October 1, 2014 / Vol. 39, No. 19 / OPTICS LETTERS
5495
Optical microcavity scanning 3D tomography Andrea Di Donato,1†,1,* Luigino Criante,†,2 Sara LoTurco,22 and Marco Farina31 1
Department of Information Engineering, Università Politecnica delle Marche, Ancona 60131, Italy 2
Center for Nano Science and Technology, Istituto Italiano di Tecnologia, Milano 20133, Italy *Corresponding author: [email protected] Received June 19, 2014; revised August 8, 2014; accepted August 18, 2014; posted August 19, 2014 (Doc. ID 214303); published September 16, 2014
A scanning optical microcavity is exploited to achieve lens-free 3D tomography of microfluidic channels. The microcavity, powered by a low-coherence source, is realized by approaching a cleaved fiber to few tens of micrometers over the sample. The interference of scattered waves inside the cavity shapes the transverse field distribution by focusing the beam and overcoming the diffraction limit due to the optical-fiber numerical aperture. The focusing effect is also preserved in the inner layers of the sample, allowing optical 3D tomography. Analysis of microfluidic channels was demonstrated through this noninvasive technique. Although the experimental setup recalls the well-known fiber-optic Fourier-domain common-path optical coherence tomography, the proposed method has intrinsic characteristics that distinguish it from the former one. © 2014 Optical Society of America OCIS codes: (140.3948) Microcavity devices; (140.3945) Microcavities; (170.4500) Optical coherence tomography; (120.3180) Interferometry; (240.3990) Micro-optical devices. http://dx.doi.org/10.1364/OL.39.005495
Optical microcavities have always played an important role in a wide range of applications and studies, due to their sensitivity and capability to confine light in small volumes. They have been applied to measure the refractive index of optical glasses [1], or to realize a wide range of displacement sensors with subnanometer resolution or biosensors [2,3]. During recent years they have attracted great attention for applications that span from quantum electrodynamics to the realization of optical sources and dynamic filters for optical communications [4]. In this work we describe an application in which an optical microcavity is scanned over a surface in order to achieve 3D optical tomography of buried microfluidic channels. Recently, developments in the field of microfluidic systems have given rise to the need for investigating the physical properties of the whole microfluidic chip (such as the 3D size and quality). In this framework the use of noninvasive techniques could play a key role where previous methods are no longer applicable [5,6]. Even though a common technique based on optical coherence tomography (OCT) could be used [7], we show here a configuration that allows better miniaturization and integrated design, enabling an easy 3D reconstruction of micrometric structures in the so-called lab on chip. The cavity is realized by approaching a cleaved optical fiber to the surface under investigation, within a working range of tens of micrometers. In this lens-free system, the diffraction limit introduced by the numerical aperture of the optical fiber is overcome thanks to the resonant behavior of the electromagnetic field inside the cavity. This phenomenon leads to a focusing effect of the transverse field distribution, which has been already exploited to achieve contrast-phase imaging [8]. Here we will show how this feature can also be extended to the inner layers of a sample, realizing a low-coherence optical tomography. In order to have a compact, miniature design and fast-scanning imaging, we carried out a common-path optical configuration. This configuration is well known in OCT, where the reference beam (coming from a partial reflector) and the reflected signal (coming from the sample) share the same optical path. A common-path low-coherence 0146-9592/14/195495-04$15.00/0
interferometer reduces the possible polarization distortion and allows a compensation for dispersion and polarization mismatch that are not due to the sample [9]. These aspects make the system suitable for phase-contrast imaging, with a low level of phase noise [10]. On the other hand, common-path OCT systems have some critical aspects, such as the position of the partial reflector, which has to be placed between the sample and the objective lens. In addition, a limited interspace between partial reflector and sample makes it difficult to use high numerical aperture objective [11]. Moreover, tuning the reference signal level turns out to be critical, thus reducing the opportunities to optimize the sensitivity. However, for all the aforementioned problems, some solutions have been reported in literature [11,12], provided that the probe is located within the coherence region. The experimental setup used in our system is described in Fig. 1, in which the optical cavity is pumped by a broadband diode laser at 850 nm (bandwidth FWHM 40 nm), which feeds a directional coupler connected to the optical probe. The reflected spectrum, coming directly from the extrinsic cavity, is acquired by an optical analyzer. The sample is placed on a piezoelectric scanner and the probe is realized through a cleaved single-mode optical fiber (NA 0.10–0.14 and MFD equal to 5.6 μm), which is moved at constant height over the sample without any
Fig. 1. Microcavity scanning tomography: experimental setup. d cavity dimension between the fiber facet and sample. © 2014 Optical Society of America
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feedback control. The piezoelectric scanner approaches the sample to the fiber down to a distance of about few tens of micrometers, realizing an extrinsic microcavity. The sample to be investigated is an empty square 2-mm-long tapered microchannel for which its central axis is located at a depth of 270 μm inside a fused silica substrate (FOCtek, JGS1). The microchannel is realized by femtosecond laser micromachining followed by wet etching with hydrofluoric acid (HF), in order to selectively remove the laser-affected zones [13]. In detail this maskless technique permits an easy prototyping and precise fabrication of 3D microstructures inside the bulk of transparent substrates, where the material is locally modified only in correspondence to the beam focus due to highly nonlinear absorption. Compared to standard lithographic techniques, which require subsequent steps of planar fabrication and layers bonding, the laser micromachining allows imprinting a complex 3D design directly, with an intrinsic accurate alignment between all the subparts of the structure. The laser irradiation is performed with the second harmonic of a regenerative mode-locked Yb:KGW femtosecond source (Pharos Light Conversion, 280 fs 1030 nm 500 kHz), selecting a pulse energy of 400 nJ and a 50× focusing objective (Mitutoyo, NA 0.42). Although the proposed optical configuration refers to that used in common-path optical tomography, the presence of a scanning microcavity as a sensing probe introduces some relevant differences. In common-path OCT systems, the acquired signal is the interference between the signal reflected from the reference surface (e.g., fiber facet or other flat surface) and from the sample. In a microcavity scanning system, the acquired signal is reflected directly from the cavity, in which a spectrum of diffracted waves from the fiber undergoes multiple reflections and interferences with waves reflected from the sample. The expression describing the complex reflected signal Γ from the optical cavity can be written as [14] Γk Ak
1 − Y k ; 1 Y k
(1)
where Z
βkr H 0 e0 2 tanφkr dkr β0 αkr φ βkr d − 2 q βkr k2 − k2r αkr xkr jykr : Y k j
∞
0
Ik Sk
1 Y Y − 2 ReY : 1 Y Y 2 ReY
(3)
This expression is quite different from the usual relation describing the output spectrum of a common-path interferometer: Ik I 1 I 2 2 Reρ;
(4)
with I 1 and I 2 being the intensities of reference and sample signals and ρ the complex degree of self-coherence, which depends on the difference between the optical paths of reference and sample signals. The function 2 ReY in Eq. (3) is the interference term that contains all the information about the scattering profile of the sample. Its value is related to the measured spectrum, through the function V k
2 ReY Sk − Ik 2 Sk Ik 1 jY j
(5)
in which Sk is the spectral intensity of the source, acquired when the optical fiber does not interact with the sample. The function V k recalls, in its form, the definition of contrast between the fringes of interferences that are used to quantify the magnitude of degree of coherence in interferometry [15]. Unlike in Eq. (4), distinguishing the reference and sample signals in the complex function Y k is not possible. The optical microcavity behaves like a multimode resonator due to the diffraction of guided modes at the distal end of the optical fiber. The reflected spectrum is shaped by multiple reflections and mutual interferences. Inside the cavity all the diffracted waves give a different contribution to the interference term, weighted by the spatial spectrum H 0 e0 of the guided mode (see [Eq. (2)]). The information about the scattering profile sz of the sample is enclosed in the phase φk of the interference term ReY k, which depends on the complex reflection coefficient Γs k. At the first order of approximation, the term Γs k is proportional to the Fourier transform of the scattering profile sz: ΓS k ejα e−y ejx ja0 kS2βk α x jy
(2)
The term H 0 is the Hankel transform of the fiberguided mode, Ak is the spectral amplitude distribution of the light source, Γ is the amplitude of the reflected signal back coupled to the fiber for each wave number k, α is the complex phase of reflected waves from the surface sample Γs ejα (with x being the real part and y being the imaginary part), whereas β and kr are the longitudinal and radial spatial frequencies, respectively. The intensity acquired by the spectrum analyzer is equal to
Skz Isz;
(6)
where we assumed a wave incident on the sample with an amplitude equal to a0 (see [Fig. 1]) [16]. According to Eq. (6), it is possible to recover the behavior of the scattering profile in-depth through the inverse Fourier transform of the contrast function V k. As proof of that, if a delta-like scattering profile is considered underneath the surface, located at a distance d from the fiber facet (see [Fig. 2]), the inverse Fourier transform of backscattered signal V k in the spatial domain leads to the same microcavity response measured in z d, but centered in z0 :
October 1, 2014 / Vol. 39, No. 19 / OPTICS LETTERS
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Fig. 2. Microcavity response in spatial domain for a delta-like scattering profile, recovered through the inverse Fourier transform.
sz δz − z0 ;
Sβ Isz e2πjβz0
αkr βkr d − z0 2 2 ReY k; d − z0 V k : 1 jY k; d − z0 j2 φ βkr d −
(7)
In detail, for each scanning point the spectral intensity of the interferometric signal, reflected from the cavity, is picked up by the optical spectrum analyzer. After the calibration, the signal Vk is converted in the k domain and then it is interpolated and analyzed in the spatial domain to define the scattering profile [14]. The peak position and amplitude of the contrast response Vk in the spatial domain are recorded for each scanning point. From the position of the peak we extract the topography information, whereas the amplitude variations allow reconstructing contrast-phase maps. The latter are correlated to the refractive-index changes and topography of the scanned surface [8]. The different behavior of the microcavity scanning system, compared with the low-coherence interferometric common-path techniques and formally described by Eqs. (3) and (4), appears quite clear when working at distances much greater than source-coherence length. Even at these distances the lateral resolution is preserved. Usually in the common-path OCT, focusing systems such as objective lens, short length of graded-index fiber, or conical fiber probe are used in order to achieve high lateral resolution [17]. Cleaved optical fibers were exploited, allowing a transverse resolution higher than 25 μm due to Gaussian-beam divergence [18]. In scanning microcavity tomography, the optical resonance itself shapes the electromagnetic field distribution inside the cavity, overcoming the diffraction due to finite numerical aperture of the fiber. The multimode interference changes the diffracted field according to the following relations, describing the amplitude of fields radiated in free space af kr and inside a microcavity ac kr [14]: af kr 1 ΓkH 0 k; kr af k; kr 1 j tanφkr : ac kr |{z} 2
Fig. 3. Numerical simulation. The sample is a multi-layered structure with n1 1.4, n2 1.5, d1 70 μm, d2 130 μm, MFD 5.2 μm, and NA 0.1–0.14. The guided mode in the optical fiber is assumed to be a linearly polarized mode (LP).
estimated to be close to 1 μm. In Fig. 3, the transverse power distribution of the electromagnetic field diffracted from the fiber tip in free space is compared with that obtained in the presence of an extrinsic resonant cavity, confirming the unexpected but significant resolution result. The latter is focused, keeping its lateral shape unchanged on each inner interface. Numerical simulations are based on Eq. (8) in which the reflection coefficient Γs of the multilayered sample is evaluated for each scattered wave. The field distribution inside the cavity is proportional to the Hankel transform H 0 of the guided mode and is also affected by the spectral characteristics αkr of the reflecting surface. Similar to all low-coherence interferometric systems, the depth resolution is mainly limited by the power spectral density Sk of the optical source. In detail, it depends on the term SkY k appearing in Eq. (3). For our system, the FWHM of response V k in the spatial domain is about 10 μm. Figure 4 describes the sample used for tomography measurements: the 3D sketch depicts the square empty microchannel fully embedded
(8)
Interference Term
The radiated field is focused by the interference term, which depends on the cavity geometry and sample reflectivity. Starting from the function ac kr we can numerically compute the power distribution over the sample surface and inside inner layers, defining in this way the transverse resolution of the system, which was
Fig. 4. (a) 3D sketch of the microfluidic square channel realized inside a glass sample, optical microscope images of the (b) cross section at the sample border, (c) top view of the channel: dashed lines highlight the zones exposed to tomography investigation; zoom view of (d) top vision; and (e) lateral vision of the central region. Dimensions taken with the optical microscope ( 3 μm) are d1 247 μm, t1 161 μm, w1 149 μm, d2 265 μm, t2 125 μm, and w2 100 μm.
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OPTICS LETTERS / Vol. 39, No. 19 / October 1, 2014
Fig. 5. (a) 3D tomography of a portion of the distal end of the square channel, plotted through an isosurface, (b) scattering profile in logarithmic scale, for a single point. Dotted line is the reference level used to display isosurface, and (c)–(e) contrast-phase maps obtained by analyzing the cavity response in each of the selected regions highlighted by double arrows in profile (b).
inside the glass; whereas the two zones exposed to investigation are highlighted by dashed lines in Fig. 4(c). The optical microscope images clearly show the square section getting larger at the two edges of the samples, due to the isotropic effect during the HF etching process to remove the internal irradiated material. As a consequence, the central region (zone 2) results are deeper and narrower than the border part (zone 1). Figure 5 shows the 3D tomography of the distal end of the channel region analyzed. As mentioned, for each scanning point, it is possible to acquire all of the scattering profile, or to study the behavior of the cavity response Vk only in a selected range of depths, obtaining topography and contrast-phase maps; the latter are attained through the analysis of the peak amplitude of the microcavity response. Contrast-phase maps have shown enhanced sensitivity compared to the topography maps [8]. The results achieved for the middle region of the channel are depicted in Fig. 6. In both scans, the optical fiber has scanned the sample at a distance of about 60 μm (microcavity dimension). In Fig. 5, the first interface of the microfluidic channel is placed at a real distance d1 248 μm below the surface, whereas the channel has height t1 of about 161 μm. The positions displayed are the optical lengths corresponding to the first and second microfluidic interface, placed at d01 ≈ 420 μm (60 μm 248 μm· sample refractive index) and 580 μm (d01 t1 .). According to Fig. 6, the real-channel dimensions d2 , t2 , and w2 are about 266, 127, and 102 μm, respectively. The above results are in good agreement with those obtained from optical microscope analysis. In summary, in this work we describe an optical microcavity scanning based system able to provide 3D optical tomography. It is a lens-free technique that exploits the confinement of light inside the cavity, which is also preserved in the inner layers of the sample. This peculiarity plays a
Fig. 6. (a) 3D tomography of the central region (zone 2), (b) scattering profile in logarithmic scale, for a single point, and (c)–(e) contrast-phase maps in each of the selected regions reported in profile (b).
key role in noninvasive analysis techniques (tomography and quality check) of lab-on-chip devices and for monitoring flowing samples through microfluidic channels. † These authors equally contributed to this work. References 1. J. H. Chen, J. R. Zhao, X. G. Huang, and Z. J. Huang, Appl. Opt. 49, 5592 (2010). 2. X. Zhou and Q. Yu, IEEE Sens. J. 11, 1602 (2011). 3. Y. Zhang, H. Shibru, K. L. Cooper, and A. Wang, Opt. Lett. 30, 1021 (2005). 4. K. J. Vahala, Nature 424, 839 (2003). 5. L. P. Yeo, S. H. Ng, Z. F. Wang, H. M. Xia, Z. P. Wang, V. S. Thang, Z. W. Zhong, and N. F. de Rooij, J. Micromech. Microeng. 20, 015017 (2010). 6. D. Hardt, B. Ganesan, M. Dirckx, G. Shoji, K. Thaker, and W. Qi, IMST Report (MIT, 2005). 7. J. Czajkowski, T. Prykäri, E. Alarousu, J. Lauri, and R. Myllylä, Proc. SPIE 7376, 737617 (2010). 8. A. Di Donato, T. Pietrangelo, A. Anzellotti, T. Monti, A. Morini, and M. Farina, Opt. Lett. 38, 5094 (2013). 9. U. Sharma, N. M. Fried, and J. U. Kang, IEEE J. Sel. Top. Quantum Electron. 11, 799 (2005). 10. X. Liu, X. Li, D.-H. Kim, I. Ilev, and J. U. Kang, Chin. Opt. Lett. 6, 899 (2008). 11. J. S. Park, Z. Chen, M. Y. Jeong, and C. S. Kim, Opt. Express 20, 1102 (2012). 12. X. Liu and J. Kang, Opt. Lett. 38, 2660 (2013). 13. S. LoTurco, R. Osellame, R. Ramponi, and K. C. Vishnubhatla, J. Micromech. Microeng. 23, 1 (2013). 14. A. Di Donato, A. Morini, and M. Farina, PIER 133, 347 (2013). 15. B. Bouma and G. Tearney, Handbook of Optical Coherence Tomography (Marcel Dekker, 2002). 16. A. Di Donato, M. Farina, D. Mencarelli, A. Lucesoli, S. Fabiani, T. Rozzi, G. M. Di Gregorio, and G. Angeloni, J. Lightwave Technol. 28, 1510 (2010). 17. K. M. Tan, M. Mazilu, T. H. Chow, W. M. Lee, K. Taguchi, B. K. Ng, W. Sibbett, C. S. Herrington, C. T. A. Brown, and K. Dholakia, Opt. Express 17, 2375 (2009). 18. X. Liu, Y. Huang, and J. U. Kang, Opt. Express 20, 16567 (2012).
5495
Optical microcavity scanning 3D tomography Andrea Di Donato,1†,1,* Luigino Criante,†,2 Sara LoTurco,22 and Marco Farina31 1
Department of Information Engineering, Università Politecnica delle Marche, Ancona 60131, Italy 2
Center for Nano Science and Technology, Istituto Italiano di Tecnologia, Milano 20133, Italy *Corresponding author: [email protected] Received June 19, 2014; revised August 8, 2014; accepted August 18, 2014; posted August 19, 2014 (Doc. ID 214303); published September 16, 2014
A scanning optical microcavity is exploited to achieve lens-free 3D tomography of microfluidic channels. The microcavity, powered by a low-coherence source, is realized by approaching a cleaved fiber to few tens of micrometers over the sample. The interference of scattered waves inside the cavity shapes the transverse field distribution by focusing the beam and overcoming the diffraction limit due to the optical-fiber numerical aperture. The focusing effect is also preserved in the inner layers of the sample, allowing optical 3D tomography. Analysis of microfluidic channels was demonstrated through this noninvasive technique. Although the experimental setup recalls the well-known fiber-optic Fourier-domain common-path optical coherence tomography, the proposed method has intrinsic characteristics that distinguish it from the former one. © 2014 Optical Society of America OCIS codes: (140.3948) Microcavity devices; (140.3945) Microcavities; (170.4500) Optical coherence tomography; (120.3180) Interferometry; (240.3990) Micro-optical devices. http://dx.doi.org/10.1364/OL.39.005495
Optical microcavities have always played an important role in a wide range of applications and studies, due to their sensitivity and capability to confine light in small volumes. They have been applied to measure the refractive index of optical glasses [1], or to realize a wide range of displacement sensors with subnanometer resolution or biosensors [2,3]. During recent years they have attracted great attention for applications that span from quantum electrodynamics to the realization of optical sources and dynamic filters for optical communications [4]. In this work we describe an application in which an optical microcavity is scanned over a surface in order to achieve 3D optical tomography of buried microfluidic channels. Recently, developments in the field of microfluidic systems have given rise to the need for investigating the physical properties of the whole microfluidic chip (such as the 3D size and quality). In this framework the use of noninvasive techniques could play a key role where previous methods are no longer applicable [5,6]. Even though a common technique based on optical coherence tomography (OCT) could be used [7], we show here a configuration that allows better miniaturization and integrated design, enabling an easy 3D reconstruction of micrometric structures in the so-called lab on chip. The cavity is realized by approaching a cleaved optical fiber to the surface under investigation, within a working range of tens of micrometers. In this lens-free system, the diffraction limit introduced by the numerical aperture of the optical fiber is overcome thanks to the resonant behavior of the electromagnetic field inside the cavity. This phenomenon leads to a focusing effect of the transverse field distribution, which has been already exploited to achieve contrast-phase imaging [8]. Here we will show how this feature can also be extended to the inner layers of a sample, realizing a low-coherence optical tomography. In order to have a compact, miniature design and fast-scanning imaging, we carried out a common-path optical configuration. This configuration is well known in OCT, where the reference beam (coming from a partial reflector) and the reflected signal (coming from the sample) share the same optical path. A common-path low-coherence 0146-9592/14/195495-04$15.00/0
interferometer reduces the possible polarization distortion and allows a compensation for dispersion and polarization mismatch that are not due to the sample [9]. These aspects make the system suitable for phase-contrast imaging, with a low level of phase noise [10]. On the other hand, common-path OCT systems have some critical aspects, such as the position of the partial reflector, which has to be placed between the sample and the objective lens. In addition, a limited interspace between partial reflector and sample makes it difficult to use high numerical aperture objective [11]. Moreover, tuning the reference signal level turns out to be critical, thus reducing the opportunities to optimize the sensitivity. However, for all the aforementioned problems, some solutions have been reported in literature [11,12], provided that the probe is located within the coherence region. The experimental setup used in our system is described in Fig. 1, in which the optical cavity is pumped by a broadband diode laser at 850 nm (bandwidth FWHM 40 nm), which feeds a directional coupler connected to the optical probe. The reflected spectrum, coming directly from the extrinsic cavity, is acquired by an optical analyzer. The sample is placed on a piezoelectric scanner and the probe is realized through a cleaved single-mode optical fiber (NA 0.10–0.14 and MFD equal to 5.6 μm), which is moved at constant height over the sample without any
Fig. 1. Microcavity scanning tomography: experimental setup. d cavity dimension between the fiber facet and sample. © 2014 Optical Society of America
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OPTICS LETTERS / Vol. 39, No. 19 / October 1, 2014
feedback control. The piezoelectric scanner approaches the sample to the fiber down to a distance of about few tens of micrometers, realizing an extrinsic microcavity. The sample to be investigated is an empty square 2-mm-long tapered microchannel for which its central axis is located at a depth of 270 μm inside a fused silica substrate (FOCtek, JGS1). The microchannel is realized by femtosecond laser micromachining followed by wet etching with hydrofluoric acid (HF), in order to selectively remove the laser-affected zones [13]. In detail this maskless technique permits an easy prototyping and precise fabrication of 3D microstructures inside the bulk of transparent substrates, where the material is locally modified only in correspondence to the beam focus due to highly nonlinear absorption. Compared to standard lithographic techniques, which require subsequent steps of planar fabrication and layers bonding, the laser micromachining allows imprinting a complex 3D design directly, with an intrinsic accurate alignment between all the subparts of the structure. The laser irradiation is performed with the second harmonic of a regenerative mode-locked Yb:KGW femtosecond source (Pharos Light Conversion, 280 fs 1030 nm 500 kHz), selecting a pulse energy of 400 nJ and a 50× focusing objective (Mitutoyo, NA 0.42). Although the proposed optical configuration refers to that used in common-path optical tomography, the presence of a scanning microcavity as a sensing probe introduces some relevant differences. In common-path OCT systems, the acquired signal is the interference between the signal reflected from the reference surface (e.g., fiber facet or other flat surface) and from the sample. In a microcavity scanning system, the acquired signal is reflected directly from the cavity, in which a spectrum of diffracted waves from the fiber undergoes multiple reflections and interferences with waves reflected from the sample. The expression describing the complex reflected signal Γ from the optical cavity can be written as [14] Γk Ak
1 − Y k ; 1 Y k
(1)
where Z
βkr H 0 e0 2 tanφkr dkr β0 αkr φ βkr d − 2 q βkr k2 − k2r αkr xkr jykr : Y k j
∞
0
Ik Sk
1 Y Y − 2 ReY : 1 Y Y 2 ReY
(3)
This expression is quite different from the usual relation describing the output spectrum of a common-path interferometer: Ik I 1 I 2 2 Reρ;
(4)
with I 1 and I 2 being the intensities of reference and sample signals and ρ the complex degree of self-coherence, which depends on the difference between the optical paths of reference and sample signals. The function 2 ReY in Eq. (3) is the interference term that contains all the information about the scattering profile of the sample. Its value is related to the measured spectrum, through the function V k
2 ReY Sk − Ik 2 Sk Ik 1 jY j
(5)
in which Sk is the spectral intensity of the source, acquired when the optical fiber does not interact with the sample. The function V k recalls, in its form, the definition of contrast between the fringes of interferences that are used to quantify the magnitude of degree of coherence in interferometry [15]. Unlike in Eq. (4), distinguishing the reference and sample signals in the complex function Y k is not possible. The optical microcavity behaves like a multimode resonator due to the diffraction of guided modes at the distal end of the optical fiber. The reflected spectrum is shaped by multiple reflections and mutual interferences. Inside the cavity all the diffracted waves give a different contribution to the interference term, weighted by the spatial spectrum H 0 e0 of the guided mode (see [Eq. (2)]). The information about the scattering profile sz of the sample is enclosed in the phase φk of the interference term ReY k, which depends on the complex reflection coefficient Γs k. At the first order of approximation, the term Γs k is proportional to the Fourier transform of the scattering profile sz: ΓS k ejα e−y ejx ja0 kS2βk α x jy
(2)
The term H 0 is the Hankel transform of the fiberguided mode, Ak is the spectral amplitude distribution of the light source, Γ is the amplitude of the reflected signal back coupled to the fiber for each wave number k, α is the complex phase of reflected waves from the surface sample Γs ejα (with x being the real part and y being the imaginary part), whereas β and kr are the longitudinal and radial spatial frequencies, respectively. The intensity acquired by the spectrum analyzer is equal to
Skz Isz;
(6)
where we assumed a wave incident on the sample with an amplitude equal to a0 (see [Fig. 1]) [16]. According to Eq. (6), it is possible to recover the behavior of the scattering profile in-depth through the inverse Fourier transform of the contrast function V k. As proof of that, if a delta-like scattering profile is considered underneath the surface, located at a distance d from the fiber facet (see [Fig. 2]), the inverse Fourier transform of backscattered signal V k in the spatial domain leads to the same microcavity response measured in z d, but centered in z0 :
October 1, 2014 / Vol. 39, No. 19 / OPTICS LETTERS
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Fig. 2. Microcavity response in spatial domain for a delta-like scattering profile, recovered through the inverse Fourier transform.
sz δz − z0 ;
Sβ Isz e2πjβz0
αkr βkr d − z0 2 2 ReY k; d − z0 V k : 1 jY k; d − z0 j2 φ βkr d −
(7)
In detail, for each scanning point the spectral intensity of the interferometric signal, reflected from the cavity, is picked up by the optical spectrum analyzer. After the calibration, the signal Vk is converted in the k domain and then it is interpolated and analyzed in the spatial domain to define the scattering profile [14]. The peak position and amplitude of the contrast response Vk in the spatial domain are recorded for each scanning point. From the position of the peak we extract the topography information, whereas the amplitude variations allow reconstructing contrast-phase maps. The latter are correlated to the refractive-index changes and topography of the scanned surface [8]. The different behavior of the microcavity scanning system, compared with the low-coherence interferometric common-path techniques and formally described by Eqs. (3) and (4), appears quite clear when working at distances much greater than source-coherence length. Even at these distances the lateral resolution is preserved. Usually in the common-path OCT, focusing systems such as objective lens, short length of graded-index fiber, or conical fiber probe are used in order to achieve high lateral resolution [17]. Cleaved optical fibers were exploited, allowing a transverse resolution higher than 25 μm due to Gaussian-beam divergence [18]. In scanning microcavity tomography, the optical resonance itself shapes the electromagnetic field distribution inside the cavity, overcoming the diffraction due to finite numerical aperture of the fiber. The multimode interference changes the diffracted field according to the following relations, describing the amplitude of fields radiated in free space af kr and inside a microcavity ac kr [14]: af kr 1 ΓkH 0 k; kr af k; kr 1 j tanφkr : ac kr |{z} 2
Fig. 3. Numerical simulation. The sample is a multi-layered structure with n1 1.4, n2 1.5, d1 70 μm, d2 130 μm, MFD 5.2 μm, and NA 0.1–0.14. The guided mode in the optical fiber is assumed to be a linearly polarized mode (LP).
estimated to be close to 1 μm. In Fig. 3, the transverse power distribution of the electromagnetic field diffracted from the fiber tip in free space is compared with that obtained in the presence of an extrinsic resonant cavity, confirming the unexpected but significant resolution result. The latter is focused, keeping its lateral shape unchanged on each inner interface. Numerical simulations are based on Eq. (8) in which the reflection coefficient Γs of the multilayered sample is evaluated for each scattered wave. The field distribution inside the cavity is proportional to the Hankel transform H 0 of the guided mode and is also affected by the spectral characteristics αkr of the reflecting surface. Similar to all low-coherence interferometric systems, the depth resolution is mainly limited by the power spectral density Sk of the optical source. In detail, it depends on the term SkY k appearing in Eq. (3). For our system, the FWHM of response V k in the spatial domain is about 10 μm. Figure 4 describes the sample used for tomography measurements: the 3D sketch depicts the square empty microchannel fully embedded
(8)
Interference Term
The radiated field is focused by the interference term, which depends on the cavity geometry and sample reflectivity. Starting from the function ac kr we can numerically compute the power distribution over the sample surface and inside inner layers, defining in this way the transverse resolution of the system, which was
Fig. 4. (a) 3D sketch of the microfluidic square channel realized inside a glass sample, optical microscope images of the (b) cross section at the sample border, (c) top view of the channel: dashed lines highlight the zones exposed to tomography investigation; zoom view of (d) top vision; and (e) lateral vision of the central region. Dimensions taken with the optical microscope ( 3 μm) are d1 247 μm, t1 161 μm, w1 149 μm, d2 265 μm, t2 125 μm, and w2 100 μm.
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OPTICS LETTERS / Vol. 39, No. 19 / October 1, 2014
Fig. 5. (a) 3D tomography of a portion of the distal end of the square channel, plotted through an isosurface, (b) scattering profile in logarithmic scale, for a single point. Dotted line is the reference level used to display isosurface, and (c)–(e) contrast-phase maps obtained by analyzing the cavity response in each of the selected regions highlighted by double arrows in profile (b).
inside the glass; whereas the two zones exposed to investigation are highlighted by dashed lines in Fig. 4(c). The optical microscope images clearly show the square section getting larger at the two edges of the samples, due to the isotropic effect during the HF etching process to remove the internal irradiated material. As a consequence, the central region (zone 2) results are deeper and narrower than the border part (zone 1). Figure 5 shows the 3D tomography of the distal end of the channel region analyzed. As mentioned, for each scanning point, it is possible to acquire all of the scattering profile, or to study the behavior of the cavity response Vk only in a selected range of depths, obtaining topography and contrast-phase maps; the latter are attained through the analysis of the peak amplitude of the microcavity response. Contrast-phase maps have shown enhanced sensitivity compared to the topography maps [8]. The results achieved for the middle region of the channel are depicted in Fig. 6. In both scans, the optical fiber has scanned the sample at a distance of about 60 μm (microcavity dimension). In Fig. 5, the first interface of the microfluidic channel is placed at a real distance d1 248 μm below the surface, whereas the channel has height t1 of about 161 μm. The positions displayed are the optical lengths corresponding to the first and second microfluidic interface, placed at d01 ≈ 420 μm (60 μm 248 μm· sample refractive index) and 580 μm (d01 t1 .). According to Fig. 6, the real-channel dimensions d2 , t2 , and w2 are about 266, 127, and 102 μm, respectively. The above results are in good agreement with those obtained from optical microscope analysis. In summary, in this work we describe an optical microcavity scanning based system able to provide 3D optical tomography. It is a lens-free technique that exploits the confinement of light inside the cavity, which is also preserved in the inner layers of the sample. This peculiarity plays a
Fig. 6. (a) 3D tomography of the central region (zone 2), (b) scattering profile in logarithmic scale, for a single point, and (c)–(e) contrast-phase maps in each of the selected regions reported in profile (b).
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