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Abstract
A conventional optical lens can be used to focus light into the target medium from outside, without disturbing the medium. The focused spot size is proportional to the focal distance in a conventional lens, resulting in a tradeoff between penetration depth in the target medium and spatial resolution. We have shown that virtual ultrasonically sculpted gradient-index (GRIN) optical waveguides can be formed in the target medium to steer light without disturbing the medium. Here, we demonstrate that such virtual waveguides can relay an externally focused Gaussian beam of light through the medium beyond the focal distance of a single external physical lens, to extend the penetration depth without compromising the spot size. Moreover, the spot size can be tuned by reconfiguring the virtual waveguide. We show that these virtual GRIN waveguides can be formed in transparent and turbid media, to enhance the confinement and contrast ratio of the focused beam of light at the target location. This method can be extended to realize complex optical systems of external physical lenses and in situ virtual waveguides, to extend the reach and flexibility of optical methods.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Light-matter interaction has been used in different applications, ranging from biological imaging and manipulation to metrology, material processing, and machine vision [1–7]. The effect of the medium on light through refraction, reflection, scattering, or absorption has been used for sensing, detection, and imaging. Moreover, light can affect the medium when it is concentrated to a high enough intensity at specific locations within the medium. Optical manipulation has been used in a wide range of applications such as optogenetic stimulation of biological events, photothermal therapy of cancer tumors, 3D printing, machining, and material processing [8–12]. A key advantage of using light, whether for probing or manipulation, is that it can penetrate through the target medium non-invasively at appropriate wavelengths. External optics methods based on a combination of optical lenses, spatial light modulators, and tunable acoustic lenses [13] have been traditionally used to shape light externally before it is launched into the target medium.
A commonly used external optical component is a single lens that can focus light into the medium from outside. Such a lens suffers from a fundamental tradeoff between the spot size of the focused beam and the depth at which it can focus light. Therefore, when focusing a Gaussian optical beam, an external lens can either achieve a long focal length with low spatial resolution or, conversely, high spatial resolution at a shallow depth.
This tradeoff can be overcome by cascading the external lens with another lens or an optical waveguide. The first lens will focus light at a short distance with a small focal spot size, and the second lens or the waveguide will relay that tightly focused beam of light over a longer distance, without compromising the spatial resolution. Unfortunately, in many practical scenarios, it is not desirable or possible to implant optical elements inside the target medium, and therefore, a two-element system cannot be implemented with the first lens outside the medium and the second lens or the relay waveguide inserted into the medium. Moreover, invasive insertion of lenses or waveguides into the medium defeats the purpose of using light as a non-invasive modality for interaction with the medium, especially in non-destructive testing of materials or imaging and stimulation of biological tissue. To alleviate this issue, external lenses with larger apertures can be used to increase the input beam diameter, to keep the spot size small for larger focal depths [14].
We have recently shown that ultrasound waves can be used to guide and pattern the trajectory of light by locally changing the refractive index of the target medium itself. Using this technique, we have demonstrated the possibility of forming in situ virtual gradient-index (GRIN) waveguides, virtual relay lenses, and spatial light modulators non-invasively [15–17]. Since ultrasound in the proper frequency range can propagate at large depths inside the medium with minimal attenuation, virtual optical components can be realized within the target medium to manipulate the trajectory of light without inserting any physical devices disturbing the medium. We have shown that these virtual optical components can be formed in transparent and scattering media such as biological tissue [15]. In this method, the virtual optical component can be reconfigured by simply changing the pattern of ultrasound waves from outside the medium. A nonlinear photoacoustic wave implementation of this idea has also been recently demonstrated for light guiding to deep biological tissue sites [18]. Transversal ultrasound guiding of light deep into scattering media has also been successfully demonstrated using this technique [19]. Ultrasound has also been used to implement acousto-optic modulators and tunable acoustic lenses to realize tunable optical elements. In such devices, ultrasonic waves actively change the local refractive index of a crystal or oil to affect light as it passes through the device [13,20].
In this paper, we demonstrate that the virtual ultrasonically sculpted GRIN waveguides can be employed to relay an externally focused beam of light through the medium, non-invasively, without compromising the spot size.
Our simulation and experimental results suggest that by sculpting the appropriate refractive index pattern within the target transparent medium, externally focused light can be relayed through multiple pitch lengths of a virtual GRIN waveguide while maintaining or even decreasing the focal spot size, thus relaxing the tradeoff between the focal distance and the spot size. Moreover, by translating and reconfiguring the pattern of ultrasonic waves, the beam of light can be confined at different locations within the medium. The spot size can also be tuned. We have also shown that virtual GRIN waveguides can be formed in scattering media to relay an externally focused beam of light through the medium. The overall contrast between the intensity of light at the focal point and the surrounding local background within the scattering medium is enhanced using this technique compared to an external physical lens with the same focal length, suggesting that the distribution of both ballistic and scattered photons is affected by the virtual waveguide. These results inspire the tantalizing notion of designing complex optical systems of multiple virtual optical elements sculpted within the target medium in tandem with external optical components to enable unprecedented control over the trajectory of light within the target medium in a non-invasive way.
2. Virtual ultrasonically sculpted optical waveguides relay an externally focused beam of light
Imagine light, generated by an external source, is to be confined and focused to a point P at a depth d inside a medium with a negligible scattering [Fig. 1(a)]. Since the light source is outside the medium, an external lens can be used to confine and focus light onto the target location inside the medium. To achieve the smallest possible focal spot using a lens, the input light must be collimated [Fig. 1(b)]. Assuming a Gaussian input beam with a diameter D impinging on an external lens with the focal distance f, the focal spot size, characterized by the Gaussian beam waist at the focal plane, can be obtained as:
where 2w is the focused Gaussian beam waist, and λ is the wavelength of light [21]. From Eq. (1), it can be seen that the focal spot size is proportional to the focal distance, f, and is inversely proportional to the input beam diameter, D. The effect of these two parameters is usually expressed in terms of the numerical aperture (NA) of the focusing lens, i.e., NA = n(D/2f), where n is the refractive index of the medium. The spatial resolution of the focused beam of light decreases with increasing the depth of penetration. To overcome this limitation and achieve a smaller focal spot and, hence, a higher spatial resolution at depth, an external lens with a shorter focal distance (i.e., larger NA) can be used to focus light shallower into the medium and, then a second lens or a waveguide can be used inside the medium to relay the tightly focused beam of light to the target point. However, inserting a second lens or waveguide into the medium will disturb the medium and is invasive. Here, we show that a virtual GRIN waveguide can be formed using ultrasound to relay the externally focused beam of light to the target location P non-invasively [Fig. 1(c)]. The external lenses are located at a distance ΔL just outside the medium.
Fig. 1. (a) Schematic illustration of a collimated beam of light from an external source and the target point P inside the medium. (b) An external lens with a long focal distance (f1) is used to focus light to the target location P at a depth d inside the medium. (c) A cascade optical system of an external lens with a short focal distance (f2) and an ultrasonically defined virtual GRIN waveguide is used to relay the focused spot through the medium to the same point P.
We used a commercial software (OpticStudio 20.1, ZEMAX LLC) to perform ray-tracing simulations and calculate the spot size using the Huygens PSF module. First, we simulated an external lens with a long focal distance (LF) of f1 = 50 mm (in air) to directly focus the input collimated beam of light through a non-scattering medium (water) with a refractive index of n = 1.333. Taking into account that the lens has to be placed outside the medium at ΔL = 11.27 mm and also the elongation of the focal distance in water (due to the higher refractive index of water compared to air), the beam was focused at a depth of d = 47.22 mm into the medium. The axial ray paths and the focused beam profile at the focal plane are shown in Fig. 2(a). A radial cross-section of the focused beam is shown in Fig. 2(b). The focused beam size, measured as the full width at half maximum, is FWHM = 12.05 µm. The FWHM is related to the beam waist radius w as FWHM ≈ w / 0.849. Next, a cascade optical system of a short focal distance (SF) external lens with f2 = 31.25 mm (in air) in tandem with an in situ virtual GRIN waveguide was simulated. The GRIN waveguide is assumed to have a cylindrically-symmetric parabolic refractive index profile along the radial direction $r$:
where ${n_0}$ is the background refractive index of the medium, Δn is the refractive index contrast, defined as the difference between the peak refractive index and ${n_0}$, Δn $= {n_{max}} - {n_0}$, and ${r_0}$ is the radius at which the parabolic profile crosses the baseline ${n_0}$. The simulation results in Fig. 2(c) show the ray paths in the cascade optical system, where the beam of light focused by the external lens is projected to a focal spot at the same depth of d = 47.22 mm along the axial direction by a virtual GRIN waveguide with a refractive index contrast of Δn = 2.3×10−3. The distance between the focal point of the external lens and the virtual GRIN waveguide was chosen to be do = 5.03 mm. A radial cross-section of the focused beam at the output plane is shown in Fig. 2(d), with the beam size of FWHM = 4.7 µm, which is much smaller than the beam size focused by the single long focal distance (LF) external lens. The focused beam size can be adjusted by changing the distance do or the ultrasound parameters, such as its frequency and amplitude. The numerical aperture of the virtual waveguide in this example is approximately 0.08.Fig. 2. (a) Optical ray tracing simulation results showing the axial beam paths through the medium and the spot size at the focal plane of a long focal distance lens. (b) The radial cross-section of the focused beam of light at point P. The diameter of the focal spot is FWHM = 12.05 µm. (c) The axial beam paths through the medium and the spot size at the focal plane of the cascade optical system of an external lens and the virtual GRIN waveguide. (d) The radial cross-section of the focused beam of light at point P using the cascade system. The diameter of the focal spot is FWHM = 4.7 µm. In these simulations, the medium is assumed to be water, d = 47.22 mm, ΔL = 11.27 mm, and do = 5.03 mm. The scale bar is 10 µm.
The smallest focal spot size can be achieved when the lens is illuminated with a collimated beam that fills the input aperture [21,22]. Theoretically, as shown in Fig. 3(a), an appropriate single lens or a cascade of lenses can be designed to focus light externally to the desired depth in an optically transparent medium and achieve the desired spot size [13,14,23]. However, this may require a large input aperture and an expanded input beam. The spot size is plotted as a function of depth in the medium in Fig. 3(b) for external single lenses with different focal lengths and different input apertures. The input aperture of the external lens is assumed to be backfilled with a collimated input beam of light, and all the lenses are held at the same distance ΔL = 11.27 mm from the boundary of the target medium. When using external lenses with larger focal distances to reach deeper into the medium, the spot size is increased. For each specific lens, increasing the input beam diameter results in decreasing the spot size. We should note that for an external lens, the spot size is linearly increased by increasing the depth [as shown in Fig. 3(b) and also by Eq. (1)], whereas it is decreased proportionally to 1/D when we increase the input beam diameter, as explained by Eq. (1). This means that compensating for the increase of the spot size by expanding the input beam becomes less effective deeper into the medium, for which larger and larger input beam diameters, i.e., D would be required [Fig. 3(b)]. Moreover, in practice, increasing the input beam size might render the use of large aperture external optics impractical for certain applications.
On the other hand, our technique can be considered as an ‘add-on’ to external optical systems. In other words, our virtual waveguide can relay the beam of light focused by an external optical lens through the medium to a deeper location [Fig. 3(a)] without increasing the spot size. The virtual lens can, theoretically, relay the externally focused beam and maintain the spot size over multiple pitch lengths of the virtual lens, as indicated by the diamond markers along the dashed horizontal line in Fig. 3(b).
Fig. 3. (a) Light is focused at different depths using the cascade optical system composed of an external lens and a virtual GRIN waveguide. External single lenses can also be used to focus light at the same depth through the medium, but the input beam diameter Di must be increased to obtain the same spot size as what can be achieved using the cascade system deep into the medium. (b) The focal spot size, defined as the FWHM of the beam at the focal plane, increases with depth for external lenses (solid lines). By increasing the input beam diameter, the focal spot size is decreased at each focal length. The focal spot size is shown for different input beam diameters of D1=3.83 mm, D2=5.67 mm, D3=7.56 mm, and D4=9.43 mm. Using our cascade optical system, the focal spot of the external input lens focused at a depth of 21.32 mm can be relayed deeper into the medium over multiple pitch lengths of the virtual waveguide (horizontal dashed line).
3. Experimental results
We performed experiments to demonstrate the concept of non-invasive relaying of an externally focused beam of light using the virtual GRIN waveguide both in a transparent and a scattering medium. First, we performed an experiment using an external aspherical lens (49102, Edmund Optics Inc.) with a short focal distance (SF) of f2 = 25 mm (in air) cascaded with a virtual ultrasonically sculpted GRIN waveguide to relay the focused beam of light to a depth of d = 47.1 mm into deionized (DI) water as a transparent medium. The lens was placed at a distance ΔL = 11.27 mm outside a container made of 3 mm thick acrylic filled with DI water [Fig. 4(a)]. Light at the wavelength of λ = 640 nm from a fiber-coupled laser (OBIS LX 640 nm, Coherent Inc.) was collimated using an adjustable fiber collimator (CFC-2X-A, Thorlabs, Inc.) before impinging on the lens, producing an input beam diameter of 0.45 mm. To form the virtual GRIN waveguide in DI water, we used a resonant ultrasonic cylindrical cavity made of a piezoelectric transducer (PZT Type II), with a wall thickness of 1.5 mm (American Piezo Ceramics, Inc.) driven by a 30 V signal at 1.512 MHz. Under these conditions, the numerical aperture of the virtual waveguide is estimated to be NA = 0.014. As shown in Fig. 4(a), the relayed focused beam of light was imaged in the transmission mode using a microscope composed of a zoom lens (VZM 600i, Edmund Optics Inc.) with a NA = 0.08 in air, directly attached to a CCD camera (BFLY-U3-50H5M-C, FLIR Systems). The zoom lens was capped with a clear optical window (WG11050-A, Thorlabs, Inc.), which was immersed in DI water. The image of the relayed confined optical beam is shown in Fig. 4(b). We also used an external aspherical lens (33-945, Edmund Optics Inc.) with a longer focal distance (LF) of f1 = 50 mm (in air) to confine light to the same depth of d = 47.1 mm into the medium. The focused beam of light at the output plane is shown in Fig. 4(c), where we can observe a much larger spot size compared with that of the cascade system. To quantify the difference, the radial cross-sections of the confined beam using the single external lens (i.e., LF) and that of the cascade system of the external lens and the virtual GRIN waveguide (i.e., SF + US) are plotted in Fig. 4(d), where we can see that the spot size in the case of a single external lens (FWHM = 53 µm) is larger than the spot size of the shorter focal distance external lens cascaded with the virtual GRIN waveguide (FWHM = 21 µm). Moreover, the peak intensity of the focused beam using the cascade system is 5.94 times higher than the peak intensity of the beam focused by the single external lens, demonstrating that the tradeoff between the focal distance and the diffraction-limited spot size [Eq. (1)] in conventional lenses is overcome using the cascade optical system. In other words, we can benefit from the strong confinement of the short focal distance external lens to achieve a tighter focus, while non-invasively relaying the focal spot through the medium using the ultrasonically defined virtual GRIN waveguide. This is a unique advantage of using ultrasonically defined virtual GRIN waveguides that enables high-resolution optical access to deep regions of the target medium non-invasively.
Fig. 4. (a) The schematic of the experimental setup showing the cascade system, where a short focal distance (SF) external lens is used to confine light through the medium, and a virtual ultrasonically defined GRIN waveguide is used to non-invasively relay the focused beam of light to a depth d into the medium. The distance between the external lens and the medium is ΔL, and the distance between the focus of the focal plane of the external lens and the virtual GRIN waveguide is do. (b) The confined beam of light at the depth d = 47.1 mm into the medium (DI water) using an aspheric lens with an effective focal distance f = 25 mm (in air) and the virtual GRIN waveguide formed at do = 13.47 mm. (c) The confined beam of light at the depth d = 47.1 mm into the medium (DI water) using a single aspheric lens with a longer effective focal distance (LF) f = 50 mm. (d) Radial cross-sections of the beams for the single external lens and the cascade system. The FWHM of the spot size for a single external lens is 53 µm, while for the cascade system, it is reduced to FWHM = 21 µm. The acquisition parameters were kept the same for both cases.
One of the advantages of the virtual GRIN waveguide is that by changing the shape and location of the ultrasound pattern, we can reconfigure the relaying effect. For example, by forming the virtual waveguide at different distances (do) from the focal plane of the external lens, the location and size of the resulting focal spot can be changed. To demonstrate this reconfigurability, we performed experiments under the same setting, as shown in Fig. 4, using a cascade system composed of the external lens (f2 = 25 mm) and the virtual GRIN waveguide. The distance do was changed by forming the same radially varying ultrasonic interference pattern at different axial distances through the medium. In each case, the imaging system was also moved accordingly to image the focal plane of the cascade system. The cross-sections of the focused beam at different axial distances are plotted in Fig. 5(a). The beam spot size (FWHM) and the peak intensity as a function of the axial distance (do) are plotted in Figs. 5(b) and 5(c), respectively. As the distance do is increased, the beam size is decreased, and the intensity of the focused and relayed beam of light is increased. To understand this trend, we should note that when the distance between the focal plane of the external lens and the virtual GRIN waveguide, which acts as a GRIN lens, is small, the relayed focal spot will be magnified through the virtual GRIN waveguide. For example, for do = 5.47 mm, the relayed focused beam spot size (FWHM = 29.8 µm) is larger than the beam spot size of light focused using the external lens (FWHM = 27 µm), whereas at do = 7.03 mm [interpolated from the experimental results in Fig. 5(b)], the relayed beam spot size matches the spot size of the external lens. By further increasing do, the relayed beam spot size is decreased until it asymptotically approaches the smallest achievable beam spot size of the virtual GRIN waveguide (i.e., FWHM = 15 µm), which was directly measured using the same collimated input beam. This limit is shown as a dashed horizontal line in Fig. 5(b). The peak intensity of light increases as the beam spot size is decreased until its rate of increase starts to slow down after do = 17.47 mm. There are two competing effects that contribute to the change of peak intensity. On the one hand, the beam spot size is decreased, and on the other hand, the coupling efficiency to the virtual GRIN waveguide is decreased as do is increased. This is mainly due to the loss of some photon flux coupled to the waveguide as do is increased, since some photons will not be captured within the acceptance angle of the GRIN waveguide beyond a certain do, due to the divergence of the incoming beam. After this point, while the decrease in the focused beam size tends to enhance the intensity of the confined beam, the loss of photon flux tends to decrease the intensity of light. The competition between these two effects results in a much slower increase in the intensity of light after do > 17.47 mm, as shown in Fig. 5(c). We expect the effect of losing photon flux becomes more dominant compared to the decrease in the spot size for larger distances do, resulting in decreasing light intensity.
Fig. 5. (a) The radial cross-section of the confined beam of light through the cascade system of a short focal distance external lens with f = 25 mm (in air) and the virtual GRIN waveguide for different distances (do) between the focal plane of the external lens and the virtual GRIN waveguide. As the distance do is increased, the spot size is decreased, and the intensity of light is increased. (b) The spot size (FWHM) decreases as the distance do is increased until it asymptotically approaches the smallest achievable spot size of the GRIN waveguide shown by the dashed line. (c) The peak intensity is a function of the distance do. The peak intensity increases as do is increased until its rate of increase starts to slow down beyond do = 17.47 mm. The error bars represent the standard deviations of the measured values in different (n = 4) experiments.
4. Effect of ultrasound parameters on the performance of the virtual relay waveguide
We showed that by changing the distance between the focal plane of the external lens and the virtual waveguide, i.e., do, the spot size and peak intensity can be tuned. The optical properties of the virtual GRIN waveguide will also directly affect the relay mechanism in the cascade system. The intensity, frequency, and pattern of ultrasound determine the optical properties of the virtual GRIN waveguide. The local density is spatially altered in response to ultrasound pressure pattern in the target medium as
where ${\rho _0}$ is the static density of the medium, $P_0$ is the initial pressure, $P(r)$ is the radial pressure generated by ultrasound, and K is the bulk modulus of the medium [24].The change of local density results in the change of local refractive index in the medium. Therefore, a spatial pattern of refractive index is sculpted in the medium because of the ultrasound pressure waves. The induced refractive index contrast is a function of the maximum and minimum local ultrasound pressure in the medium, which depends on the peak electric potential that drives the ultrasonic transducer. The refractive index as a function of the local density can be obtained from the Lorentz-Lorenz relation [25–27] as
where $n(r )$ is the refractive index as a function of radius, A is the molar refractivity of the medium, $\rho (r )$ is the density, and M is the molecular weight. Combining Eqs. (3) and (4), we obtain the explicit nonlinear relationship between the local refractive index of the medium and the ultrasonic pressure asIn the case of our virtual ultrasonic waveguide, the peak ultrasonic pressure change is usually very small and therefore, the refractive index contrast, i.e., Δn can be approximated as a linear function of the pressure difference, i.e., P - P0 as described in one of our previous publications [16]. For the experiments in this paper, we used a cylindrical ultrasonic cavity to generate a cylindrically symmetric refractive index profile in the medium, for which the radial refractive index profile is not a perfect parabolic function as we used in our simulations and, in fact, follows a Bessel profile along the radial direction [17]. The amplitude of this spatial refractive index profile can be approximated as
where ${J_m}$ is the mth-order of Bessel function of the first kind, ${k_r}$ is the radial wavenumber, m is the azimuthal mode number, n0 is the background refractive index, and Δn is the refractive index contrast. This refractive index profile is shown in Fig. 6(a) for two different cases. If the input beam diameter is large enough so that it extends to the first trough of the Bessel function next to the central peak, then the overall refractive index contrast will be even larger and equal to ∼1.4 Δn. In this paper, we are assuming that the input beam diameter is smaller than the central peak of the Bessel function, such that the refractive index contrast is equal to Δn with respect to the background index n0.The input aperture for the virtual waveguide can therefore be calculated from ${J_m}({{k_r}r} )= 0$ as
where $2.40483$ is the first zero-crossing of the zeroth-order Bessel function of the first kind. The radial wavenumber ${k_r}$ is related to ultrasound frequency, f and the speed of ultrasound in the medium, ${c_s}$ as ${k_r} = \frac{{2\pi f}}{{{c_s}}}$ . Therefore, the input aperture can be expressed asFig. 6. (a) Refractive index profile of the virtual waveguide for two different frequencies of ultrasound and two different refractive index contrast (Δn) values. Parabolic profiles (dashed lines) are used in simulations to approximate the Bessel profiles generated by the cylindrical transducers. D1 and D2 indicate the diameters of the input collimated beams used as input for the two virtual waveguides. (b) Simulated ray paths for the parabolic profiles in (a), showing that the larger frequency and Δn result in a shorter pitch. (c) The focal spot size defined as the FWHM decreases nonlinearly with increasing Δn. (d) The frequency determines the width of the refractive index profile and the input aperture for a fixed Δn. (e) The ¼ pitch length (location of the first focus) is a function of frequency and Δn.
By increasing the frequency of ultrasound, the refractive index profile is squeezed and the input aperture is decreased, as shown in Fig. 6(a). Increasing the frequency contributes to reducing the first quarter pitch of the virtual GRIN waveguide, defined as the distance between the input collimated beam and the first focal point along the virtual waveguide. The first quarter pitch of the virtual GRIN waveguide for two frequencies of 3 MHz and 5 MHz is shown in Fig. 6(b).
As discussed earlier, the ultrasound peak pressure affects the refractive index contrast. The diffraction-limited spot size of the virtual waveguide as a function of refractive index contrast Δn is shown in Fig. 6(c). By increasing the refractive index contrast, the spot size is decreased nonlinearly. This behavior can be explained based on the definition of numerical aperture for a GRIN waveguide with a parabolic refractive index profile, i.e., NA=$\sqrt {{{({{n_0} + \Delta n} )}^2} - {n_0}^2} $, which is a nonlinear function of refractive index contrast. Since the focal spot size is inversely proportional to the numerical aperture, we expect the nonlinear behavior shown in Fig. 6(c).
We performed these simulations using the ray-tracing and Huygens PSF modules of the ZEMAX OpticStudio software. In these simulations, the Bessel profiles were approximated by parabolic profiles [dashed curves in Fig. 6(a)], defined by Eq. (2), and the diameter of the input collimated beam D was set to match the distance between the two crossing points of the parabolic profile at ${n_0}$, which is the input aperture of the virtual waveguide.
Increasing the frequency of ultrasound will decrease the input aperture [Fig. 6(d)], as well as the location of the first focus, i.e., the first quarter pitch, as shown in Fig. 6(e). Therefore, the focal spot size along the waveguide remains constant for the parabolic profile with a constant Δn. Higher Δn and higher frequency both result in a larger refractive index gradient along the radial direction [Fig. 6(a)] and, as a result, the quarter pitch length will be decreased [Fig. 6(e)]. Using both of these parameters, one can precisely design the axial location of focal points along the waveguide.
In addition to characterizing the behavior of the cascade optical system in transparent water, we performed experiments in a scattering medium composed of Intralipid 20% mixed with DI water. The performance of the single external lens with a long focal distance (LF) was compared with the performance of the cascade system composed of the shorter focal distance (SF) external lens and the virtual GRIN waveguide in a turbid medium. An external lens with a focal distance of f2 = 25 mm (in air) in tandem with the virtual GRIN waveguide formed by ultrasound waves at the frequency of 1.512 MHz under the same conditions for experiments in DI water was used to focus light at a depth of d = 47.1 mm through the scattering medium. The distance between the focal plane of the external lens and the ultrasonically defined GRIN waveguide was set to do = 13.47 mm. The reduced scattering coefficient of the Intralipid solution was measured as µ′s = 0.93 cm-1 using the Oblique Incidence Reflectometry (OIR) method [28]. Therefore, light is confined through the depth of the scattering medium with an optical thickness of OT = 4.38 TMFP. The confined beam of light at d = 47.1 mm is shown in Fig. 7(a). We also performed an experiment using a single long focal distance f1 = 50 mm (in air) external lens to directly focus light through the scattering medium at the same depth of d = 47.1 mm [Fig. 7(b)]. The radial cross-sections of the focused beams are plotted in Fig. 7(c) both for the case of the single external lens with a long focal distance (LF) and the cascade system (SF + US). The contrast ratio (i.e., the peak to background intensity ratio) for the case of the cascade system is measured to be 2.86, which is more than two times larger than the contrast ratio for the case of the single external lens with a contrast of 1.31. This shows that using the ultrasonic virtual waveguide in the cascade system, we can confine and focus light more effectively through the scattering medium, whereas the multiple scattering events in the medium almost overwhelms the beam of light focused by the single LF external lens. Also, as shown in Fig. 7(c), the beam focused by the cascade system is much narrower than that of the beam focused by the LF single external lens, showing that the same input light can be confined through the scattering medium with a much higher spatial resolution using the virtual ultrasonic GRIN waveguide.
Fig. 7. (a) The confined beam profile of light at the optical depth of 4.38 TMFP into the scattering medium using a cascade system of an aspheric lens with a short focal distance (SF) f = 25 mm (measured in air) and the virtual GRIN waveguide formed at do = 13.47 mm. (b) The confined beam of light using a single aspheric lens with a longer focal distance (LF) f = 50 mm (measured in air). (c) Radial cross-sections of the beams for the single external lens and the cascade system.
We performed another set of experiments in a medium made of a higher concentration of Intralipid with a slightly higher reduced scattering coefficient of µ′s = 1.05 cm-1. The optical beam profiles through the medium with an optical thickness of OT = 4.95 TMFP are shown in Figs. 8(a) and 8(b). The single external lens (LF) cannot focus light to the same depth anymore [Fig. 8(a)] due to the overwhelming effects of scattering in the medium, compared to the focusing power of the external lens, while the cascade system can still confine light through the medium [Fig. 8(b)]. The radial cross-sections of the beams are plotted in Fig. 8(c), where a contrast ratio of 1.6 can be achieved for the cascade system. These results demonstrate the power of the ultrasonic virtual waveguide to effectively confine and relay light through a thick scattering medium with higher contrast and spatial resolution compared to the single external lens.
Fig. 8. (a) The confined beam profile of light at the optical depth of 4.95 TMFP into the scattering medium using a cascade system of an external aspheric lens with a short focal distance (SF) f = 25 mm (measured in air) and the virtual GRIN waveguide formed at do = 13.47 mm. (b) The confined beam of light using a single external aspheric lens with a longer focal distance (LF) f = 50 mm (measured in air). (c) Radial cross-sections of the beams for the single external lens and the cascade system. The brightness of the experimental images (a) and (b) has been increased by 40% for better visualization.
5. Discussion
In this paper, we showed that ultrasonically sculpted virtual GRIN waveguides can be formed deep in a medium, non-invasively, for in situ relaying and focusing of light. In particular, we demonstrated that when the virtual waveguide is used in tandem with an external lens, the focal distance can be extended without increasing the spot size. In this cascade system of a short focal distance external lens and a virtual ultrasonically formed GRIN waveguide, the tightly focused beam of light by the external lens at shallow depth is relayed to a deeper region within the medium.
We should note that a similar relaying effect could be achieved by placing a second physical optical element, such as a GRIN lens or an optical fiber, into the medium to extend the focal distance of the whole optical system without compromising the spot size. However, implanting such a physical optical element would involve disruption of the medium. Another way to achieve the desired focal spot at the desired focal depth into the medium is to design an appropriate external optical system by expanding the collimated input optical beam or forming a diverging input optical beam before it is focused by the external lens into the medium. In such external optical system designs, the focal spot size linearly increases with the focal depth and is inversely proportional to the input beam diameter (D) of the external lens that focuses light into the medium. Therefore, the spot size can be decreased proportionally to 1/D to compensate for the effect of increasing the focal depth. However, compensating for the increase of the spot size by expanding the input beam becomes less effective and more challenging for larger depths into the medium. Moreover, implementing such optical systems with large input apertures might not be practical for some applications.
We have demonstrated that ultrasonically sculpted virtual GRIN waveguides can relay an externally focused beam of light through the medium over multiple pitch lengths of the waveguide without increasing the spot size. In such a cascade optical system, delivering light at depth does not require increasing the input beam diameter to the external lens that focuses light into the medium. The virtual ultrasonically-defined GRIN waveguide can be considered as a flexible and effective add-on to external optical systems, which can be defined into the medium non-invasively.
More complex optical systems, including compound lenses made of a cascade of physical elements, have been designed for microscopy and photography [29,30]. Inspired from these established designs, the method presented in this paper based on ultrasonically formed virtual waveguides can be extended to design even more complex optical systems composed of multiple virtual optical waveguides and external physical elements to shape light within the target medium. A unique advantage of our ultrasonically sculpted optical waveguide is that it can be formed inside the target medium where a physical waveguide or a GRIN lens should not be inserted invasively.
An additional advantage of the presented technique is reconfigurability that opens up new opportunities for in situ non-invasive beam steering and beam forming inside the target medium. We demonstrated that the size and peak intensity of the focused beam of light can be tuned by reconfiguring the ultrasound pattern. This idea can be extended, for example, by combining the cascading method presented in this paper with our previous work on spatial beam forming [17] to split a single externally focused beam of light into multiple focused beams of light deep inside the medium.
The comparison with a single long focal distance external lens to focus light through the medium at the same depth as the cascade system showed that the achieved level of confinement and the peak intensity of the focused beam are much higher when the cascade optical system is used. To ensure a fair comparison, the input beam diameters were kept the same. If the input beam diameter is increased, the spot size is decreased both for the cascade system and the single external lens. However, the behavior of the two systems is different. In the cascade system, the reduced spot size can be relayed intact at any desired depth, whereas in the case of the external optics, the reduced spot size can be projected to a certain depth, beyond which the input beam diameter has to increase even further. Increasing the input beam diameter comes at the cost of higher aberrations at the focal plane. Of course, external physical lenses have been optimized (e.g., aspherical lenses) to minimize aberrations. In the case of the virtual ultrasonically sculpted waveguide discussed in this paper, the aberrations might be more pronounced when the input beam diameter is increased. These aberrations can be compensated by using adaptive optical techniques [31] or by optimizing the ultrasound pattern to sculpt a refractive index profile that minimizes spherical aberrations. Moreover, for specific applications, only a certain range of input beam diameters can be practical. For example, if an array of closely spaced optical channels is used to access the medium (e.g., for mapping or imaging), the size of the input beam for each channel needs to be kept small enough to enable dense integration without interference. For a given input beam diameter, the virtual GRIN waveguide needs to be formed properly such that the coupling of light from the external lens to the GRIN waveguide is optimized. An important advantage of the cascade optical system is that the output beam size and intensity can be tuned by changing the ultrasound pattern, independently of the input beam size, whereas for the single external lens with a long focal distance, the output beam size and peak intensity only depend on the input beam diameter.
We demonstrated that the presented technique can also be used in scattering media to effectively confine and focus light at depth. We showed that using the cascade optical system, light can be confined and focused through a scattering medium (OT > 4 TMFP) with a noticeable contrast ratio of 2.86 between peak intensity and background. Moreover, we showed that the virtual GRIN waveguide can enable confinement of light through a scattering medium (OT = 4.95 TMFP), where the long focal distance external lens fails to confine light due to the overwhelming effect of scattering. Understanding the specific mechanisms that contribute to the observed enhancement of the contrast ratio for the light confined in turbid media using the cascade optical system needs further investigation. One of the possible contributing factors can be the geometrical path difference between the cascade optical system and the long focal distance external lens, which affects the number and distribution of both ballistic and scattered photons that reach the focal point and the surrounding regions. Also, some of the scattered photons can be guided and confined through the virtual GRIN waveguide towards the focal point, potentially contributing to enhancing the overall light throughput at the target location. We have recently developed a simulation tool that can be used to study the effect of ultrasonic guiding of light in a scattering medium [32].
To achieve effective coupling of light to the virtual GRIN waveguide, the numerical aperture of the external lens has to be smaller or equal to the numerical aperture of the virtual waveguide. Moreover, the relayed spot size is determined by the smaller numerical aperture in this cascade system. Higher numerical apertures and smaller spot sizes can be achieved by increasing the peak intensity of ultrasound to achieve a higher refractive index contrast in the medium. However, for some applications, the peak intensity of ultrasound has to be within defined safety limits. In biological applications, the maximum amount of ultrasonic intensity is constrained by certain safety limits, such as Thermal Index (TI) and Mechanical Index (MI) [33]. Therefore, the achievable numerical aperture and the spot size would be limited. In certain scenarios, for example, when focusing a very small spot size at a shallow depth, an external lens might be preferred, whereas in other scenarios involving focusing of light deeper into the medium, the cascade system might be more effective.
Funding
National Science Foundation (NSF) (1730147, 1935849); Carnegie Mellon University James Sprague Presidential Graduate Fellowship (M.G.S.); Sybiel Berkman Foundation (gift).
Disclosures
The authors declare that there are no conflicts of interest.
References
1. R. Horstmeyer, H. Ruan, and C. Yang, “Guidestar-assisted wavefront-shaping methods for focusing light into biological tissue,” Nat. Photonics 9(9), 563–571 (2015). [CrossRef]
2. W. Yang, L. Carrillo-Reid, D. S. Peterka, and R. Yuste, “Two-photon holographic optogenetics of neural circuits (Conference Presentation),” Proc. SPIE 9690, 96902Q (2016). [CrossRef]
3. E. S. Boyden, “A history of optogenetics: the development of tools for controlling brain circuits with light,” F1000 Biol. Rep. 3, 11 (2011). [CrossRef]
4. W. M. Steen and J. Mazumder, Laser Material Processing: Fourth Edition (Springer, London, 2010).
5. C. Schubert, M. C. Van Langeveld, and L. A. Donoso, “Innovations in 3D printing: A 3D overview from optics to organs,” Br. J. Ophthalmol. 98(2), 159–161 (2014). [CrossRef]
6. S. G. Narasimhan, S. K. Nayar, B. Sun, and S. J. Koppal, “Structured light in scattering media,” in Proceedings of the IEEE International Conference on Computer Vision (2005), I, pp. 420–427.
7. S. K. Nayar and S. G. Narasimhan, “Vision in bad weather,” in Proceedings of the IEEE International Conference on Computer Vision (IEEE, 1999), 2, pp. 820–827.
8. K. Deisseroth, “Optogenetics,” Nat. Methods 8(1), 26–29 (2011). [CrossRef]
9. J. B. Vines, J. H. Yoon, N. E. Ryu, D. J. Lim, and H. Park, “Gold nanoparticles for photothermal cancer therapy,” Front. Chem. 7(APR), 167 (2019). [CrossRef]
10. H. Quan, T. Zhang, H. Xu, S. Luo, J. Nie, and X. Zhu, “Photo-curing 3D printing technique and its challenges,” Bioact. Mater. 5(1), 110–115 (2020). [CrossRef]
11. A. K. Dubey and V. Yadava, “Laser beam machining-A review,” Int. J. Mach. Tools Manuf. 48(6), 609–628 (2008). [CrossRef]
12. G. Padmanabham and R. Bathe, “Laser Materials Processing for Industrial Applications,” Proc. Natl. Acad. Sci., India, Sect. A 88(3), 359–374 (2018). [CrossRef]
13. S. Y. Kang, M. Duocastella, and C. B. Arnold, “Variable optical elements for fast focus control,” Nat. Photonics 14(9), 533–542 (2020). [CrossRef]
14. R. Kingslake and R. B. Johnson, Lens Design Fundamentals (Elsevier Inc., 2010).
15. M. Chamanzar, M. G. Scopelliti, J. Bloch, N. Do, M. Huh, J. Iafrati, V. S. Sohal, M.-R. R. Alam, M. M. Maharbiz, D. Seo, J. Iafrati, V. S. Sohal, M.-R. R. Alam, and M. M. Maharbiz, “Ultrasonic sculpting of virtual optical waveguides in tissue,” Nat. Commun. 10(1), 92 (2019). [CrossRef]
16. M. G. Scopelliti and M. Chamanzar, “Ultrasonically sculpted virtual relay lens for in situ microimaging,” Light: Sci. Appl. 8(1), 65 (2019). [CrossRef]
17. Y. Karimi, M. G. Scopelliti, N. Do, M.-R. Alam, and M. Chamanzar, “In situ 3D reconfigurable ultrasonically sculpted optical beam paths,” Opt. Express 27(5), 7249–7265 (2019). [CrossRef]
18. A. Ishijima, U. Yagyu, K. Kitamura, A. Tsukamoto, I. Sakuma, and K. Nakagawa, “Nonlinear photoacoustic waves for light guiding to deep tissue sites,” Opt. Lett. 44(12), 3006 (2019). [CrossRef]
19. M. N. Cherkashin, C. Brenner, G. Schmitz, and M. R. Hofmann, “Transversally travelling ultrasound for light guiding deep into scattering media,” Commun. Phys. 3(1), 1–11 (2020). [CrossRef]
20. T. S. Francis and X. Yang, Introduction to Optical Engineering (Cambridge University Press, 1997).
21. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley-Interscience, 2007).
22. H. Zappe, Fundamentals of Micro-Optics (Cambridge University Press, 2010).
23. M. Duocastella and C. B. Arnold, “Enhanced depth of field laser processing using an ultra-high-speed axial scanner,” Appl. Phys. Lett. 102(6), 061113 (2013). [CrossRef]
24. M. Jelali and A. Kroll, Hydraulic Servo-Systems: Modelling, Identification and Control (Springer Science & Business Media, 2012).
25. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Elsevier, 2013).
26. L. Weiss, A. Tazibt, A. Tidu, and M. Aillerie, “Water density and polarizability deduced from the refractive index determined by interferometric measurements up to 250 MPa,” J. Chem. Phys. 136(12), 124201 (2012). [CrossRef]
27. E. McLeod and C. B. Arnold, “Mechanics and refractive power optimization of tunable acoustic gradient lenses,” J. Appl. Phys. 102(3), 033104 (2007). [CrossRef]
28. L. Wang and S. L. Jacques, “Use of a laser beam with an oblique angle of incidence to measure the reduced scattering coefficient of a turbid medium,” Appl. Opt. 34(13), 2362 (1995). [CrossRef]
29. S. F. Ray and W. C. Gates, “Applied Photographic Optics,” J. Photogr. Sci. 42(4), 140 (1994). [CrossRef]
30. Y. Zhang and H. Gross, “Systematic design of microscope objectives. Part I: System review and analysis,” Adv. Opt. Technol. 8(5), 313–347 (2019). [CrossRef]
31. C. Choi, K. D. Song, S. Kang, J. S. Park, and W. Choi, “Optical imaging featuring both long working distance and high spatial resolution by correcting the aberration of a large aperture lens,” Sci. Rep. 8(1), 1–12 (2018). [CrossRef]
32. A. Pediredla, Y. K. Chalmiani, M. G. Scopelliti, M. Chamanzar, S. Narasimhan, and I. Gkioulekas, “Path tracing estimators for refractive radiative transfer,” ACM Trans. Graph. 39(6), 241 (2020).
33. Z. Izadifar, P. Babyn, and D. Chapman, “Mechanical and Biological Effects of Ultrasound: A Review of Present Knowledge,” Ultrasound Med. Biol. 43(6), 1085–1104 (2017). [CrossRef]
