Time Domain Full Field Optical Coherence Tomography Microscopy

Abstract

This chapter mostly describes the main characteristics of the full field OCM (FFOCM) systems with an emphasis on the main differences with other systems described in this book: the use of spatially and temporally incoherent sources and large numerical aperture objectives that leads to a 3D ultra-high spatial resolution. We also show that FFOCM can be associated to other imaging modalities such as fluorescence and elasticity in order to increase its sensitivity and specificity when used as a diagnosis tool. Finally FFOCM is shown to successfully match the requirements of a rigid endoscopic system.

Appendix

1.1 FFOCM: Signals and Noises

(The calculation is performed for a single pixel of the camera and the notations are indicated on the Fig. 25.18)

Fig. 25.18

Representation of the various light levels impinging the various parts of the interferometer

• (a) Negligible incoherent and stray light level (N _inch < αN ₀)

  Let a N ₀ be the number of photoelectrons generated from the reference arm αN ₀ ∼ N _sat.

  Let R N ₀ be the number of photoelectrons generated from the object slice under examination (R <<α).

  Let N _inch be the number of stray light- and incoherent light-induced photoelectrons (N _inch <<N _sat).

  For a π-shifted two-phase detection (+ or −),

  $$ \begin{array}{l}{I}_{+}=\alpha {N}_0+R{N}_0+{N}_{\mathrm{inch}}+2\sqrt{\alpha {N}_0R{N}_0} \cos \varphi \\ {}{I}_{-}=\alpha {N}_0+R{N}_0+{N}_{\mathrm{inch}}-2\sqrt{\alpha {N}_0R{N}_0} \cos \varphi \end{array} $$

  The measured signal is

  $$ \begin{array}{l}S={I}_{+}-{I}_{-}\\ {}\kern0.48em =4\sqrt{\alpha {N}_0R{N}_0 \cos \varphi}\end{array} $$

  We usually take its absolute value:

  $$ \begin{array}{l}\left\langle \left| \cos \varphi \right|\right\rangle =\left\langle \sqrt{{ \cos}^2\varphi}\right\rangle \\ {}\kern2.16em =\left\langle \sqrt{\frac{1+ \cos \left(2\varphi \right)}{2}}\right\rangle \\ {}\kern2.16em =1/\sqrt{2}\end{array} $$

  Please note that this is true for a random scattering sample and not for a mirror sample.

  The signal is then

  $$ \begin{array}{l}S=4\sqrt{\alpha {N}_0R{N}_0/2}\\ {}\kern0.48em =4\sqrt{N_{\mathrm{sat}}R{N}_0/2}\end{array} $$

  The shot noise being

  $B=\sqrt{N_{\mathrm{sat}}}=\sqrt{\alpha {N}_0}$ (the reference being the major signal)

  The signal-to-noise ratio is

  $$ \begin{array}{l}\mathrm{S}/\mathrm{B}=4\sqrt{R{N}_0/2}\\ {}\kern1.2em =4\sqrt{R{N}_{\mathrm{sat}}/2\alpha}\end{array} $$

  The limit of detection (signal = noise) corresponds to $4\sqrt{R_{\min }{N}_{\mathrm{sat}}/2\alpha =1}$, so that

  $$ {R}_{\min }=\frac{\alpha }{8{N}_{\mathrm{sat}}} $$

  Typical numerical value

  Using typical values of α = 0.16 N _sat = 200,000 for silicon cameras, one gets

  R _min = 10⁻⁷ or 70 dB for a two-image acquisition, 1 Mpixels, 75 processed images/s.

  Averaging 100 images takes a few more than 1 s and leads to R_min = 10⁻⁹ or 90 dB.

• (b) Significant incoherent and stray light level (N _inch = or >αN ₀)

  We have then N_sat = αN ₀ + N _inch

  The signal is

  $$ \begin{array}{l}S=4\sqrt{\alpha {N}_0R{N}_0/2}\\ {}\kern0.48em =4\sqrt{\left({N}_{\mathrm{sat}}-{N}_{\mathrm{inch}}\right)R{N}_0/2}\end{array} $$

  The shot noise is $B=\sqrt{N_{\mathrm{sat}}}$

  The signal-to-noise ratio is

  $$ \begin{array}{l}\mathrm{S}/\mathrm{B}=4\sqrt{\frac{\left({N}_{\mathrm{sat}}-{N}_{\mathrm{inch}}\right)R{N}_0}{2{N}_{\mathrm{sat}}}}\\ {}\kern1.60em =4\sqrt{\frac{\alpha {N}_0R{N}_0}{2{N}_{\mathrm{sat}}}}\end{array} $$

  The limit of detection (signal = noise) corresponds to $4\sqrt{\frac{\left({N}_{\mathrm{sat}}-{N}_{\mathrm{inch}}\right){R}_{\min }{N}_0}{2{N}_{\mathrm{sat}}}}=1$

  $$ {R}_{\min }=\frac{1}{8.\left({N}_{\mathrm{sat}}-{N}_{\mathrm{inch}}\right)}\cdot \frac{N_{\mathrm{sat}}}{N_0} $$
  $$ {R}_{\min }=\frac{\alpha {N}_{\mathrm{sat}}}{8.{\left({N}_{\mathrm{sat}}-{N}_{\mathrm{inch}}\right)}^2} $$