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Multiplexed 3D imaging using wavelength encoded spectral interferometry: a proof of principle
Optics Communications 222 (2003) 127–136 www.elsevier.com/locate/optcom
Multiplexed 3D imaging using wavelength encoded spectral interferometry: a proof of principle L. Froehly a,*, S. Nieto Martin a, T. Lasser a, C. Depeursinge a, F. Lang b a
LOB/Bio-E/EPFL, Ecublens CH1015 Lausanne, Switzerland b C.H.U.V. CH1015 Lausanne, Switzerland
Received 24 February 2003; received in revised form 19 May 2003; accepted 26 May 2003
Abstract Based on spectral interferometry and wavelength multiplexing, we demonstrate that a combination of these methods could lead to a 3D imaging system. This system has the advantage of a one-dimensional geometry transmission channel and a limited scanning need. This kind of imaging system could, of course, be of a great interest in small size endoscope with 3D imaging capability and a small channel cross-section (less than 1 mm). Theoretical analysis as well as experimental proof of principle is presented in the following paper. 2003 Elsevier Science B.V. All rights reserved. PACS: 42.40.K; 82.80.C; 87.62 Keywords: Spectral interferometry; 3D imaging; Wavelength encoding; Endoscopy
1. Introduction During these last 20 years significant progress have been done in the treatment of benign as well as of malignant, tracheal and bronchial stenoses. The importance of stenoses dimensions knowledge appears to be a key factor to take the correct therapeutic decision and to perform it accurately [1]. This accurate dimensioning should be done in the pre-therapeutic endoscopic step. The importance of 3D imaging endoscopic systems is then
*
Corresponding author. Tel.: +41-21-693-7773; fax: +41-21693-3701. E-mail address: luc.froehly@epfl.ch (L. Froehly).
obvious and lot of efforts are realised today in order to develop chirurgical suitable endoscopes. Some of the requirements for such an endoscope should be, regarding scientific arguments: • True-scale 3D imaging. • High resolution: less than 0.5 mm in all dimensions. • Compatible with rigid and flexible bronchoscopy. The actual state of the art in 3D endoscopy is split into two main categories of methods: the socalled direct methods for which a probe measures the distances to the organ wall and the indirect measurement techniques where a set of 2D slices are performed and post-computed in order to reconstruct a 3D volume. The indirect measurement
0030-4018/03/$ - see front matter 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0030-4018(03)01611-0
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techniques will not be pointed out as they involve Magnetic Resonance Imaging techniques or radiological measurement which are expensive, time consuming and somehow for the radiological one the healthiness is not proved. We will be more interested in the other category of method. One of the more interesting and promising method in this area is Endoscopic Optical Coherence Tomography [2]. As this method is based on the well known OCT concept, advantages are those of OCT: that is a decoupling between axial and transverse resolution, a high sectioning ability depending on the temporal coherence correlation length of the light radiation. Anyway even if an OCT parallelisation is widely studied since a few years it is not directly convertible to the endoscopic purpose, as we need small catheter sizes (in the millimetre range). This means that a 2D or 3D scanning is still necessary with all the non-convenience it supposes for a rapid and precise in vivo imaging technique. Recently an endoscopic set up, which overcomes some of the upper-presented problems, has been published [3]. This system is actually directly based on two main concepts that are the wavelength encoding [4,5] and the confocal microscopy. The mixing of the two techniques allows a line transmission rather than a single point. This will lead to a reduction of the scanning from 3 dimensions down to 2. A recent patent application claims [6] a heterodyne detection for an increased s=n ratio purpose. The patent also suggests the possibility to use group or phase delay as a tool enabling 3D measurements with a limitation of the scanning needs, but without any more details. The 3D imaging technique we will describe here has similarities with the one described by Tearney [6] but involves quite different concepts. In our set-up we will especially focus on the group delay measurement or the frequency shift in the backscattered signal using spectral interferometry [7] and we will experimentally show that this will lead to the need of only a one dimensional scanning system. The paper will be decomposed as followed: in a first part we will describe the experimental set-up and the second part will then be devoted to the experiment presentation as well as to the reached result. A
qualitative description of the working principles will then follow in a third part. The fourth part will be a discussion about the limitations of the imaging system especially speaking about the lateral and depth resolution that we will see to be tightly connected by a Fourier relationship. Finally the last section will be devoted to the mathematical analysis in terms of Ôtemporal impulse responseÕ and Ôspectral responseÕ which are, of course, Fourier transformed of each others.
2. Experimental set-up The imaging set-up basic elements are shown in Fig. 1. S is a white light, or at least, broadband source. The L1 , L2 lens system expands the nearly collimated beam coming from S. After transmission and reflection at beamsplitter (B.S.) the two beams will be recombined: one of them is coming from the interferometer reference arm of optical path length 2l0 . The other one comes from the Ôendoscopic armÕ 1 of optical path length 2l1 . This latter arm includes a spectroscopic probe consisting of a low resolution diffraction grating G1 (or a dispersive prism as well), located in the front focal plane of a lens L3 , the object (O) to be imaged occupying the back focal region of the lens. This special configuration is necessary for two reasons: first we realise a telecentric system what gives a reference optical path surface which is a plane rather than a sphere section. The second reason is that of course we need each ÔwavelengthÕ to be focused on the sample so that the object need to be at the L3 image focus. We should notice that, in the reference arm, a lens L4 , identical to L3 , is necessary to balance the dispersion between the two interferometer arms. In the common path before spectral analysis, as we will see in Section 4, a
1 In the single fiber endoscopic configuration, an optical single or multimode fiber of arbitrary length will take place between (B.S.) and G1 carrying the signal. For a symmetry purpose a similar fiber should be inserted in the reference arm. Furthermore a rotating Wollaston–Dove prism may be inserted between L3 and (O) for image scanning purposes. Nevertheless these improvements will not be considered in the following Ôproof of principleÕ presentation.
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Fig. 1. Experimental set-up for encoded imaging. L1 to L7 are lenses; with L1 and L2 used as a beam expander whereas L5 and L7 coupled with the pinhole (P) are used for a filtering purpose. (B.S.) is a 50/50 cube beam splitter. G1 and G2 are gratings with G1 the encoding grating with a low resolution and G2 the high resolution grating. (O) is the scattering sample under test.
spatial Fourier filtering is necessary to select only those components from the backscattered light that are significant for the imaging purpose. This will be done with a Kepler telescope constituted of lenses L5 and L6 in the focal plane of which is placed a pinhole (P). Then, after beam recombination and the filtering, a high resolution spectroscope performs the spectral analysis of the spectral interference field; this spectroscope consists of a powerful diffraction grating G2 followed by an aberration-free, large sized collimating lens L7 . A CCD image detector (D) records and stores the spectral interference pattern. It will be shown in the next that we have to analyse locally the spectral interference pattern so that a further Short Time Fourier Transform (S.T.F.T), or a wavelet analysis of it is a well-adapted method for this purpose. This analysis will directly yields to an image (O0 ) of the object (O) with its geometrical shape dðxÞ. The transverse and longitudinal resolutions Dx and Dd of the image will be discussed in a further section.
3. Experiment and result An elementary optical structure was manufactured as test object (O) (Fig. 2(a)): a piece of strongly backscattering tape was covered on one
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Fig. 2. Experimental results (a) is the test object constituted of a piece of back-scattering sample partly covered with a glass plate of refractive index n. (b) The resulting spectrum recorded on the detector (D). (c) The wavelet transform of previous graph. The broken line is a smoothed average of the wavelet transform with, in this case, the help of the Ôa prioriÕ knowledge of the profile. Further development will correct this ÔartefactÕ. The obtained experimental step gives anyway the correct depth value of the glass plate within the resolution limit range. This could allow us to conclude to the imaging system validity.
half of its length by a parallel glass plate; tape length was 1 cm, glass plate thickness was 1 mm. The light source (S) was a 100-nm bandwidth femtosecond TiSa laser. The approximative illuminated groove numbers of diffraction gratings (G1 ) and (G2 ) were, respectively, p1 2400 and p2 36,000. In such conditions the recorded spectral pattern in detector plane (D) looked like Fig. 2(b), whereas its Ôwindowed Fourier analysisÕ or, here, Wavelet transform, reconstructed the object reflectivity and geometrical shape as seen in Fig. 2(c). This figure shows: • the expected speckle structure of image light fields backscattered from object (O); • the effect of the average rectilinear shape of the tape along the transverse x coordinate; • the effect of the glass plate thickness, which breaks the average straight line, shifting it by an amount that was checked to be proportional to 2ne along the longitudinal axis dðxÞ; • the longitudinal resolution limit dd of this image, which was also, fortunately, in the order of 2ne in these experimental conditions. This resolution limit and ways to improve it will be discussed in an other section. Although image quality clearly deviated here from the required level for medical or biological endoscopy, this is, to our knowledge, the first
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information transmission on both transverse and longitudinal features of an optical scattering structure through a one-dimensional channel geometry which could be directly converted to a fibered endoscopic set-up. 3.1. Remark on the signal-to-noise ratio problem The light power required on the sample to get an image with a sufficient signal-to-noise (s=n) ratio has not yet been investigated in this paper as the main idea was to show and explain this new imaging system principles without being really concerned by this true problem: that is the reason why we have chosen a highly back scattering sample as a test one and in this case as well as with a simple scattering sheet of paper we do not report any sensitivity problem. Of course this s=n study will be, now an important part of the next step of our work. We should notice anyway that this problem will strongly depend on the output optic we will use. Indeed, the output lens L3 will determine the field depth of the measurement and also the collection angle for the backscattered light. This will then be determined by the imaging system characteristics needed for a certain application. We also could imagine to use a highly sensitive CCD camera rather than a classical one as we do now. An other interesting and encouraging point is described in a recent publication [8]. Leigteb et al. demonstrates that spectral OCT systems, or FDOCT, (an other recent way of calling spectral interferometry) could have a higher sensitivity than OCT systems (around 7 dB) especially in situations with low light levels. One reason presented in the paper is the fact that in FDOCT optical energy is collected rather than optical power leading to a different detector use with different noise expression.
4. Working principle Before giving a more accurate mathematical description of the successive signal processing steps, leading to a discussion on various possible improvements in image accuracy and quality as well as on mechanical compactness of the device,
Fig. 3. Sample illumination shape. Here is shown the enlightened line of length a f Dh0 due to the diffraction angle range which depends on the source bandwidth Dk with Dh0 ðN Dk= cosðh0 ÞÞ.
we will now qualitatively explain how this imaging device works. The basic idea is depicted in Fig. 3 and could be described as follows: light diffraction through a low resolution diffraction grating (G1 ) and focusing through the lens L2 illuminates a straight line segment of object (O). This focused spectral pattern has a length a f Dh0 , where Dh0 ðN ðk2 k1 Þ= cosðh0 ÞÞ with N equal to G1 grooves spatial frequency and k1 , k2 are the minimum and maximum wavelengths of illuminating light spectrum (k1 ¼ 1=r1 ; k2 ¼ 1=r2 ). h0 is the grating (G1 ) output diffraction angle and f the lens focal length. Thus each of the various points of the object line receives and scatters a given ÔwavelengthÕ (blue side of the spectrum for lower diffraction angles and red for larger ones). Part of the radiation will then be backscattered through (G1 ). According to the inverse optical path law, those components will give the various colours well-collimated directions, being parallel to that one of the incident beam. After the grating (G1 ), others components will become Ôquasi plane wavesÕ with a diffraction angle different from zero so that they will be focused out of the filtering pinhole (P) (Fig. 1) and not contribute to the image viewed by the detector. 2 Due
2
We may notice that in the endoscopic fibred set-up the filtering is not necessary as it could be automatically performed by the entrance of a single mode fiber.
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to the local reflectivity rðxÞ and geometrical shape dðxÞ of (O), each ÔcolourÕ carries an x dependent attenuation rðxÞ and temporal delay dðxÞ=c. Reflection on beamsplitter B.S. directs this parallel beam towards G2 , which performs a high-resolution spectral analysis. Spectral display of this beam on photodetector (D) reproduces the reflectivity distribution rðxÞ along coordinate k, with a given scaling factor: this corresponds to the Ôimage spectrogramÕ published by Friesem et al. [9], Lacourt et al. [4], Bartelt [5] earlier. Nevertheless the reference beam introduces new features in this Ôimage spectrogramÕ, giving it an interference structure. Indeed each colour did cross the same optical path 2l0 during its round trip between beamsplitter (B.S.) and mirror (M), whereas its optical path varied by an amount dðkÞ, as k exhibits a quasi-linear variation with coordinate x in the back focal volume of lens (L2 ). Such a dependence of the optical path difference between the interferometer arms, with the radiation wavelength will have the following consequence on the spectrum detected by (D): this spectrum will be modulated by a nearly sinusoidal interference pattern (Ôchannelled spectrumÕ) of local periodicity dk ¼ 1 1 ðk2 =2Þðl0 dðkÞ l1 Þ or dr ¼ 12ðl0 dðkÞ l1 Þ according to a classical property of two-beams white-light interferometry. Therefore the detected intensity distribution contains informations on both reflectivity rðxÞ and geometrical shape dðxÞ of the illuminated part of the object (O). In next section, a more detailed mathematical analysis will show how these informations will be, respectively, encoded as contrast and spacing of the spectral interference pattern: recovering rðxÞ and dðxÞ from these interferences will then be achieved by performing a local frequency analysis in successive parts of the channel spectrum, which lengths are much smaller than ÔaÕ, of the whole spectrum; the use of wavelet transformation is, of course, a suitable way to perform the information recovering. Our choice was to use Morlet wavelet as the mother wavelet in reason of the good correlation between this wavelet and the local channel spectra shape. Clearly, transmission of reflectivity and geometrical shape of a three dimensional structure instead of a two-dimensional one, considering this first presented set-up, will require the
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introduction of a scanning component inside this imaging channel. The mechanical scanning could be replaced by a more elaborated transmission line consisting of a one-dimensional array of identical optical fibres, in place of the above-considered single fibre. This kind of set-up has been previously described by Lacourt et al. [4] for two dimensional images transmission.
5. Image resolution We will now discuss the resolution limit problem for the presented system. For 3D imaging system the lateral as well as the transverse resolution have to be considered. 5.1. Transverse resolution (Fig. 4) DlT The N number of independent samples resolved by the spectral encoding stage (G1 , L2 ) in the object (O) may be calculated in a straightforward way: N is just equal to the number of spectral samples resolved by the spectrometer. As the whole light source bandwidth is Dr ¼ ðr2 r1 Þ, N will then be given by N ¼ ðDr=dr1 Þ, where dr1 denotes the spectrometer resolving power. In the ultimate case
Fig. 4. Resolution limits. On the upper part of the Figure is shown the transverse resolution, which is just equivalent to the grating G1 resolution limit. The down part explains the longitudinal resolution limit: the G2 grating must have a resolution much higher than G1 in order to resolve the local channelled spectra of each dr1 channel.
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where the spectrometer works at its Ôintrinsic resolving powerÕ, that is under single spatial mode illumination (single mode fiber channel or aberration-free collimated beam), with aberration-free 1 collimating optics L2 , this limit is dr1 ðk1 kp1 Þ where k denotes the average optical wavelength, k1 being the working diffraction order of (G1 ), and p1 the illuminated (G1 ) groove number. In our experiment these parameters take the following values k ¼ 800 nm, as k2 750 nm, k1 850 nm hence Dr ’ 0:16 lm1 , k1 ¼ 1, p1 3000 (groove spatial frequency ¼ 600 mm1 illuminated on approximately 5 mm) yielding 1 dr1 ¼ ðk1 kp1 Þ 0:4 mm1 and N 160=0:4 ¼ 400 resolved spectral samples inside the illuminating light bandwidth: this also will be about the number of independent pixels across the transmitted image. The whole image size being approximately 1 cm, the transverse image resolution DlT will be about 3 102 mm. 5.2. Longitudinal resolution DlL The image reconstruction accuracy along the longitudinal coordinate will be directly proportional to each transverse sample (or pixel) spectral bandwidth dr1 of this image; indeed measurements of optical path differences between the two interferometer arms through channelled spectrum metrology will have their precision ruled by the linewidth of each elementary channelled spectrum modulating each image pixel, that is just dr1 . The resolution asked to grating G2 is to exhibit enough large angular dispersion and spectral resolution dr2 for a satisfactory imaging of these channelled spectra (C.S.1, C.S.2, C.S.3) as shown in Fig. 4. In our experiment, the longitudinal resolution DlL 1 was then close to ð0:4Þ ¼ 2:5 mm, what corresponds rather well to the observed uncertainty (2 mm) on the glass plate optical thickness. Improving this optical path measurement accuracy is quite possible at the cost of lower transverse resolution: that is a larger dr1 , i.e., less transversally resolved samples within a given spectral bandwidth Dr of the light source. Of course, the broader this bandwidth, the larger the number of imaged samples will be. Nevertheless an Ôuncertainty relationshipÕ will always connect the number
N ¼ ðDr=dr1 Þ of transverse image samples to the optical depth uncertainty ð1=dr1 Þ ¼ dd, ðN =ddÞ ¼ Dr being a constant for a given light source or in terms of DlL , DlT it will lead to N DlT DlL . For instance white light illumination (Dr 1000 mm1 ) and 0.1 mm longitudinal resolution are compatible with transmission of N 100 pixels along the transverse coordinate x (e.g., 104 pixels of a two dimensional scene, when some scanning device or fiber array will be used in combination with the spectroscopic channel). 5.3. Remark about the measurement range We will briefly discuss here the rather straightforward connection between the resolution problem and the measurement range in both dimensions. As it has been shown in Section 5.1 N, the number of spectral samples resolved by the spectrometer, is also the number of independent spectral channels in the image and also the number of laterally resolved points. Each of these channels also encodes the local depth. This leads to the fact that there are as many laterally resolved points than longitudinal ones. Of course it remains anyway that two points will not be depth resolved if the depth difference between them is smaller than the depth resolving power ð1=dr1 Þ ¼ dd.
6. Mathematical analysis The above discussions about working principles were given in spectral terms only, as encoding parts of the considered set-up are usual spectroscopes and two beam interferometer. This image transmission channel may be completely characterised by its spectral responses at each of the detector pixels. The spectral response at each pixel is Fourier Transformed of the Temporal Impulse Response. Due to linearity and time-invariance, of the imaging channel with respect to optical field amplitude the impulse response of a sequence of components (or optical paths) is the convolution of the individual impulse responses of each component (or optical paths). The impulse response of the reference arm, just after transmission through beamsplitter (B.S.), is
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2l0 h0 ðtÞ ¼ d t ; c
" ð6:1Þ
where d is now, denoting the Dirac distribution, whereas this letter was representing differential increments of spectral or geometric variables in previous sections; but it will lead to no confusion in the next. The impulse response corresponding to one round trip of light between (B.S.) and input of diffraction grating (G1 ) over length 2AB is 2AB hðABÞ ðtÞ ¼ d t : ð6:2Þ c The impulse response corresponding to path (BC) between the input plane of (G1 ) and a given point C of object (O) results from the convolution of: BC hðBCÞ ðtÞ ¼ d t ð6:3Þ c with the temporal impulse response hG1 ðt; h0 Þ of (G1 ) along the h0 direction of diffraction. In Fig. 5 we recall and show the main features of the temporal impulse response in the case of an excitation with the as shortest as possible white light optical pulse. Under a uniform illumination by a Dirac impulse (G1 ) would transmit a Ôtemporal impulse responseÕ written as:
hG1 ðt; h0 Þ ¼
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!
rect
t ct I 0 T ðh Þ kðh0 Þ
!# ð6:4Þ
the constant multiplying factor being omitted, where the wavetrain length T ðh0 Þ is the optical 0 delay sþs ¼ p1 gðsin h þ sin h0 Þ between longer and c shorter diffracted rays, with g being the (G1 ) grating period, and kðh0 Þ ¼ gðsin h þ sin h0 Þ is the periodicity of the pulse train scattered from each grating groove. After backscattering of light by object (O) at point C with local reflectivity rðh0 Þ and back transmission through grating (G1 ), the impulse response at point B may be written as hB ðt; h0 Þ ¼ rðh0 Þ hðBCÞ ðtÞ hðBCÞ ðtÞ h i ð6:5Þ
hG1 ðt; h0 Þ hG1 ðt; h0 Þ : Therefore the temporal impulse response hA ðt; h0 Þ of the whole spectroscopic imaging system including the interferometer set-up, when considered at the recombination point A on beam splitter (B.S.) takes the following form: h i hA ðt; h0 Þ ¼ hB ðt; h0 Þ hðABÞ ðtÞ þ Fh0 ðtÞ; ð6:6Þ where F is a photometric factor depending on a filter which allows interferometer amplitude balancing between two arms. When illuminated by a broadband collimated 2 beam, with a power spectrum jH0 ðmÞj , the device 2 0 transmits a filtered spectrum jHA0 ðm; h Þj at its re2 2 0 combination point A: jHA0 ðm; h Þj ¼ jH0 ðmÞj 0 2 0 jHA ðm; h Þj , where HA ðm; h Þ is the impulse response hA ðt; h0 Þ Fourier Transform. Hence the complete expression of filtered spectral amplitude (" X HA0 ðt; h0 Þ ¼ sinc p m nm0 ðh0 Þ n
0
T ðh Þ
2
# exp ð j2ptsÞrðh0 Þ )
þ F expð j2pts0 Þ H0 ðtÞ
Fig. 5. Grating temporal impulse response calculation. Here is shown the temporal impulse response of a grating as depending on geometrical parameters (angle of incidence and enlightened area) as well as on grating parameters (grooves number per mm).
ð6:7Þ
where m0 ðh0 Þ ¼ ðc=kðh0 ÞÞ, s ¼ 2ðAB þ BCÞ=c ¼ 2ðl1 þ dðh0 Þ=c and s0 ¼ 2l0 =c, where sinc function denotes the cardinal sine function sin cðxÞ ¼ sinx x and the optical path 2ðAB þ BCÞ being noted
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2ðl1 þ dðxÞÞ or 2ðl1 þ dðh0 ÞÞ as there is a correspondence between h0 and x through the relation tan h0 ¼ fx (f is the L2 focal length) and with dðh0 Þ being the Ôoptical depthÕ of object (O). Let us now consider the expression (6.7). We will briefly show in the following that this expression could be reduced to the next one: 2 1 sin c p m m0 ðh0 Þ T ðh0 Þ HA0 ðt; h0 Þ ¼ 2j expð j2ptsÞrðh0 Þ þ F expð j2pts0 Þ jH0 ðtÞj;
ð6:8Þ
where • only positive frequencies are considered • we end up with n ¼ 1 as other terms will be filtered out by the band-limited input pulse spectrum. Indeed, as shown in Fig. 6, the H0 ðmÞ term is the Fourier transform of the input optical pulse eðtÞ. As we are dealing with an optical pulse it is modulated with a sine term of frequency mp . Its Spectrum will then exhibit a positive part as well as a negative one due to the convolution by the Sine Fourier Transform given by ð1=2jÞdðv mp Þ dðv þ mp ÞÞ. Furthermore, it is interesting to compare the recurrence length of the frequency response, given by m0 ðh0 Þ ¼ ðc=kðh0 ÞÞ, with the
spectrum width. As an example we consider the case of a grating with a frequency of N ¼ 600 grooves per mm what gives a groove periodicity of g ¼ 2 lm. The maximum recurrence length in this case is when h0 ¼ 90. The maximum pulse train period will then be t¼
c 3 108 ¼ 1014 Hz: g 2 106
Our TiSa laser source has a bandwidth of about 100 nm at 800 nm. The corresponding frequency bandwidth is Dm ¼
cDk ð3 108 Þð100 109 Þ ¼ ¼ 2 108 Hz: 800 109 k2
We clearly see that only the first order sinc (for n ¼ 1 and n ¼ 1) will be within the spectrum bandwidth, other orders could then be neglected. Finally it is clear that no information loss will result if only positive frequencies are considered and the 1=2j term could also be omitted as a simple constant multiplying factor of the whole expression. The power spectrum of backscattered light from object point C will then becomes in a quite classical way: 2 0 2 HA ðt; h0 Þ ¼ jH0 ðtÞj IðtÞ !# " l0 l1 dðh0 Þ 1 þ vðtÞ cos 4pt c ð6:9Þ with 14 0 2 sinpT ðh0 Þ m t0 ðh0 Þ A þ F 2; IðtÞ ¼ rðh0 Þ @ pT ðh0 Þ m t0 ðh0 Þ ð6:10Þ 12 , sin 2pT ðh0 Þ m t0 ðh0 Þ 0 @ vðtÞ ¼ 2Frðh Þ A IðtÞ 2pT ðh0 Þ m t0 h0 0
ð6:11Þ Fig. 6. Simplification of HA0 ðt; h0 Þ. The reason of the HA0 ðt; h0 Þ expression simplification is shown step by step as a consequence of the Fourier components products in comparison with their frequency bandwidth.
2 jHA0 ðt; h0 Þj
is shown in Fig. 7 The A graph of 2 whole power spectrum jHA00 ðtÞj received on the second, high resolution spectroscope (G2 , L7 ) is nothing but the intensity superposition of all the
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Fig. 7. The shape of jHA0 ðt; h0 Þj2 in the case of a single point illumination. The registered spectrum in the case of a single illumination is depicted and the link between the shape and the calculated expression jHA0 ðt; h0 Þj2 is shown.
components jHA0 ðt; h0 Þj2 backscattered by all the illuminated object parts: Z 2 00 2 0 0 H ðtÞ ¼ H t; h dh0 : A A ðDh0 Þ
Graphs of such a superposition will be seen in Fig. 8 for different samples simulations. In Fig. 8(a) could be seen on the top (1) the obtained spectrum for a flat sample with dðh0 Þ ¼ constant ¼ 0:8 mm. The down graph (2) is the wavelet transform of
previous graph, which clearly shows an average constant shape around 0.8 mm. In Fig. 8(b) the sample is now tilted angularly with a dðh0 Þ (or dðkÞ as there is a relation between both variables) following a Ôlinear lawÕ dðkÞ ¼ 40 102 k 2:5 103 with dðkÞ in mm for k in m. This leads to dðh0 Þ varying between 0.5 and 0.9 mm as clearly shown on the wavelet transform (Fig. 8(b) (4)) of the graph presented in Fig. 8(b) (3). For each of this simulation 10 points have been considered for a
Fig. 8. Simulation results. (a) The result of the computed jHA00 ðtÞj2 , which is the integration of jHA0 ðt; h0 Þj2 over 10 points in the case of a ÔflatÕ scattering sample at a constant distance of about 0.8 mm. The upper graph is the spectrum and the down graph is the wavelet transform of it, showing a constant average line around 0.8 mm. (b) The result of the computed in the case of a tilted scattering sample with a distance following the law dðkÞ ¼ 40 102 k 2:5 103 with dðkÞ in mm if k is in meter. The upper graph is the spectrum and the down graph is the wavelet transform of it, showing the linear average variation between 0.5 and 0.8 mm.
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grating G1 with 600 grooves per mm enlightened on 0.3 mm with a source spectral bandwidth of about 100 nm around 800 nm.
system will also be of interest in order to compare this system with EOCT s=n capabilities. All these next steps will be presented in further publications.
7. Conclusion and outlooks
Acknowledgements
A new imaging set-up has been demonstrated which allows recording and transmission of 3D scattered images through a one-dimensional channel. The first prototype was presented to be a single line acquisition system. This requires, for a whole 3D image recording, a one-dimensional scanning, which could be performed by different ways we also suggested in the paper. Furthermore, the resolution problem has also been analysed through the spectral analysis of the system and a strong dependence between the transverse and the longitudinal resolution was shown to be a Fourier uncertainty relationship. Despite this, the theoretically achievable resolution could be satisfactory for the tracheal and bronchial stenoses dimensioning endoscopy, what requires a lateral and longitudinal resolution of about 0.2 mm. Finally, although the presented set-up was not yet an endoscopic one we have tried to show how this system could be implemented in a, somewhat, straightforward way to the endoscopic purpose. The next steps of our works will be to increase the spectral image contrast by decreasing of the signal DC component. The other point will be to implement a scanning-less 3D imaging system using a different geometry but the same concepts than those presented in this paper. An implementation of an heterodyne detection version of our
First of all we would like to thank the Swiss National Science Foundation for his financial support to this program with subside number 3200062028. We also would express our gratitude to M.Liebling, from the Biomedical Imaging Laboratory (Bio-E, EPFL), about its help in our wavelet understanding. And finally we would like to thank I. Verrier and G. Brun from St. Etienne University (France) for their interest in this project.
References [1] G.T. Simpson, M.S. Strong, G.B. Healy, S.M. Shapsay, C.W. Vaughan, Ann. Otol. Rhinol. Laryngol. 91 (1982) 384. [2] G.J. Tearney, M.E. Brezinski, B.E. Bouma, S.A. Boppart, C. Pitris, J.F. Southern, J.G. Fujimoto, Science 276 (5321) (1997) 2037. [3] G.J. Tearney, M. Shishkov, B.E. Bouma, Opt. Lett. 27 (6) (2002) 412. [4] P. Boni Lacourt, Opt. Commun. 27 (1) (1978) 57. [5] H.O. Bartelt, Opt. Commun. 27 (3) (1978) 365. [6] G.J. Tearney, J. Guillermo, B.E. Bouma, E. Brett, Webb, H. Robert, United States Patent No. 6, 341, 036. [7] C. Froehly, A. Lacourt, J.C. Vienot, Nouv. Rev. Optique 4 (4) (1973) 183. [8] R. Leitgeb, C.K. Hitzenberger, A.F. Fercher, Opt. Exp. 11 (8) (2003) 889. [9] A.A. Friesem, U. Levy, Opt. Lett. 2 (1978) 133.
Multiplexed 3D imaging using wavelength encoded spectral interferometry: a proof of principle L. Froehly a,*, S. Nieto Martin a, T. Lasser a, C. Depeursinge a, F. Lang b a
LOB/Bio-E/EPFL, Ecublens CH1015 Lausanne, Switzerland b C.H.U.V. CH1015 Lausanne, Switzerland
Received 24 February 2003; received in revised form 19 May 2003; accepted 26 May 2003
Abstract Based on spectral interferometry and wavelength multiplexing, we demonstrate that a combination of these methods could lead to a 3D imaging system. This system has the advantage of a one-dimensional geometry transmission channel and a limited scanning need. This kind of imaging system could, of course, be of a great interest in small size endoscope with 3D imaging capability and a small channel cross-section (less than 1 mm). Theoretical analysis as well as experimental proof of principle is presented in the following paper. 2003 Elsevier Science B.V. All rights reserved. PACS: 42.40.K; 82.80.C; 87.62 Keywords: Spectral interferometry; 3D imaging; Wavelength encoding; Endoscopy
1. Introduction During these last 20 years significant progress have been done in the treatment of benign as well as of malignant, tracheal and bronchial stenoses. The importance of stenoses dimensions knowledge appears to be a key factor to take the correct therapeutic decision and to perform it accurately [1]. This accurate dimensioning should be done in the pre-therapeutic endoscopic step. The importance of 3D imaging endoscopic systems is then
*
Corresponding author. Tel.: +41-21-693-7773; fax: +41-21693-3701. E-mail address: luc.froehly@epfl.ch (L. Froehly).
obvious and lot of efforts are realised today in order to develop chirurgical suitable endoscopes. Some of the requirements for such an endoscope should be, regarding scientific arguments: • True-scale 3D imaging. • High resolution: less than 0.5 mm in all dimensions. • Compatible with rigid and flexible bronchoscopy. The actual state of the art in 3D endoscopy is split into two main categories of methods: the socalled direct methods for which a probe measures the distances to the organ wall and the indirect measurement techniques where a set of 2D slices are performed and post-computed in order to reconstruct a 3D volume. The indirect measurement
0030-4018/03/$ - see front matter 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0030-4018(03)01611-0
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techniques will not be pointed out as they involve Magnetic Resonance Imaging techniques or radiological measurement which are expensive, time consuming and somehow for the radiological one the healthiness is not proved. We will be more interested in the other category of method. One of the more interesting and promising method in this area is Endoscopic Optical Coherence Tomography [2]. As this method is based on the well known OCT concept, advantages are those of OCT: that is a decoupling between axial and transverse resolution, a high sectioning ability depending on the temporal coherence correlation length of the light radiation. Anyway even if an OCT parallelisation is widely studied since a few years it is not directly convertible to the endoscopic purpose, as we need small catheter sizes (in the millimetre range). This means that a 2D or 3D scanning is still necessary with all the non-convenience it supposes for a rapid and precise in vivo imaging technique. Recently an endoscopic set up, which overcomes some of the upper-presented problems, has been published [3]. This system is actually directly based on two main concepts that are the wavelength encoding [4,5] and the confocal microscopy. The mixing of the two techniques allows a line transmission rather than a single point. This will lead to a reduction of the scanning from 3 dimensions down to 2. A recent patent application claims [6] a heterodyne detection for an increased s=n ratio purpose. The patent also suggests the possibility to use group or phase delay as a tool enabling 3D measurements with a limitation of the scanning needs, but without any more details. The 3D imaging technique we will describe here has similarities with the one described by Tearney [6] but involves quite different concepts. In our set-up we will especially focus on the group delay measurement or the frequency shift in the backscattered signal using spectral interferometry [7] and we will experimentally show that this will lead to the need of only a one dimensional scanning system. The paper will be decomposed as followed: in a first part we will describe the experimental set-up and the second part will then be devoted to the experiment presentation as well as to the reached result. A
qualitative description of the working principles will then follow in a third part. The fourth part will be a discussion about the limitations of the imaging system especially speaking about the lateral and depth resolution that we will see to be tightly connected by a Fourier relationship. Finally the last section will be devoted to the mathematical analysis in terms of Ôtemporal impulse responseÕ and Ôspectral responseÕ which are, of course, Fourier transformed of each others.
2. Experimental set-up The imaging set-up basic elements are shown in Fig. 1. S is a white light, or at least, broadband source. The L1 , L2 lens system expands the nearly collimated beam coming from S. After transmission and reflection at beamsplitter (B.S.) the two beams will be recombined: one of them is coming from the interferometer reference arm of optical path length 2l0 . The other one comes from the Ôendoscopic armÕ 1 of optical path length 2l1 . This latter arm includes a spectroscopic probe consisting of a low resolution diffraction grating G1 (or a dispersive prism as well), located in the front focal plane of a lens L3 , the object (O) to be imaged occupying the back focal region of the lens. This special configuration is necessary for two reasons: first we realise a telecentric system what gives a reference optical path surface which is a plane rather than a sphere section. The second reason is that of course we need each ÔwavelengthÕ to be focused on the sample so that the object need to be at the L3 image focus. We should notice that, in the reference arm, a lens L4 , identical to L3 , is necessary to balance the dispersion between the two interferometer arms. In the common path before spectral analysis, as we will see in Section 4, a
1 In the single fiber endoscopic configuration, an optical single or multimode fiber of arbitrary length will take place between (B.S.) and G1 carrying the signal. For a symmetry purpose a similar fiber should be inserted in the reference arm. Furthermore a rotating Wollaston–Dove prism may be inserted between L3 and (O) for image scanning purposes. Nevertheless these improvements will not be considered in the following Ôproof of principleÕ presentation.
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Fig. 1. Experimental set-up for encoded imaging. L1 to L7 are lenses; with L1 and L2 used as a beam expander whereas L5 and L7 coupled with the pinhole (P) are used for a filtering purpose. (B.S.) is a 50/50 cube beam splitter. G1 and G2 are gratings with G1 the encoding grating with a low resolution and G2 the high resolution grating. (O) is the scattering sample under test.
spatial Fourier filtering is necessary to select only those components from the backscattered light that are significant for the imaging purpose. This will be done with a Kepler telescope constituted of lenses L5 and L6 in the focal plane of which is placed a pinhole (P). Then, after beam recombination and the filtering, a high resolution spectroscope performs the spectral analysis of the spectral interference field; this spectroscope consists of a powerful diffraction grating G2 followed by an aberration-free, large sized collimating lens L7 . A CCD image detector (D) records and stores the spectral interference pattern. It will be shown in the next that we have to analyse locally the spectral interference pattern so that a further Short Time Fourier Transform (S.T.F.T), or a wavelet analysis of it is a well-adapted method for this purpose. This analysis will directly yields to an image (O0 ) of the object (O) with its geometrical shape dðxÞ. The transverse and longitudinal resolutions Dx and Dd of the image will be discussed in a further section.
3. Experiment and result An elementary optical structure was manufactured as test object (O) (Fig. 2(a)): a piece of strongly backscattering tape was covered on one
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Fig. 2. Experimental results (a) is the test object constituted of a piece of back-scattering sample partly covered with a glass plate of refractive index n. (b) The resulting spectrum recorded on the detector (D). (c) The wavelet transform of previous graph. The broken line is a smoothed average of the wavelet transform with, in this case, the help of the Ôa prioriÕ knowledge of the profile. Further development will correct this ÔartefactÕ. The obtained experimental step gives anyway the correct depth value of the glass plate within the resolution limit range. This could allow us to conclude to the imaging system validity.
half of its length by a parallel glass plate; tape length was 1 cm, glass plate thickness was 1 mm. The light source (S) was a 100-nm bandwidth femtosecond TiSa laser. The approximative illuminated groove numbers of diffraction gratings (G1 ) and (G2 ) were, respectively, p1 2400 and p2 36,000. In such conditions the recorded spectral pattern in detector plane (D) looked like Fig. 2(b), whereas its Ôwindowed Fourier analysisÕ or, here, Wavelet transform, reconstructed the object reflectivity and geometrical shape as seen in Fig. 2(c). This figure shows: • the expected speckle structure of image light fields backscattered from object (O); • the effect of the average rectilinear shape of the tape along the transverse x coordinate; • the effect of the glass plate thickness, which breaks the average straight line, shifting it by an amount that was checked to be proportional to 2ne along the longitudinal axis dðxÞ; • the longitudinal resolution limit dd of this image, which was also, fortunately, in the order of 2ne in these experimental conditions. This resolution limit and ways to improve it will be discussed in an other section. Although image quality clearly deviated here from the required level for medical or biological endoscopy, this is, to our knowledge, the first
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information transmission on both transverse and longitudinal features of an optical scattering structure through a one-dimensional channel geometry which could be directly converted to a fibered endoscopic set-up. 3.1. Remark on the signal-to-noise ratio problem The light power required on the sample to get an image with a sufficient signal-to-noise (s=n) ratio has not yet been investigated in this paper as the main idea was to show and explain this new imaging system principles without being really concerned by this true problem: that is the reason why we have chosen a highly back scattering sample as a test one and in this case as well as with a simple scattering sheet of paper we do not report any sensitivity problem. Of course this s=n study will be, now an important part of the next step of our work. We should notice anyway that this problem will strongly depend on the output optic we will use. Indeed, the output lens L3 will determine the field depth of the measurement and also the collection angle for the backscattered light. This will then be determined by the imaging system characteristics needed for a certain application. We also could imagine to use a highly sensitive CCD camera rather than a classical one as we do now. An other interesting and encouraging point is described in a recent publication [8]. Leigteb et al. demonstrates that spectral OCT systems, or FDOCT, (an other recent way of calling spectral interferometry) could have a higher sensitivity than OCT systems (around 7 dB) especially in situations with low light levels. One reason presented in the paper is the fact that in FDOCT optical energy is collected rather than optical power leading to a different detector use with different noise expression.
4. Working principle Before giving a more accurate mathematical description of the successive signal processing steps, leading to a discussion on various possible improvements in image accuracy and quality as well as on mechanical compactness of the device,
Fig. 3. Sample illumination shape. Here is shown the enlightened line of length a f Dh0 due to the diffraction angle range which depends on the source bandwidth Dk with Dh0 ðN Dk= cosðh0 ÞÞ.
we will now qualitatively explain how this imaging device works. The basic idea is depicted in Fig. 3 and could be described as follows: light diffraction through a low resolution diffraction grating (G1 ) and focusing through the lens L2 illuminates a straight line segment of object (O). This focused spectral pattern has a length a f Dh0 , where Dh0 ðN ðk2 k1 Þ= cosðh0 ÞÞ with N equal to G1 grooves spatial frequency and k1 , k2 are the minimum and maximum wavelengths of illuminating light spectrum (k1 ¼ 1=r1 ; k2 ¼ 1=r2 ). h0 is the grating (G1 ) output diffraction angle and f the lens focal length. Thus each of the various points of the object line receives and scatters a given ÔwavelengthÕ (blue side of the spectrum for lower diffraction angles and red for larger ones). Part of the radiation will then be backscattered through (G1 ). According to the inverse optical path law, those components will give the various colours well-collimated directions, being parallel to that one of the incident beam. After the grating (G1 ), others components will become Ôquasi plane wavesÕ with a diffraction angle different from zero so that they will be focused out of the filtering pinhole (P) (Fig. 1) and not contribute to the image viewed by the detector. 2 Due
2
We may notice that in the endoscopic fibred set-up the filtering is not necessary as it could be automatically performed by the entrance of a single mode fiber.
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to the local reflectivity rðxÞ and geometrical shape dðxÞ of (O), each ÔcolourÕ carries an x dependent attenuation rðxÞ and temporal delay dðxÞ=c. Reflection on beamsplitter B.S. directs this parallel beam towards G2 , which performs a high-resolution spectral analysis. Spectral display of this beam on photodetector (D) reproduces the reflectivity distribution rðxÞ along coordinate k, with a given scaling factor: this corresponds to the Ôimage spectrogramÕ published by Friesem et al. [9], Lacourt et al. [4], Bartelt [5] earlier. Nevertheless the reference beam introduces new features in this Ôimage spectrogramÕ, giving it an interference structure. Indeed each colour did cross the same optical path 2l0 during its round trip between beamsplitter (B.S.) and mirror (M), whereas its optical path varied by an amount dðkÞ, as k exhibits a quasi-linear variation with coordinate x in the back focal volume of lens (L2 ). Such a dependence of the optical path difference between the interferometer arms, with the radiation wavelength will have the following consequence on the spectrum detected by (D): this spectrum will be modulated by a nearly sinusoidal interference pattern (Ôchannelled spectrumÕ) of local periodicity dk ¼ 1 1 ðk2 =2Þðl0 dðkÞ l1 Þ or dr ¼ 12ðl0 dðkÞ l1 Þ according to a classical property of two-beams white-light interferometry. Therefore the detected intensity distribution contains informations on both reflectivity rðxÞ and geometrical shape dðxÞ of the illuminated part of the object (O). In next section, a more detailed mathematical analysis will show how these informations will be, respectively, encoded as contrast and spacing of the spectral interference pattern: recovering rðxÞ and dðxÞ from these interferences will then be achieved by performing a local frequency analysis in successive parts of the channel spectrum, which lengths are much smaller than ÔaÕ, of the whole spectrum; the use of wavelet transformation is, of course, a suitable way to perform the information recovering. Our choice was to use Morlet wavelet as the mother wavelet in reason of the good correlation between this wavelet and the local channel spectra shape. Clearly, transmission of reflectivity and geometrical shape of a three dimensional structure instead of a two-dimensional one, considering this first presented set-up, will require the
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introduction of a scanning component inside this imaging channel. The mechanical scanning could be replaced by a more elaborated transmission line consisting of a one-dimensional array of identical optical fibres, in place of the above-considered single fibre. This kind of set-up has been previously described by Lacourt et al. [4] for two dimensional images transmission.
5. Image resolution We will now discuss the resolution limit problem for the presented system. For 3D imaging system the lateral as well as the transverse resolution have to be considered. 5.1. Transverse resolution (Fig. 4) DlT The N number of independent samples resolved by the spectral encoding stage (G1 , L2 ) in the object (O) may be calculated in a straightforward way: N is just equal to the number of spectral samples resolved by the spectrometer. As the whole light source bandwidth is Dr ¼ ðr2 r1 Þ, N will then be given by N ¼ ðDr=dr1 Þ, where dr1 denotes the spectrometer resolving power. In the ultimate case
Fig. 4. Resolution limits. On the upper part of the Figure is shown the transverse resolution, which is just equivalent to the grating G1 resolution limit. The down part explains the longitudinal resolution limit: the G2 grating must have a resolution much higher than G1 in order to resolve the local channelled spectra of each dr1 channel.
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where the spectrometer works at its Ôintrinsic resolving powerÕ, that is under single spatial mode illumination (single mode fiber channel or aberration-free collimated beam), with aberration-free 1 collimating optics L2 , this limit is dr1 ðk1 kp1 Þ where k denotes the average optical wavelength, k1 being the working diffraction order of (G1 ), and p1 the illuminated (G1 ) groove number. In our experiment these parameters take the following values k ¼ 800 nm, as k2 750 nm, k1 850 nm hence Dr ’ 0:16 lm1 , k1 ¼ 1, p1 3000 (groove spatial frequency ¼ 600 mm1 illuminated on approximately 5 mm) yielding 1 dr1 ¼ ðk1 kp1 Þ 0:4 mm1 and N 160=0:4 ¼ 400 resolved spectral samples inside the illuminating light bandwidth: this also will be about the number of independent pixels across the transmitted image. The whole image size being approximately 1 cm, the transverse image resolution DlT will be about 3 102 mm. 5.2. Longitudinal resolution DlL The image reconstruction accuracy along the longitudinal coordinate will be directly proportional to each transverse sample (or pixel) spectral bandwidth dr1 of this image; indeed measurements of optical path differences between the two interferometer arms through channelled spectrum metrology will have their precision ruled by the linewidth of each elementary channelled spectrum modulating each image pixel, that is just dr1 . The resolution asked to grating G2 is to exhibit enough large angular dispersion and spectral resolution dr2 for a satisfactory imaging of these channelled spectra (C.S.1, C.S.2, C.S.3) as shown in Fig. 4. In our experiment, the longitudinal resolution DlL 1 was then close to ð0:4Þ ¼ 2:5 mm, what corresponds rather well to the observed uncertainty (2 mm) on the glass plate optical thickness. Improving this optical path measurement accuracy is quite possible at the cost of lower transverse resolution: that is a larger dr1 , i.e., less transversally resolved samples within a given spectral bandwidth Dr of the light source. Of course, the broader this bandwidth, the larger the number of imaged samples will be. Nevertheless an Ôuncertainty relationshipÕ will always connect the number
N ¼ ðDr=dr1 Þ of transverse image samples to the optical depth uncertainty ð1=dr1 Þ ¼ dd, ðN =ddÞ ¼ Dr being a constant for a given light source or in terms of DlL , DlT it will lead to N DlT DlL . For instance white light illumination (Dr 1000 mm1 ) and 0.1 mm longitudinal resolution are compatible with transmission of N 100 pixels along the transverse coordinate x (e.g., 104 pixels of a two dimensional scene, when some scanning device or fiber array will be used in combination with the spectroscopic channel). 5.3. Remark about the measurement range We will briefly discuss here the rather straightforward connection between the resolution problem and the measurement range in both dimensions. As it has been shown in Section 5.1 N, the number of spectral samples resolved by the spectrometer, is also the number of independent spectral channels in the image and also the number of laterally resolved points. Each of these channels also encodes the local depth. This leads to the fact that there are as many laterally resolved points than longitudinal ones. Of course it remains anyway that two points will not be depth resolved if the depth difference between them is smaller than the depth resolving power ð1=dr1 Þ ¼ dd.
6. Mathematical analysis The above discussions about working principles were given in spectral terms only, as encoding parts of the considered set-up are usual spectroscopes and two beam interferometer. This image transmission channel may be completely characterised by its spectral responses at each of the detector pixels. The spectral response at each pixel is Fourier Transformed of the Temporal Impulse Response. Due to linearity and time-invariance, of the imaging channel with respect to optical field amplitude the impulse response of a sequence of components (or optical paths) is the convolution of the individual impulse responses of each component (or optical paths). The impulse response of the reference arm, just after transmission through beamsplitter (B.S.), is
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2l0 h0 ðtÞ ¼ d t ; c
" ð6:1Þ
where d is now, denoting the Dirac distribution, whereas this letter was representing differential increments of spectral or geometric variables in previous sections; but it will lead to no confusion in the next. The impulse response corresponding to one round trip of light between (B.S.) and input of diffraction grating (G1 ) over length 2AB is 2AB hðABÞ ðtÞ ¼ d t : ð6:2Þ c The impulse response corresponding to path (BC) between the input plane of (G1 ) and a given point C of object (O) results from the convolution of: BC hðBCÞ ðtÞ ¼ d t ð6:3Þ c with the temporal impulse response hG1 ðt; h0 Þ of (G1 ) along the h0 direction of diffraction. In Fig. 5 we recall and show the main features of the temporal impulse response in the case of an excitation with the as shortest as possible white light optical pulse. Under a uniform illumination by a Dirac impulse (G1 ) would transmit a Ôtemporal impulse responseÕ written as:
hG1 ðt; h0 Þ ¼
133
!
rect
t ct I 0 T ðh Þ kðh0 Þ
!# ð6:4Þ
the constant multiplying factor being omitted, where the wavetrain length T ðh0 Þ is the optical 0 delay sþs ¼ p1 gðsin h þ sin h0 Þ between longer and c shorter diffracted rays, with g being the (G1 ) grating period, and kðh0 Þ ¼ gðsin h þ sin h0 Þ is the periodicity of the pulse train scattered from each grating groove. After backscattering of light by object (O) at point C with local reflectivity rðh0 Þ and back transmission through grating (G1 ), the impulse response at point B may be written as hB ðt; h0 Þ ¼ rðh0 Þ hðBCÞ ðtÞ hðBCÞ ðtÞ h i ð6:5Þ
hG1 ðt; h0 Þ hG1 ðt; h0 Þ : Therefore the temporal impulse response hA ðt; h0 Þ of the whole spectroscopic imaging system including the interferometer set-up, when considered at the recombination point A on beam splitter (B.S.) takes the following form: h i hA ðt; h0 Þ ¼ hB ðt; h0 Þ hðABÞ ðtÞ þ Fh0 ðtÞ; ð6:6Þ where F is a photometric factor depending on a filter which allows interferometer amplitude balancing between two arms. When illuminated by a broadband collimated 2 beam, with a power spectrum jH0 ðmÞj , the device 2 0 transmits a filtered spectrum jHA0 ðm; h Þj at its re2 2 0 combination point A: jHA0 ðm; h Þj ¼ jH0 ðmÞj 0 2 0 jHA ðm; h Þj , where HA ðm; h Þ is the impulse response hA ðt; h0 Þ Fourier Transform. Hence the complete expression of filtered spectral amplitude (" X HA0 ðt; h0 Þ ¼ sinc p m nm0 ðh0 Þ n
0
T ðh Þ
2
# exp ð j2ptsÞrðh0 Þ )
þ F expð j2pts0 Þ H0 ðtÞ
Fig. 5. Grating temporal impulse response calculation. Here is shown the temporal impulse response of a grating as depending on geometrical parameters (angle of incidence and enlightened area) as well as on grating parameters (grooves number per mm).
ð6:7Þ
where m0 ðh0 Þ ¼ ðc=kðh0 ÞÞ, s ¼ 2ðAB þ BCÞ=c ¼ 2ðl1 þ dðh0 Þ=c and s0 ¼ 2l0 =c, where sinc function denotes the cardinal sine function sin cðxÞ ¼ sinx x and the optical path 2ðAB þ BCÞ being noted
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2ðl1 þ dðxÞÞ or 2ðl1 þ dðh0 ÞÞ as there is a correspondence between h0 and x through the relation tan h0 ¼ fx (f is the L2 focal length) and with dðh0 Þ being the Ôoptical depthÕ of object (O). Let us now consider the expression (6.7). We will briefly show in the following that this expression could be reduced to the next one: 2 1 sin c p m m0 ðh0 Þ T ðh0 Þ HA0 ðt; h0 Þ ¼ 2j expð j2ptsÞrðh0 Þ þ F expð j2pts0 Þ jH0 ðtÞj;
ð6:8Þ
where • only positive frequencies are considered • we end up with n ¼ 1 as other terms will be filtered out by the band-limited input pulse spectrum. Indeed, as shown in Fig. 6, the H0 ðmÞ term is the Fourier transform of the input optical pulse eðtÞ. As we are dealing with an optical pulse it is modulated with a sine term of frequency mp . Its Spectrum will then exhibit a positive part as well as a negative one due to the convolution by the Sine Fourier Transform given by ð1=2jÞdðv mp Þ dðv þ mp ÞÞ. Furthermore, it is interesting to compare the recurrence length of the frequency response, given by m0 ðh0 Þ ¼ ðc=kðh0 ÞÞ, with the
spectrum width. As an example we consider the case of a grating with a frequency of N ¼ 600 grooves per mm what gives a groove periodicity of g ¼ 2 lm. The maximum recurrence length in this case is when h0 ¼ 90. The maximum pulse train period will then be t¼
c 3 108 ¼ 1014 Hz: g 2 106
Our TiSa laser source has a bandwidth of about 100 nm at 800 nm. The corresponding frequency bandwidth is Dm ¼
cDk ð3 108 Þð100 109 Þ ¼ ¼ 2 108 Hz: 800 109 k2
We clearly see that only the first order sinc (for n ¼ 1 and n ¼ 1) will be within the spectrum bandwidth, other orders could then be neglected. Finally it is clear that no information loss will result if only positive frequencies are considered and the 1=2j term could also be omitted as a simple constant multiplying factor of the whole expression. The power spectrum of backscattered light from object point C will then becomes in a quite classical way: 2 0 2 HA ðt; h0 Þ ¼ jH0 ðtÞj IðtÞ !# " l0 l1 dðh0 Þ 1 þ vðtÞ cos 4pt c ð6:9Þ with 14 0 2 sinpT ðh0 Þ m t0 ðh0 Þ A þ F 2; IðtÞ ¼ rðh0 Þ @ pT ðh0 Þ m t0 ðh0 Þ ð6:10Þ 12 , sin 2pT ðh0 Þ m t0 ðh0 Þ 0 @ vðtÞ ¼ 2Frðh Þ A IðtÞ 2pT ðh0 Þ m t0 h0 0
ð6:11Þ Fig. 6. Simplification of HA0 ðt; h0 Þ. The reason of the HA0 ðt; h0 Þ expression simplification is shown step by step as a consequence of the Fourier components products in comparison with their frequency bandwidth.
2 jHA0 ðt; h0 Þj
is shown in Fig. 7 The A graph of 2 whole power spectrum jHA00 ðtÞj received on the second, high resolution spectroscope (G2 , L7 ) is nothing but the intensity superposition of all the
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Fig. 7. The shape of jHA0 ðt; h0 Þj2 in the case of a single point illumination. The registered spectrum in the case of a single illumination is depicted and the link between the shape and the calculated expression jHA0 ðt; h0 Þj2 is shown.
components jHA0 ðt; h0 Þj2 backscattered by all the illuminated object parts: Z 2 00 2 0 0 H ðtÞ ¼ H t; h dh0 : A A ðDh0 Þ
Graphs of such a superposition will be seen in Fig. 8 for different samples simulations. In Fig. 8(a) could be seen on the top (1) the obtained spectrum for a flat sample with dðh0 Þ ¼ constant ¼ 0:8 mm. The down graph (2) is the wavelet transform of
previous graph, which clearly shows an average constant shape around 0.8 mm. In Fig. 8(b) the sample is now tilted angularly with a dðh0 Þ (or dðkÞ as there is a relation between both variables) following a Ôlinear lawÕ dðkÞ ¼ 40 102 k 2:5 103 with dðkÞ in mm for k in m. This leads to dðh0 Þ varying between 0.5 and 0.9 mm as clearly shown on the wavelet transform (Fig. 8(b) (4)) of the graph presented in Fig. 8(b) (3). For each of this simulation 10 points have been considered for a
Fig. 8. Simulation results. (a) The result of the computed jHA00 ðtÞj2 , which is the integration of jHA0 ðt; h0 Þj2 over 10 points in the case of a ÔflatÕ scattering sample at a constant distance of about 0.8 mm. The upper graph is the spectrum and the down graph is the wavelet transform of it, showing a constant average line around 0.8 mm. (b) The result of the computed in the case of a tilted scattering sample with a distance following the law dðkÞ ¼ 40 102 k 2:5 103 with dðkÞ in mm if k is in meter. The upper graph is the spectrum and the down graph is the wavelet transform of it, showing the linear average variation between 0.5 and 0.8 mm.
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grating G1 with 600 grooves per mm enlightened on 0.3 mm with a source spectral bandwidth of about 100 nm around 800 nm.
system will also be of interest in order to compare this system with EOCT s=n capabilities. All these next steps will be presented in further publications.
7. Conclusion and outlooks
Acknowledgements
A new imaging set-up has been demonstrated which allows recording and transmission of 3D scattered images through a one-dimensional channel. The first prototype was presented to be a single line acquisition system. This requires, for a whole 3D image recording, a one-dimensional scanning, which could be performed by different ways we also suggested in the paper. Furthermore, the resolution problem has also been analysed through the spectral analysis of the system and a strong dependence between the transverse and the longitudinal resolution was shown to be a Fourier uncertainty relationship. Despite this, the theoretically achievable resolution could be satisfactory for the tracheal and bronchial stenoses dimensioning endoscopy, what requires a lateral and longitudinal resolution of about 0.2 mm. Finally, although the presented set-up was not yet an endoscopic one we have tried to show how this system could be implemented in a, somewhat, straightforward way to the endoscopic purpose. The next steps of our works will be to increase the spectral image contrast by decreasing of the signal DC component. The other point will be to implement a scanning-less 3D imaging system using a different geometry but the same concepts than those presented in this paper. An implementation of an heterodyne detection version of our
First of all we would like to thank the Swiss National Science Foundation for his financial support to this program with subside number 3200062028. We also would express our gratitude to M.Liebling, from the Biomedical Imaging Laboratory (Bio-E, EPFL), about its help in our wavelet understanding. And finally we would like to thank I. Verrier and G. Brun from St. Etienne University (France) for their interest in this project.
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