{2} k}{d ω^{2}}

f = \sqrt{1 + \frac{4 d^{2}}{τ_{0}^{4}} {(\frac{d^{2} k}{d ω^{2}})}^{2}},

l_{C, dispersed} = l_{C} \cdot f

G (τ) = 〈 E (t) \cdot E (t + τ) 〉 = \int_{- \infty}^{\infty} G (ω) e^{- i ω π} d ω;

G (τ + τ_{0}) = \int_{- \infty}^{\infty} G (ω) e^{- i Φ_{0}} e^{- i ω τ} d ω

G_{Disp} (τ + τ_{0}) = \int_{- \infty}^{\infty} G (ω) e^{- i (Φ_{0} + Φ_{Disp})} e^{- i ω τ} d ω .

Φ_{Disp} (ω - ω_{0}) = k (ω - ω_{0}) z

= k (ω_{0}) z + k^{(1)} (ω_{0}) (ω - ω_{0}) z + k^{(2)} (ω_{0}) \frac{{(ω - ω_{0})}^{2}}{2} z + k^{(3)} (ω_{0}) \frac{{(ω - ω_{0})}^{3}}{6} z + \dots

v_{g} = {[k^{(1)} (ω_{0})]}^{- 1}

E (t) = \int_{- \infty}^{\infty} \hat{E} (ω) e^{- i ω t} d ω,

I_{I T} (τ) = G (τ) = 〈 E (t + τ) E^{*} (t) 〉 = \int_{- \infty}^{\infty} d ω G (ω) e^{- i ω τ}

I_{I T} (τ) = 〈 E (t + τ) {E^{*}}_{Disp} (t) 〉 = \int_{- \infty}^{\infty} d ω G (ω) e^{- i ω τ} e^{- i Φ_{Disp} (ω)} .

I_{Comp} (τ) = F T {I_{local} (ω) \exp (i Φ_{local} (ω)} .

I_{Comp} (τ) = I_{local} (τ) \otimes K_{local} (τ)

K (t) = a e^{- t^{2} ⁄ τ_{0}^{2}} e^{i ω_{0} t} .

n (λ) = \sqrt{1 + B_{1} \frac{λ^{2}}{λ^{2} - C_{1}} + B_{2} \frac{λ^{2}}{λ^{2} - C_{2}} + B_{3} \frac{λ^{2}}{λ^{2} - C_{3}}}

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Equations (18)

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