Talbot bands in the theory and practice of optical coherence tomography A. Gh. Podoleanu Applied Optics Group, School of Physical Sciences, University of Kent, CT2 7NH, Canterbury, UK
Presentation is based on results obtained during supervision of research performed by: PhD students Daniel Woods and Michael Hughes and postdoctoral researcher Adrian Bradu
Content Justification of study Conventional channelled spectrum OCT or Fourier domain-oct, or camera based Talbot bands Comments on their value for FD-OCT
Broadband optical source CHANNELLED SPECTRUM [OR FOURIER DOMAIN (FD)] OCT Reference Mirror L1 BS RB X OB Object Mirror Z CCD Diffraction Grating INTERFEROMETER+ SPECTROMETER L2 Electrical Spectrum Analyzer
DISADVANTAGES OF SPECTRAL DOMAIN OCT (i) Visibility decay with depth; (iii) Mirror terms (same spectrum modulation for positive and negative OPD values); (ii) Dynamic focus not possible.
Mirror terms in OCT equivalent to OPD sign ambiguity in sensing FD-OCT image from a real-valued spectral interferogram consists of two overlapped images that are symmetrical with respect to the zero plane (zero-phase delay) of the interferometer; The zero plane must be positioned outside the imaged sample which reduces the imaging depth range to less than half.
So far, the decay of sensitivity with depth was considered a consequence of the limited spectral resolution of the spectrometer We will show here that this has a more fundamental origin We will also show that the curve of sensitivity with depth and the mirror terms are inter-related
Phase-shifting methods allow acquisition of the complex spectral interferogram to be obtained in a multiframe sequence. However, in practice the complex conjugate rejection ratio is limited by the accuracy of the phase steps and also by the mechanical stability of the interferometer and the sample during the acquisition time of the frame sequence.
In Y. Yasuno, S. Makita, T. Endo, G. Aoki, H. Sumimura, M. Itoh, and T. Yatagai, "One-shotphase-shifting Fourier domain optical coherence tomography by reference wavefront tilting, two phase shifted spectra are recorded simultaneously on different lines of an area detector. requires an area detector which reduces speed performance and the light efficiency is critical.
Using a 3x3 splitter
3x3 fiber couplers The detection of the complex interferogram using 3x3 fiber couplers as phase-shifting elements allows simultaneous detection of the real and imaginary components of the spectral interferogram. This requires two separate detectors for acquiring the quadrature components. Slight misalignments in matching the spectrometers (in the case of broadband FD-OCT) and uneven wavelengthdependent splitting ratios in the coupler limit the suppression of the complex conjugate artifacts.
Opt. Express, 14/ 4, (2006), 1487-1496 Beating frequency generated by two AO frequency shifters Beating signal frequency is the same for all wavelengths allowing for achromatic complex reconstruction Complex system, calculations required
R. A. Leitgeb, C. K. Hitzenberger, and A. F. Fercher, Phase shifting algorithm to achieve high speed long depth range probing by frequency domain optical coherence tomography, Opt. Lett. 28, 2201-2003 (2003). E. Götzinger, M. Pircher, R. Leitgeb, and C. Hitzenberger, "High speed full range complex spectral domain optical coherence tomography," Opt. Express 13, 583-594 (2005),
Cancellation methods All previous methods to reduce the mirror terms are cancellation methods They require several measurements which are either performed: simultaneously, such as in the case of a 3x3 coupler method, or the heterodyne method using frequency shifters or sequentially, such as in the case of phase shifting.
Using several measurements, the mirror term are reduced or eliminated. Therefore they require stability of the parameters. If the sample moves (the eye), instabilities, errors, mismatches lead to limited attenuation of the mirror term The reported maximum complex conjugate rejection ratio is 20 db 50 db.
TALBOT BANDS Talbot Bands describe a curious effect, [F. Talbot, An experiment on the interference of light, Philos. Mag. 10, 364 (1837)], when a beam from a white light source was analyzed spectrally using a prism. By inserting a glass plate halfway into the beam, dark bands were produced when the plate was inserted so as to intercept a certain side of the beam only. If the plate was inserted halfway into the beam from a direction corresponding to the red end of the dispersed spectrum dark bands (Talbot Bands) were observed. If the plate was inserted from the opposite side of the beam (corresponding to the blue end of the spectrum), no bands were observed
Historic G. B. Airy, The Bakerian Lecture on the theoretical explanation of an apparent new polarity in light, Phil. Trans. R. Soc. London 130, 225-244, (1840), dispelled any curious effects as mere results of interference which enhances the amplitude of some wavelengths and reduces the amplitude of others.
DEMONSTRATION OF TALBOT BANDS SPECTROMETER SLD BEAM TOWARDS THE GRATING CCD LENS SFM GRATING 2011
Optical beam Glass plate
Glass plate Intersecting the Right of the beam no bands are obtained
Glass plate Intersecting the Left of the beam, Talbot Bands are obtained
Difference between Left and Right side of an optical beam Modulation of the spectrum arises when the glass plate intersects a certain side of the beam!?!?!?!?
Properties of Talbot Bands The density of bands is proportional to the thickness of glass and its index of refraction. Talbot bands exhibit a symmetric-triangular relationship between band visibility and plate thickness: A maximum visibility is observed at a thickness dependent on the beam width; Visibility goes to zero near a plate thickness of zero and also as the thickness exceeds twice the value at which the maximum occurs.
VISIBILITY OF TALBOT BAND 1 l c Optimum thickness Maximum thickness OPD t opt 2t opt VISIBILITY RESTRICTED TO ONE SIGN OF OPD ONLY
By diffraction (dispersion), coherence length increases In the Bragg maximum of order n, the difference in path from one grating line to the next is nλ: For an incoming beam exciting N grating lines, the diffracted beam is made of N pulses, delayed in between by nλ. This equates to an increase in the coherence length of the diffracted wave from l c to nnλ. A. Gh. Podoleanu, Unique interpretation of Talbot Bands and Fourier domain white light interferometry Opt. Express, 2007, Vol. 15,. 9867-9876
Right Left Right lc Extended Extended wavetrain wavetrain length length Tilt of the wavefront Left
CL DIFFRACTION GRATING Ω β No glass plate, No overlap, no bands -1 0 A``
CL DIFFRACTION GRATING Ω β Thin plate Some overlap, bands of intensity proportional to the amount of overlap -1 0 A``
CL DIFFRACTION GRATING Ω β Optimum thickness Total overlap, Maximum intensity bands -1 0 A``
CL DIFFRACTION GRATING Ω β Thickness larger than optimum, Less intensity -1 0 A``
CL DIFFRACTION GRATING Ω β So thick, as the wavetrains exhibit no overlap -1 0 A``
CL DIFFRACTION GRATING Ω β OPD =0 No overlap, no bands -1 0 A``
CL DIFFRACTION GRATING Ω β Inserted in the blue side, wavetrains become even more separated, so no bands -1 0 A``
Sides of the beam where Talbot Broadband optical source bands appear CL A Undelayed Beam Delayed Beam DG A` Ω β SL Red -1 1 0 Blue Π A`` Observer eye
OW RW When the object and reference beams overlap, the diffracted wavetrains are totally superposed R L R L 2Nλ
Diffraction grating For OPD = 0 Wavetrains do not superpose L R NO BANDS ARE OBSERVED N λ N λ L R TALBOT BANDS ARE OBTAINED Introducing a glass plate to intersect half of the beam to delay the Left part, brings the two diffracted waves into overlap c t> 0 N λ
BOS Modified Michelson interferometer to produce Talbot bands RM Screen Screen X RB OM Z OB L1 BS RW OW D D max D 0 CCD D min Diffraction grating L2 ESA N N A. Podoleanu, J. Pure and Applied Optics, 6, 413-424, (1997). A. Podoleanu, J. Pure and Applied Optics, 7, 517-536, (1998).
BOS Modified Michelson interferometer to produce Talbot bands RM Screen Screen X RB OM Z OB D max D CCD L1 BS RW OW D 0 D min Diffraction grating L2 ESA N N A. Podoleanu, J. Pure and Applied Optics, 6, 413-424, (1997). A. Podoleanu, J. Pure and Applied Optics, 7, 517-536, (1998).
Spectral interferometer sensitive to both positive and negative OPD Mirror terms or all classical FD-OCT configurations, where the two object and reference beams are superposed in their way toward the spectrometer
Spectral interferometer sensitive to positive OPD only No mirror terms Modified Michelson interferometer D. Woods, A. Podoleanu, Controlling the shape of Talbot bands visibility, Optics Express, 2008, Vol. 16, No. 13 9654-9670.
I sin Mβ sin β 2 0 [ ] = β= bksinθ)/2. I 1 2λ/b λμ/b 0 λ/mb 2λ/b Sinθ WIDTH OF THE DIFFRACTED PEAK The larger the number of grating lines M, the longer the diffracted wave-train length, the larger the axial range according to Talbot Bands Theory The larger M, the narrower the diffracted spot on the CCD
VISIBILITY OF TALBOT BAND 1 l c Optimum thickness Maximum thickness OPD Mλ 2Mλ= 2 λ= 2wavetrain length
APPLICATIONS TO OCT ATTENUATION TO TOTAL ELIMINATION OF MIRROR TERMS, NO CALCULATION, NO MULTIPLE STEPS REQUIRED BETTER PROFILE FOR THE SENSITIVITY DECAY WITH DEPTH, MAXIMUM MOVED TOWARDS THE MIDDLE OF THE RANGE APPLICATIONS IN SENSING RECOGNIZING WHICH PATH LENGTH IS LONGER
Spectrometer resolution Number of grating lines M, period b I = I sin M β sin β 2 2 0 sin c ( α )[ ] Zeroes of the pattern: λ sinθ m = ± [1,2,...(M 1),(M + 1),(M + Mb 2),...]
Spectrometer resolution Number of CCD pixels A 2 ( ξ ) 2 2 2 w ξ sin ( z ) = exp ln 2 ξ 2 ξ = π / 2).( z / z ) Z = λ /(4 λ) ( RD RD w = δλ / λ λ is the range of wavelengths seen by a single pixel δλ is resolution of the spectrometer Z.Hu, Y. Pan, & A.M.Rollins, Analytical model of spectrometer-based two-beam spectral interferometry, Appl. Opt. 46, 8499-8505 (2007).
Experimental results Total attenuation of the mirror term A. Podoleanu, D. Woods, Power-efficient Fourier domain optical coherence tomography setup for selection in the optical path difference sign using Talbot bands, Opt. Lett. 32, 2300-2302 (2007).
DG Spectrometer CCD PC SLD L2 DC BS MO1 TS1 Sxy L1 3 1 2 MO2 Reference Position Sample TS2
Lateral beam displacem ent: Channelled Spectrum Intensity (db) 40 20 0-20 -40-60 Novel method Old method 0m m 2.74m m -2.7 4m m Novel method -80-4 -2 0 2 4 Path difference (mm )
Images
Each diffracted wavefront has a Right (R) and a Left (L) part c t 2Nλ Nλ 0 Nλ 2Nλ Zero overlap Partial overlap Maximum overlap Partial overlap Zero overlap CLASSIC CASE OW R L R L RW RW OW R L R L OW RW R L R L OW R L RW R L R L RW OW R L Screens in place p=1 Talbot bands 2Nλ c t<0 2Nλ c t<0 2Nλ No overlap No overlap Zero overlap Maximum overlap Zero overlap L L L L L OW OW OW OW OW R R R R Nλ R RW RW RW RW RW 2Nλ c t>0 2Nλ c t>0 Screens in place p=-1 Talbot bands c t<0 c t<0 R L RW Nλ Nλ OW c t<0 Nλ R OW L c t<0 RW Nλ A. Gh. Podoleanu, Unique interpretation of Talbot Bands and Fourier domain white light interferometry Opt. Express, 2007, Vol. 15,. 9867-9876 Nλ R RW Nλ Nλ OW L Nλ R c t>0 Nλ OW RW L c t>0 Nλ c t>0 R Zero overlap Maximum overlap Zero overlap No overlap No overlap OW c t>0 RW L Nλ
M. Hughes, A. Podoleanu, Electronics Letters, 45, 2, 182 183 (2009)
Electronics Letters, 45, 2, 182 183 (2009)
REFLECTIVE SPECTROMETER TO AVOID DISPERSION BETWEEN THE TWO LATERALLY SHIFTED BEAMS Bradu A, Podoleanu A. Gh., Attenuation of mirror image and enhancement of the signal-to-noise ratio in a Talbot bands optical coherence tomography system, J. Biomed Optics, 16(7), 076010, July 2011, 076010-1 to 076010-10.
One shot display of sensitivity of the FD-OCT set-up measured as a function of depth Using a tilted piece of paper
Further reading A. Podoleanu, S. Taplin, D. J. Webb, D. A. Jackson, J. Pure and Applied Optics, 6, 413-424, (1997). A. Podoleanu, S. Taplin, D. J. Webb, D. A. Jackson, J. Pure and Applied Optics, 7, 517-536, (1998). odoleanu, D. Woods, Opt. Lett. 32, 2300-2302 (2007). A. Gh. Podoleanu, Opt. Express, 2007, Vol. 15,. 9867-9876 D. Woods, A. Podoleanu, Optics Express, 2008, Vol. 16, No. 13 9654-9670. A. Bradu, A. Podoleanu, J. Biomed Opt, 2011, 16, 7, 076010-1 to 10. Michelson modified interferometer using screens Theoretical model for TB of multimode lasers B scan OCT image from finger with no mirror terms Temporal short pulse description of similarity between TBs and OCT Visibility profile with depth is the correlation of power distribution profile in the two beams Demonstration of sensitivity improvement on images from tissue, with the peak of sensitivity moved inside the tissue
Leverhulme Trust European Research Council, Advanced fellowship to A.Podoleanu ACKNOWLEDGMENTS Engineering and Physical Sciences Research Council Biotechnology and Biological Sciences Research Council New York Eye and Ear Infirmary European Commission, Marie Curie Actions Ophthalmic Technologies Inc., Toronto Ariba New York Pfizer, Sandwich, Kent Superlum Moscow University of Kent