Non-Hermitian Topological Photonics

Formation of Weyl exceptional rings.

Recently, there has been a great deal of interest in topological systems across many fields due to their ability to realize non-reciprocal transport even in the presence of disorder. Implicit in this statement is the assumption that both the underlying system and possible types of disorder are energy-conserving, i.e. Hermitian. However, one of the most important features of photonic systems is their ability to break Hermiticity through material gain or absorption, and through radiative outcoupling. This feature allows for the creation of lasers, for example, and more recently has enabled photonic systems to realize a variety of more exotic non-Hermitian phenomena, such as exceptional points. Given the potential uses of topologically protected transport in photonic devices, such as in optical interconnects, it is important to explore the consequences of non-Hermitian perturbations on topological systems, both to understand the limits of topological protection in the presence of gain and loss, but also to discover the new phenomena possible at the intersection of these two fields.

In my own work, I have made a few advances in the rapidly expanding field of non-Hermitian topological photonics. First, in collaboration with Prof. Shanhui Fan, I analytically proved that when gain and loss are added to Weyl materials, the Weyl points in the system's spectrum expand into one-dimensional contours of exceptional points, on which the Berry charge of the original Weyl point is exactly preserved. These topologically charged contours of exceptional points are called Weyl exceptional rings, and Fermi arc surface states still connect pairs of oppositely charged rings. Moreover, we then demonstrated that these Weyl exceptional rings allow for a new class of topological phase transition, as the strength of the non-Hermiticity is increased, rings with opposite charge can merge together, annihilating their Berry charge [1].

Bipartite helical array realizing a Weyl exceptional ring.

Then, in collaboration with Prof. Mikael Rechtsman, I experimentally realized the first Weyl exceptional ring in any system, using a helical waveguide array [2]. To add the non-Hermiticity to the system, breaks were added to some of the waveguides, yielding a mechanism for tunable radiative out-coupling. It is also interesting to note that Weyl exceptional rings are the first known distributed source of Berry flux, which runs counter to the usual intuition that sources or sinks of Berry charge act like magnetic monopoles of Berry curvature in the Brillouin zone of a system.

Bound states in the continuum

In an open system, a state is considered to be bound if all of the energy put into it remains spatially localized and does not radiate away to infinity. Conversely, a state for which any initial energy eventually will radiate away is called a resonance. Typically, a sufficient criteria for determining whether or not a state is bound is if its frequency lies within the continuum of frequencies available to the scattering channels; if the frequency lies outside of this continuum, the state is bound. However, it is possible to find radiationless states whose frequencies lie within the continuum of scattering channels. These peculiar modes have been coined bound states in the continuum (BICs), and exist in a range of different physical systems.

Lines of BICs in an embedded photonic crystal slab.

Previously, most studies of BICs focused on creating these bound modes by carefully engineering the device in which the BICs are spatially localized. However, recently in collaboration with Profs. Chia Wei Hsu and Mikael Rechtsman, I discovered that an equally reasonable way to create systems supporting BICs is to design the environment surrounding the device, and leave the device itself unchanged [3]. Moreover, this new methodology allows for some control over the scattering states, which can be used to realize some exotic phenomena. For example, in photonic systems, the scattering channels are usually plane waves in a homogeneous medium, and as such the two polarizations of light are degenerate. But, by designing the environment of the system to be a photonic crystal, this scattering channel degeneracy is broken, enabling us to realize continuous lines of BICs, rather than isolated BICs.

Control over pairs of polarization states

Polarization flow of a complex birefringent material on the Poincare sphere.

As polarization is one of the fundamental properties of light, devices and materials which manipulate light's polarization are important for many applications, such as optical communication networks and sensors. Typically, given a particular input state of polarization, it is always possible to find a set of birefringent components which can rotate that initial state into any desired output polarization state. However, once this transformation is chosen, it completely determines the rotation of any input state. In particular, when using traditional (Hermitian) birefringent materials, this transformation always preserves the interior angle between any two polarization states.

Instead, in collaboration with Prof. Shanhui Fan, I discovered that by using birefringent materials which also possessed gain and loss along different optical axes, we could manipulate pairs of input polarization states independently [4]. These so called complex birefringent materials can realize both packing and separation of polarization states, and may have uses in new multiplexing schemes in optical communications. Moreover, the analytic theory underpinning these devices also shows how the eigenvectors of the dielectric tensor form an axis about which polarization flows on the Poincaré sphere, yielding an insightful method for visualizing these phenomena.

Designing an incoherent light source

Design and experiment of a chaotic cavity laser producing speckle-free output.

A prominent challenge in many optical imaging techniques, such as optical coherence tomography, is finding a light source that is both bright and spatially incoherent to minimize speckle. Traditionally, continuous wave lasers have been unable to fill this role due to speckle, yielding intractable artifacts in the resulting signal. As part of my doctoral thesis, I worked on using a new, semi-classical laser theory, called the steady-state ab initio laser theory (SALT) [5], to design a laser cavity that is optimized for multi-mode, speckle-free, continuous wave emission for use in optical imaging experiments. We accomplished this by choosing a cavity in which the ray dynamics of the lasing modes are chaotic, thus delocalizing the modes and yielding reduced mode competition from spatial hole-burning, while simultaneously producing many passive cavity modes with similar Q-values. Devices built based on this prediction have been fabricated and tested by Prof. Hui Cao's group at Yale, and the experimental results were found to match the theoretical predictions for the optimal cavity [6].

Developing and testing a theory of the quantum laser linewidth

Since the development of the quantum limited laser linewidth due to phase fluctuations from spontaneous emission events by Schawlow and Townes, many corrections have been subsequently discovered. Traditionally, these corrections have all been considered to be mutually independent, and include the Petermann factor, due to the non-orthogonality of the cavity modes, the incomplete inversion factor, and the bad-cavity correction, a linewidth reduction due to dispersion effects. Recently, a new method of calculating the intrinsic laser linewidth in terms of quantum fluctuations around the SALT steady- state was developed [7], which demonstrated that the Petermann factor and bad-cavity corrections to the Schawlow-Townes linewidth are intertwined, and can only be separated in certain limits.

Quantitative verification of our quantum limited laser linewidth theory.

Following on this success, and in collaboration with Dr. Adi Pick and Profs. Steven Johnson and Douglas Stone, we were able to derive a complete analytic theory of the quantum-limited laser linewidth that inherently included all of the previously known corrections, again demonstrating that in general these effects are not simply multiplicative [8]. This new theory also includes the Henry alpha-factor, which is a significant source of noise in semiconductor lasers. To verify this theory, I wrote a finite-difference time-domain (FDTD) code which simulated the Maxwell-Bloch equations coupled to the Langevin noise equations of the gain medium. Using this algorithm, we were able to quantitatively verify our analytical theory without the use of any fitting parameters [9].


[1] A. Cerjan, M. Xiao, L. Yuan, and S. Fan, "Effects of non-Hermitian perturbations on Weyl Hamiltonians with arbitrary topological charges," Phys. Rev. B 97, 075128 (2018).

[2] A. Cerjan, S. Huang, M. Wang, K. P. Chen, Y. D. Chong, and M. C. Rechtsman, "Experimental realization of a Weyl exceptional ring," Nat. Photonics 13, 623 (2019).

[3] A. Cerjan, C. W. Hsu, and M. C. Rechtsman, "Bound States in the Continuum through Environmental Design," Phys. Rev. Lett. 123, 023902 (2019).

[4] A. Cerjan and S. Fan, "Achieving Arbitrary Control over Pairs of Polarization States Using Complex Birefringent Metamaterials," Phys. Rev. Lett. 118, 253902 (2017).

[5] H. E. Türeci, A. D. Stone, and B. Collier, "Self-consistent multimode lasing theory for complex or random lasing media," Phys. Rev. A 74, 043822 (2006).

[6] B. Redding, A. Cerjan, X. Huang, M. L. Lee, A. D. Stone, M. A. Choma, and H. Cao, "Low-Spatial Coherence Electrically-Pumped Semiconductor Laser for Speckle-Free Full-Field Imaging," Proc. Natl. Acad. Sci. USA 112, 1304-1309 (2015).

[7] Y. D. Chong and A. D. Stone, "General linewidth formula for steady-state multimode lasing in arbitrary cavities," Phys. Rev. Lett. 109 063902 (2012).

[8] A. Pick, A. Cerjan, D. Liu, A. W. Rodriguez, A. D. Stone, Y. D. Chong, and S. G. Johnson, "Ab-initio multimode linewidth theory for arbitrary inhomogeneous laser cavities," Phys. Rev. A 91, 063806 (2015).

[9] A. Cerjan, A. Pick, Y. D. Chong, S. G. Johnson, and A. D. Stone, "Quantitative test of general theories of the intrinsic laser linewidth," Opt. Express 23, 28316 (2015).