Breaking the tradeoff between confinement and focal distance using virtual ultrasonic optical waveguides

Conventional optical lenses have been used to focus light from outside without disturbing the medium. The focused spot size is proportional to the focal distance in a conventional lens, resulting in a fundamental tradeoff between depth of penetration in the target medium and spatial resolution. We have shown that virtual ultrasonically sculpted graded-index (GRIN) optical waveguides can be formed in the target medium to guide and steer light without disturbing the medium. Here, we demonstrate that such virtual waveguides can relay and tune an externally focused beam of light through the medium without compromising the spot size, thus breaking the tradeoff between the depth of penetration and spatial resolution in conventional physical lenses. We show that the virtual GRIN waveguides can be formed in transparent as well as in turbid media to provide an unprecedented enhancement of spot size and contrast ratio of the confined beam of light. This method can be used to realize more complex optical systems of external physical lenses and in situ virtual waveguides to extend the reach and flexibility of optical methods.


Introduction:
Light-matter interaction has been used in many different applications, ranging from biological imaging and manipulation to metrology, material processing, and machine vision 1-7 . Through the interaction of light with matter, a signature of the medium may affect light, which can be used for sensing, detection, or 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. A key advantage of using light, whether for probing or manipulation, is that it can penetrate through materials non-invasively at the appropriate wavelength range. Different optical components such as lenses, spatial light modulators and waveguides have been used to shape the trajectory of light. Tunable external optical components such as optical modulators, tunable lenses and gratings have also been realized based on electro-optic or acousto-optic effects to reconfigure the pattern of light before it is launched into the target medium [8][9][10][11] . The ability to manipulate the trajectory of light within the target medium would expand the power and flexibility of optical methods. Invasive insertion of traditional optical components such as lenses or waveguides into the medium defeats the purpose of using light as a non-invasive modality for interaction with the medium, especially in the case of non-destructive testing of materials or imaging and stimulation of biological tissue. 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 medium. Using this technique, we have demonstrated the possibility of forming in situ virtual gradedindex (GRIN) waveguides, virtual relay lenses, and spatial light modulators non-invasively [12][13][14] . Since ultrasound in the proper frequency range can propagate deep with minimal attenuation in the medium, the virtual optical components can be realized within the target medium to manipulate the trajectory of light without implanting any physical devices to disturb the medium. We have shown that these linear virtual optical components can be formed in transparent as well as scattering media such as biological tissue 12 . 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 15 . Transversal ultrasound guiding of light deep into scattering media has also been successfully demonstrated using this technique 16 .
Here, we demonstrate that virtual ultrasonically sculpted GRIN waveguides can be used to relay an externally focused beam of light through the medium non-invasively without compromising the spot size. Our simulation results suggest that by sculpting the appropriate refractive index pattern within the target medium, light focused by an external lens can be relayed through multiple pitch lengths of a GRIN waveguide, while maintaining or even decreasing the focal spot size. Therefore, the tradeoff between the focal distance and the focal spot size in traditional lenses is broken using this method. Moreover, by translating the pattern of ultrasonic waves, one can relay the beam of light to different locations within the medium and tune the confined beam spot size. We show that virtual GRIN waveguides can also be formed in scattering media to relay an externally focused beam of light through the medium. Using this technique, the overall contrast between the focal region and the surrounding area at the target location deep within the scattering medium is enhanced, suggesting that the distribution of ballistic and scattered photons will be affected. 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 devices to enable unprecedented control over the trajectory of light within the target medium in a non-invasive way.
Breaking the tradeoff between the spot size and the focal distance using ultrasonically sculpted virtual GRIN optical waveguides Imagine light generated by an external source is supposed to be confined and focused to a point P at a depth d inside a medium with a negligible level of scattering (Fig. 1a). Since the light source is outside the medium, an external lens can be used to confine and focus light to the target location inside the medium. To achieve the smallest possible focal spot using a lens, the input light must be collimated (Fig. 1b). Given the input beam diameter D and the focal distance f, which should be slightly larger than d, i.e., f = d+L so that the external lens can be placed just outside the medium, the focal spot size, characterized by the beam waist at the focal plane assuming negligible aberrations, can be obtained as 17

=
, (Eq.1) where 2w is the focused beam diameter, defined as the full width at half maximum (FWHM) of the beam, and λ is the wavelength of light. The focal spot size is proportional to the focal distance and therefore, there is a tradeoff between the depth of penetration and the spatial resolution of the focused beam of light. To break this tradeoff and achieve a smaller focal spot and, hence, a higher spatial resolution at depth, an external lens with a shorter focal distance can be used to focus light shallower into the medium and, then a virtual GRIN waveguide can be formed using ultrasound to relay the focused beam of light to the target location P (Fig. 1c). 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 performed numerical simulations using a ray tracing analysis in ZEMAX OpticStudio. 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. Given a L = 11.27mm to place the lens outside the medium and the elongation of the focal distance in water, the beam was focused at the depth d = 47.22mm. The axial ray paths and the focused beam profile at the focal plane are shown in Fig. 2a. A radial cross section of the focused beam is shown in Fig. 2b. The focused beam diameter was measured as 2w = 55.34μm. Next, a cascade optical system of a short focal distance (SF) external lens with f2 = 25 mm (in air) in tandem with an in situ ultrasonically sculpted virtual GRIN waveguide was simulated. The in situ ultrasonically sculpted virtual GRIN waveguide was defined by a parabolic refractive index profile along the radial direction with a maximum refractive index contrast of Δn = 6×10 -4 . The simulation results in Fig. 2c show the ray paths along the axial direction in the cascade optical system, where the beam of light focused by the external lens is relayed along the axial direction by the virtual GRIN waveguide to the same depth d = 47.22mm, for which the distance between the focal point of the external lens and the virtual GRIN lens, i.e., do had to be do =13.47mm. A radial cross section of the focused beam at the output plane is shown in Fig. 2d, where the beam diameter is 2w = 9.1μm, which is much smaller than the beam diameter focused by the single (LF) external lens. The focused beam diameter can be adjusted by changing the parameters of ultrasound such as its frequency, amplitude or the distance do. The beam diameter of the focused beam is 2w= 55.34μm. 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 beam diameter of the focused beam is 2w= 9.1μm. In these simulations, the medium is assumed to be water, d = 47.22mm, L = 11.27mm, and do = 13.47mm.

Experiments:
We performed experiments to demonstrate the concept of non-invasively relaying an externally focused beam of light using the virtual GRIN waveguide both in a transparent and also in 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 L = 11.27 mm outside a container made of 3 mm thick acrylic filled with DI water (Fig. 3a). Light at the wavelength of  = 640 nm from a fiber-coupled laser (OBIS LX 640nm, Coherent Inc.) was collimated using an adjustable fiber collimator (CFC-2X-A, Thorlabs, Inc.) before impinging on the lens. As shown in Fig. 3a, the focused beam of light was imaged in the transmission mode using a microscope composed of a zoom lens (VZM 600i, Edmund Optics Inc.) 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.) and was immersed in DI water. To form the virtual GRIN waveguide in DI water, we used a cylindrical piezoelectric transducer made of PZT Type II, with a thickness = 1.5mm (American Piezo Ceramics, Inc.) driven by a 30V signal at 1.512 MHz. The relayed confined optical beam is shown in Fig. 3b. 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 = 47.1mm into the medium. The focused beam of light at the output plane is shown in Fig. 3c, where we can observe a much larger spot size compared with the cascade system. To quantify the difference, the radial cross sections of the confined beam using the single external lens (i.e., LF) and the cascade system of the external lens and the virtual GRIN waveguide (i.e., SF+US) are plotted in Fig. 3d, where we can see that the beam diameter in the case of a single external lens (FWHM = 53 µm) is larger than the beam diameter of the shorter focal distance external lens cascaded with the virtual GRIN waveguide (FWHM = 20 µm). Moreover, the peak intensity of the focused beam using the cascade system is 6.98 times higher than the peak intensity of the beam focused by the single external lens, demonstrating that the trade-off between the focal distance and the diffraction-limited spot size (Eq. 1) in conventional lenses is broken 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. 3 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 L 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 = 13.47mm. 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 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 function. For example, by forming the virtual waveguide at different distances (do) from the focal plane of the external lens, the location and the resulting focused beam diameter can be changed. To demonstrate this reconfigurability, we performed experiments under the same setting as shown in Fig. 3 using a cascade system, composed of the external lens (f2 = 25mm) 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. The cross sections of the focused beam of light are plotted at different axial distances in Fig. 4a. The beam diameter (FWHM) and the peak intensity as a function of the axial distance (do) are plotted in Fig.4b and 4c respectively. As the distance do is increased, the beam diameter 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 lens. For example, for do = 5.47mm, the relayed focused beam diameter (FWHM = 29.8µm) is larger than the beam diameter of light focused using the external lens (FWHM = 27µm), whereas at do = 7.03 mm (interpolated from experimental results in Fig. 4b), the relayed beam diameter matches the beam diameter of the external lens. By further increasing do, the relayed beam diameter is decreased, until it asymptotically approaches the diffraction limited beam diameter of the virtual GRIN lens (i.e., FWHM = 15µm), which was directly measured using the same collimated input beam. This limit is shown as a dashed horizontal line in Fig. 4b. The peak intensity of light increases as the beam diameter is decreased until its rate of increase starts to slow down after do = 17.47mm. There are two competing effects that contribute to the change of output intensity. On one hand, the beam diameter 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 input 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 > 17.47mm as shown in Fig. 4c. We expect the effect of losing photon flux becomes dominant compared to the decrease in the beam diameter for larger distances do, resulting in a decreasing the light intensity. Fig. 4 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 beam diameter is decreased and the intensity of light is increased. b) The beam diameter decreases as the distance do is increased until it asymptotically approaches the diffraction limited spot size of the GRIN waveguide shown by the dashed line. c) The peak intensity as 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 = 17.47mm. The error bars represent the standard deviations of the measured values in different (n=4) experiments.
We also performed experiments in a scattering medium composed of Intralipid 20% mixed in DI water to compare the performance of the single external lens with a long focal distance (LF) and the cascade system composed of the shorter focal distance (SF) external lens and the virtual GRIN waveguide. 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 = 47.1mm through the scattering medium. The distance between the focal plane of the external lens and the ultrasonically defined GRIN waveguide was set to do = 13.47mm. The reduced scattering coefficient of the Intralipid solution was measured as µ′s = 0.93 cm -1 using the Oblique Incidence Reflectometry (OIR) method 18 . 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 = 47.1mm is shown in Fig.  5a. 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 = 47.1mm (Fig. 5b). The radial cross sections of the focused beams are plotted for both cases of the single external lens with a long focal distance (LF) and the cascade system (SF+US) in Fig. 5c. The contrast ratio (i.e., the peak to background 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 the ultrasonic virtual waveguide 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 LF single external lens. The central peak focused by the cascade system is much narrower than the beam focused by the LF single external lens, showing that light can be confined through the scattering medium with a much higher spatial resolution using the virtual ultrasonic GRIN waveguide. We also performed another set of similar experiments in a medium with a slightly higher scattering coefficient of µ′s = 1.05 cm -1 made of Intralipid solution at a slightly higher concentration. The optical beam profiles through the medium with an optical thickness of OT = 4.95 TMFP are shown in Fig. 6a and 6b. The single external lens (LF) cannot focus light anymore (Fig. 6a) due to the overwhelming effects of scattering through the medium, while the cascade system can still focus light through the medium (Fig. 6b).
The radial cross sections of the beams are plotted in Fig. 6c, 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.

Discussion and Conclusion:
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 manipulation of light.
In particular, we demonstrated that when the virtual waveguide is used in tandem with an external lens, the tradeoff between the focal distance and the spot size in conventional lenses will be broken. 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 a shallow depth is relayed to a deeper region within the medium. Compound lenses consisting of a cascade of physical elements have been designed for microscopy and photography 19,20 . Inspiring from these established designs, one can use the method presented in this paper based on ultrasonically formed virtual waveguides, to design complex optical systems composed of a series of virtual optical waveguides to shape light within the target medium. A unique advantage of our ultrasonically sculpted optical waveguide is that it can be formed where a physical waveguide or a GRIN lens cannot be placed non-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 the peak intensity of the focused beam of light can be tuned by reconfiguring the pattern of ultrasound. This idea can be extended for example, by combining the cascading method presented in this paper with our former work on spatial beam forming 14 , one can imagine to split a single externally focused beam of light into multiple focused beams 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 diameter was 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 at the expense of increasing aberrations, which might be compensated by using adaptive optical techniques 21 . Moreover, for each specific application, 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 small 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 diameter, whereas for the single external lens with 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 rather thick scattering medium (OT > 4.38 TMFP) with a noticeable contrast ratio of 2.86 between the confined beam and the 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 of the confined light through 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 the enhancement of the optical contrast at the target location.
We expect that the observed unprecedented advantage over conventional external optics for non-invasive manipulation of light in the target medium with high contrast and high spatial resolution will enable a plethora of applications involving light delivery and potentially optical imaging.