1. Center for Biomedical Optics and New Laser Systems Modeling OCT using the extended Huygens-Fresnel principle Peter E. Andersen Optics and Fluid Dynamics Department Risø National Laboratory E-mail peter.andersen@risoe.dk

2. P. E. Andersen - 7/5/2016 Optics and Fluid Dynamics Department RISØ Modeling the OCT geometry  Theoretical analysis – based on the extended Huygens-Fresnel principle, mutual coherence functions, and the ABCD-matrix formalism, – valid for single and multiple scattering regimes simultaneously.  References – L. Thrane et al., J. Opt. Soc. Am A 17, 484-490 (2000). – R. F. Lutomirski and H. T. Yura, Appl. Opt. 10, 1652-1658 (1971). – H. T. Yura and S. G. Hanson, J. Opt. Soc. Am. A 4, 1931-1948 (1987).

3. P. E. Andersen - 7/5/2016 Optics and Fluid Dynamics Department RISØ Modeling the OCT geometry  Assumptions – Gaussian input beams, – strong forward scattering, i.e. negligible backward scattering; »valid for most tissues, – diffuse backscattering from tissue discontinuity, – focal point is located at the tissue discontinuity, – tissue bulk absorption neglected, – polarization effects excluded; »may be included if needed.

4. P. E. Andersen - 7/5/2016 Optics and Fluid Dynamics Department RISØ Sample beam arm

5. P. E. Andersen - 7/5/2016 Optics and Fluid Dynamics Department RISØ  The mean square heterodyne signal current – conversion from optical power to current, – the mutual coherence functions, – vectors are two-dimensional vectors transverse to the optical axis. Detector current (ac)

6. P. E. Andersen - 7/5/2016 Optics and Fluid Dynamics Department RISØ Input optical fields  Input sample and reference fields – optical power, – 1/e intensity radius of input beams, – focal length, –

7. P. E. Andersen - 7/5/2016 Optics and Fluid Dynamics Department RISØ  Reflected sample field in the mixing plane (lens) – is the sample field in the discontinuity plane. Mixing plane (1)

8. P. E. Andersen - 7/5/2016 Optics and Fluid Dynamics Department RISØ Mixing plane (2)  Insert and combine to get the mutual coherence function  Propagation to and from discontinuity is statistically independent

9. P. E. Andersen - 7/5/2016 Optics and Fluid Dynamics Department RISØ  Sample field at discontinuity – assume diffuse backscattering from the tissue discontinuity; – the field at two different points on the discontinuity is uncorrelated, – therefore, we need only to find the intensity of the sample field at the discontinuity (before propagating the field back to the mixing plane). Sample field

10. P. E. Andersen - 7/5/2016 Optics and Fluid Dynamics Department RISØ  The intensity in the discontinuity plane – the ABCD-matrix formalism is used. Focal plane intensity contains the scattering effects

11. P. E. Andersen - 7/5/2016 Optics and Fluid Dynamics Department RISØ Radial focal plane intensity

12. P. E. Andersen - 7/5/2016 Optics and Fluid Dynamics Department RISØ Intensity plot

13. P. E. Andersen - 7/5/2016 Optics and Fluid Dynamics Department RISØ Intensity plot

14. P. E. Andersen - 7/5/2016 Optics and Fluid Dynamics Department RISØ  The Green’s function – trick: propagation through the medium has been separated into homogeneous space propagation multiplied by the scattering term.  The Green’s function for homogeneous space Green’s function

15. P. E. Andersen - 7/5/2016 Optics and Fluid Dynamics Department RISØ Scattering of the medium  The scattering term – :the scattering coefficient, – :the lateral coherence length (see Appendix), – :, – the “shower curtain” effect is included through the lateral coherence length.

16. P. E. Andersen - 7/5/2016 Optics and Fluid Dynamics Department RISØ Appendix: lateral coherence  Calculating the lateral coherence length for the focused sample beam – the integration is taken from discontinuity (s=0) to mixing plane (s=d+z).

17. P. E. Andersen - 7/5/2016 Optics and Fluid Dynamics Department RISØ  The matrix element B  The scattering coefficient Appendix: lateral coherence

18. P. E. Andersen - 7/5/2016 Optics and Fluid Dynamics Department RISØ Appendix: lateral coherence  Solving the integral

19. P. E. Andersen - 7/5/2016 Optics and Fluid Dynamics Department RISØ Appendix: lateral coherence  For large phase variances, the scattering term may be written – where the above integration has been used for identifying the lateral coherence length  o, – it can be compared to the spatial coherence length known from Young’s experiment.

20. P. E. Andersen - 7/5/2016 Optics and Fluid Dynamics Department RISØ  The mean square heterodyne signal current (lens plane) Signal current (ac)

21. P. E. Andersen - 7/5/2016 Optics and Fluid Dynamics Department RISØ Plotting  – 1: diffuse reflectance (shower curtain effect included), – 2: diffuse back- scatter (no shower curtain effect), – 3: specular reflect- ance (no shower curtain effect), – 4: single back- scatter.

22. P. E. Andersen - 7/5/2016 Optics and Fluid Dynamics Department RISØ The shower curtain effect (1)  Increasing the distance yields: – smoother wave front  clearer image, – less intensity  poorer signal.  Examples: – beautiful woman (handsome man) taking a shower, – ocean floor observed from airplane, – optimization of OCT system; »i.e. distance between lens and medium.

23. P. E. Andersen - 7/5/2016 Optics and Fluid Dynamics Department RISØ The shower curtain effect (2)  Inherent effect of the beam propagation – often overlooked for modeling.

24. P. E. Andersen - 7/5/2016 Optics and Fluid Dynamics Department RISØ Experimental results (cuvette) L. Thrane et al., J. Opt. Soc Am. A 17, 484-490 (2000). g  0.927 z = 0.5 mm f  16 mm w 0  125  m

25. P. E. Andersen - 7/5/2016 Optics and Fluid Dynamics Department RISØ Novel OCT model  The analytical model may be used – for system analysis (in combination with noise considerations), – for optimization, – for image improvement; »correcting the reflection for attenuation due to scattering: true-reflection algorithm.

26. P. E. Andersen - 7/5/2016 Optics and Fluid Dynamics Department RISØ Monte Carlo simulations  Monte Carlo code – extension of code by Wang and Jacques; »S. L. Jacques and L. Wang, Chap. 4 in Optical-Thermal Response of Laser-Irradiated Tissue (eds. A. J. Welch and M. J. C. van Gemert), Plenum Press, New York, 1995,  OCT Monte Carlo – includes true sample beam Gaussian focal point spread, – includes diffuse reflection at discontinuity, – detection in conjugate planes, »A. Tycho, T. M. Jørgensen, H. T. Yura, P. E. Andersen, “Derivation of a Monte Carlo method for modeling heterodyne detection in optical coherence tomography systems”, submitted to in Appl. Opt. 2002.

27. P. E. Andersen - 7/5/2016 Optics and Fluid Dynamics Department RISØ MCS: Comparison to EHF

28. P. E. Andersen - 7/5/2016 Optics and Fluid Dynamics Department RISØ Summary of model  The model – general model, – valid for both single and multiple scattering, – includes the shower curtain effect, – easy to apply to other geometries (due to the ease-of-use of the ABCD-matrix formalism).  System performance – multiple scattering light plays an important role for the performance of the OCT system.  Outlook – imaging performance may be improved by correction for the scattering (true-reflection algorithm).

29. P. E. Andersen - 7/5/2016 Optics and Fluid Dynamics Department RISØ Summary  Optical coherence tomography – introduced a theoretical model applicable to OCT systems, – discussed the shower curtain effect, – compares favorably to histopathology (?).  Clinical applications – ophthalmology; »retinal diseases and nerve fiber layer, – dermatology; »skin cancer and skin diseases, – endoscopes.

30. P. E. Andersen - 7/5/2016 Optics and Fluid Dynamics Department RISØ OCT: suggested reading  Optical coherence tomography (OCT) »D. Huang et al., Science 254, 1178 (1991). »E. A. Swanson et al., NATO ASI Series vol. 325, Kluwer Acad. Publishers, Dordrecht, 1996. »J. M. Schmitt, IEEE Selected Topics in Quantum Electronics 5, 1205 (1999). »L. Thrane, H. T. Yura, and P. E. Andersen, J. Opt. Soc. Am. A 17, 484-490 (2000).  Extended Huygens-Fresnel principle »R. F. Lutomirski and H. T. Yura, Appl. Opt. 10, 1652 (1971). »H. T. Yura, Optica Acta 26, 627 (1979).