Being a young researcher, my research interests are continually expanding and evolving. Therefore, the content of this page is not meant to be a restrictive list of topics I am interested in. On the contrary, it is only meant to give a brief overview of my technical background and convey my prior exposure to the subjects listed below.

My research interests overarching the topics listed below are, quantum and classical communication, estimation and detection theories, and the theoretical foundations of optical imaging. In addition, I have great interest in signal representation theory (bases, frames etc.), stochastic process theory and quantum computing.

(Please note that this page is currently undergoing some changes, so fonts and figures will receive an upgrade soon)

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The biphoton source---a source generating a pair of photons in an entangled quantum state---has been utilized in a number of optical imaging applications, such as optical coherence tomography, ghost imaging, holography and lithography. However, a debate continues as to whether the imaging characteristics of such imagers should be attributed to the entanglement property of the photons. This thesis sets out to consider optical imaging in the framework of Gaussian-state light fields, which encompasses thermal light---the common source used in traditional, classical optical imagers--- and biphoton-state light as special instances, thereby permitting a unified treatment of classical and non-classical sources. Within this framework, the fundamental property of the biphotons that leads to an image is the phase-sensitive coherence between the two photon fields, rather than their entanglement per se .

Therefore, coherence theory for phase-sensitive light fields is an indispensable foundation to understanding the characteristics of the images obtained via such sources. Nonetheless, optical coherence theory has almost exclusively been concerned with phase-insensitive correlations, despite decades of theoretical and experimental work on the generation and applications of light with phase-sensitive correlations. We begin in this thesis, to develop such a theory for scalar, classical fields.

Because classical fields may also have phase-sensitive coherence, it is of interest to explore new imaging configurations using classical phase-sensitive light sources, to determine whether advantages, previously thought to be in the sole realm of biphoton imagers, may be achieved without resorting to non-classical sources. We report such a new configuration for optical coherence tomography, which utilizes a phase-conjugate amplifier in conjunction with a Michelson interferometer to detect interference between two classical light fields with a non-zero phase-sensitive cross correlation. This imaging configuration---which we call phase-conjugate optical coherence tomography (PC-OCT)---shares the same factor-of-two axial resolution improvement and cancellation of even-order dispersion terms that are the key features of quantum optical coherence tomography, but without the necessity for non-classical signal and reference beams.

Does light with orbital angular momentum offer fundamental advantages in free-space communication? As part of a seed grant, we studied line-of-sight, free-space (vacuum) propagation with Gaussian transmitter and receiver pupils. A singular-value decomposition of the propagation kernel reveals degeneracy in its eigenvalues. Labeling the eigenvalues in decreasing order with no loss of generality, we find that the eigenvalues are \eta^\ell (\ell=1,2,3...), and the $\ell^{th}$ distinct eigenvalue spans an eigenspace of dimension \ell. An orthonormal basis for the eigenspace associated with this eigenvalue is given by (m,n) Laguerre-Gaussian modes (which have orbital angular momentum) where $m+n+1=\ell$, or by (m,n) Hermite-Gaussian modes (which have no orbital angular momentum), where again $m+n+1=\ell$. Because channel capacity performance is only a function of the power coupling between the transmitter aperture and the receiver aperture (i.e. is only a function of the eigenvalues and not the particular eigenfunctions), we conclude that orbital angular momentum does not play a role in determining the channel capacity for classical information transmission over the free-space channel shown above.

The analytic eigenvalue distribution obtained from this decomposition (with a Gaussian aperture approximation) was used to extend some single-spatial-mode channel capacity analysis into the multiple-spatial-mode regime by a colleague in our group, resulting in power-filling of a terrace-shaped volume.

Most atmospheric optical links are set up to operate in the far-field power transfer regime, in which diffraction spread is the dominant effect on the beam, resulting in very weak power coupling between the transmitter and the receiver. However, it is also possible to establish geometries such that the link operates in the near-field regime, where, in absence of turbulence, it is possible to focus the beam onto the receiver with almost perfect power coupling. Work on performance of near-field atmospheric optical communication systems is scarce in existing literature, perhaps due to increased complexity in prescribed models.

In this work, we obtain error probability bounds for uncoded transmission, when pulse-position modulation or on-off keying in combination with a coherent or direct detection receiver is employed. Furthermore, we derive bounds on the ergodic capacity for coded transmission over fast-fading channels utilizing either local oscillator shot-noise dominated coherent detection (Gaussian noise statistics) or shot-noise limited direct detection (Poisson noise statistics).

We find that the worst-case performance, given the knowledge of the first and second moment of the power coupling between the transmitter and the receiver (which is random due to atmospheric turbulence), has a probability-of-error floor as a function of signal power. This floor can be avoided if further assumptions are made on the distribution of the fading, such as unimodality. On the hand, with coded transmission, the turbulence effects can be mitigated to achieve non-zero ergodic capacity. We note that this capacity can be approached only in the limit where symbols are sent over "channels" that have independent fading coefficients (i.e. fast fading channels).

Shannon's famous channel-capacity theorem tells us the maximum data rate at which one can reliably communicate over a channel for which we have a---possibly statistical---input/output description. This description however, is a classical description of the channel. What if we are trying to communicate classical information (i.e. a set of classical messages) over a channel with a quantum mechanical description? Holevo, Schumaker and Westmoreland answered this question with the infamous HSW theorem.

Let us make the classical/quantum distinction in describing a channel more concrete. Light is a popular physical-layer information carrier in this era. From a classical standpoint, light is simply a function of space and time (this function can be deterministic or stochastic), governed by Maxwell's equations and the constitutive relations of the medium of interest. Bit information is encoded in laser light (coherent fields) via modulation, and is often decoded through coherent or direct detection photodetectors. The noise in the channel then is the noise in the photodetection process plus any additional uncertainty in the received optical field. If we assume the latter to not exit (i.e. if we assume the channel is a pure, deterministic loss, but nothing else), the only noise contribution is from the photodetection event. In the semi-classical interpretation of photodetection, the noise is due to random electron releases from the photosensitive material.

This is a fundamental departure from a quantum mechanical description of light. In quantum mechanics, an optical field in free-space is an operator field of space and time. The state of this field operator, determines the state of the optical field. A channel is described as an input/output map that descibes how this input state of the field is changed to another state at the output. Bit information is encoded in the field by modulating the state of the field and the decoding is done by performing a measurement on the received optical field. In quantum mechanics, measurements are inherently noisy, and the fundamental noise in quantum channels is due to this uncertainty; not due to the operating principles of a particular measurement device (such as a photodetector).

There has been much work on the quantum channel capacity of optical communication channels, using the quantum mechanical description for light (these are often referred to as Bosonic communication channels). It has been shown that for a single spatial mode pure-loss channel, although it is optimal to encode information in classical laser light (coherent-state fields), the optimal measurement at the receiver is not any of the popular coherent or direct detection schemes employed today (and so far we don't really know what the physical realization of the optimal measurement is!).

Unfortunately, for other Bosonic channels of interest, the channel capacity results rely on several "minimum output entropy" conjectures; an open problem in the field of quantum information theory that has attracted a lot of recent attention (including myself).

My contribution to the project in our group has been to show that two minimum output entropy conjectures employed in "Bosonic broadcast channels" are equivalent when all input states are restricted to Gaussian states. The proof relies on Symplectic transformations on the optical fields to decouple different modes and the fact that Von Neumann entropy is invariant to these transformations. A pictorial depiction of the essential equivalence is shown in the figures above.