Ultrasound-mediated Optical Imaging and Focusing in Scattering Media

Because of its non-ionizing and molecular sensing nature, light has been an attractive tool in biomedicine. Scanning an optical focus allows not only high-resolution imaging but also manipulation and therapy. However, due to multiple photon scattering events, conventional optical focusing using an ordinary lens is limited to shallow depths of one transport mean free path (lt\u27), which corresponds to approximately 1 mm in human tissue. To overcome this limitation, ultrasonic modulation (or encoding) of diffuse light inside scattering media has enabled us to develop both deep-tissue optical imaging and focusing techniques, namely, ultrasound-modulated optical tomography (UOT) and time-reversed ultrasonically encoded (TRUE) optical focusing. While UOT measures the power of the encoded light to obtain an image, TRUE focusing generates a time-reversed (or phase-conjugated) copy of the encoded light, using a phase-conjugate mirror to focus light inside scattering media beyond 1 lt\u27. However, despite extensive progress in both UOT and TRUE focusing, the low signal-to-noise ratio in encoded-light detection remains a challenge to meeting both the speed and depth requirements for in vivo applications. This dissertation describes technological advancements of both UOT and TRUE focusing, in terms of their signal detection sensitivities, operational depths, and operational speeds. The first part of this dissertation describes sensitivity improvements of encoded-light detection in UOT, achieved by using a large area (∼5 cm × 5 cm) photorefractive polymer. The photorefractive polymer allowed us to improve the detection etendue by more than 10 times that of previous detection schemes. It has enabled us to resolve absorbing objects embedded inside diffused media thicker than 80 lt\u27, using moderate light power and short ultrasound pulses. The second part of this dissertation describes energy enhancement and fluorescent excitation using TRUE focusing in turbid media, using photorefractive materials as the phase-conjugate mirrors. By using a large-area photorefractive polymer as the phase-conjugate mirror, we boosted the focused optical energy by ~40 times over the output of a previously used photorefractive Bi12SiO20 crystal. Furthermore, using both a photorefractive polymer and a Bi12SiO20 crystal as the phase-conjugate mirrors, we show direct visualization and dynamic control of TRUE focus, and demonstrate fluorescence imaging in a thick turbid medium. The last part of this dissertation describes improvements in the scanning speed of a TRUE focus, using digital phase-conjugate mirrors in both transmission and reflection modes. By employing a multiplex recording of ultrasonically encoded wavefronts in transmission mode, we have accelerated the generation of multiple TRUE foci, using frequency sweeping of both ultrasound and light. With this technique, we obtained a 2-D image of a fluorescent target centered inside a turbid sample having a thickness of 2.4 lt\u27. Also, by gradually moving the focal position in reflection mode, we show that the TRUE focal intensity is improved, and can be continuously scanned to image fluorescent targets in a shorter time

Jordan totient quotients

The Jordan totient $J_k(n)$ can be defined by $J_k(n)=n^k\prod_{p\mid n}(1-p^{-k})$. In this paper, we study the average behavior of fractions $P/Q$ of two products $P$ and $Q$ of Jordan totients, which we call Jordan totient quotients. To this end, we describe two general and ready-to-use methods that allow one to deal with a larger class of totient functions. The first one is elementary and the second one uses an advanced method due to Balakrishnan and P\'etermann. As an application, we determine the average behavior of the Jordan totient quotient, the $k^{th}$ normalized derivative of the $n^{th}$ cyclotomic polynomial $\Phi_n(z)$ at $z=1$, the second normalized derivative of the $n^{th}$ cyclotomic polynomial $\Phi_n(z)$ at $z=-1$, and the average order of the Schwarzian derivative of $\Phi_n(z)$ at $z=1$.Comment: 16 page