THE criticism on the passage quoted from p. 3 of the book by Profs. Harkness and Morley (NATURE, February 23, p. 347) turns on the fact that, in dealing with number divorced from measurement, the authors have used the phrase “an infinity of objects” without an explicit statement of its meaning. I am not sure that I understand the passage in their letter which refers to this point; but it seems to me to imply that the distinction between “finite” and “infinite” is one which does not require definition. This is not the only accepted view. It is not, for instance, the view taken in Herr Dedekind's book, “Was sind und was sollen die Zahlen.” As regards the opening sentences of Chapter xv., the authors have apparently misunderstood the point of my objection. With the usually received definition of convergence of an infinite product, Π(1-αn), if convergent, is different from zero. So far as the passage quoted goes, Π(1-αn) might be zero; and it is therefore not shown to be convergent, if the usual definition of convergence be assumed. As to the passage quoted from p. 232, I must express to the authors my regret for having overlooked the fact that the particular rearrangement, there made use of, has been fully justified in Chapter viii. Whether Log x is or is not, at the beginning of Chapter iv., defined by means of a string and a cone, will be obvious to any one who will read the whole passage (p. 46, line 16, to p. 47, line 9) leading up to the definition.

Theory of Functions

The ability to conduct experiments at length scales and temperatures at which interesting and potentially useful quantum-mechanical phenomena emerge in condensed-matter or atomic systems is now commonplace. In optics, though, the weakness with which photons interact with each other makes exploring such behaviour more difficult. Here we describe an optical system that exhibits strongly correlated dynamics on a mesoscopic scale. By adding photons to a two-dimensional array of coupled optical cavities each containing a single two-level atom in the photon-blockade regime, we form dressed states, or polaritons, that are both long-lived and strongly interacting. Our results predict that at zero temperature the system will undergo a characteristic Mott insulator (excitations localized on each site) to superfluid (excitations delocalized across the lattice) quantum phase transition. Moreover, the ability to couple light to and from individual cavities of this system could be useful in the realization of tuneable quantum simulators and other quantum-mechanical devices.

Quantum phase transitions of light

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After over two decades of exploration, tunable diode lasers are beginning to find significant applications, driven largely by the huge demand for bandwidth that is guiding many developments in the optical fiber communication business today. In the paper, some of the history and key developments that have led to the technologies available today are reviewed from the perspective of the author. After discussion of some of the early work, the focus shifts to widely tunable diode lasers, which would appear to be key enablers for future dense wavelength-division multiplexing and optical switching and networking systems. The distinguishing characteristics of the current technological alternatives are summarized.

Monolithic tunable diode lasers

Semiconductor amplifiers and lasers with tapered gain regions provide the highest-brightness all-semiconductor sources of near infrared continuous wave (CW) power and have many useful applications. Simple models of these devices are described. Their applications, device performance and brightness limitations are reviewed1.

Semiconductor amplifiers and lasers with tapered gain regions

A semiconductor optical integrated circuit for wave division multiplexing has a semiconductor waveguide layer, a succession of diffraction grating points in the waveguide layer along a predetermined diffraction grating contour, a semiconductor diode array in the waveguide layer having plural optical ports facing the succession of diffraction grating points along a first direction, respective semiconductor diodes in the array corresponding to respective ones of a predetermined succession of wavelengths, an optical fiber having one end thereof terminated at the waveguide layer, the one end of the optical fiber facing the succession of diffraction grating points along a second direction, wherein the diffraction grating points are spatially distributed along the predetermined contour in such a manner that the succession of diffraction grating points diffracts light of respective ones of the succession of wavelengths between the one end of the optical fiber and corresponding ones of the optical ports.

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Wavelength-division multiplexed optical integrated circuit with vertical diffraction grating

The lateral modes of broad area lasers are investigated theoretically. The nonlinear interaction between optical field and effective refractive index leads to a saturable nonlinearity in the governing field equation, so that self-modulated solutions are found to be stable with increased current injection above saturation intensity. We derive approximate analytical solutions for traveling wave fields within the broad area laser. The field amplitude consists of a small ripple superimposed on a large dc value. Matching fields at the boundary determines the modulation depth and imparts an overall phase curvature to the traveling wave mode. There are multiple lateral modes for a given set of operating conditions, and modes with successively more lobes in the ripple have greater overall phase curvature. In contrast to the linear problem, several lateral modes can achieve the same modal gain, for a given injected current density, by saturating the gain to different extent. Thus, these modes would exhibit slightly different optical powers.

Self-stabilized nonlinear lateral modes of broad area lasers

GaAs heterostructure lasers with unstable resonator cavities were demonstrated for the first time with both curved mirrors fabricated by etching. Typical output powers of 0.35 W were observed in a stable, highly coherent lateral mode. The laser operated stably in a single longitudinal mode over a large range of injection currents. The external quantum efficiency was 70% of that of a similar laser with both mirror facets cleaved implying good output coupling of the energy from the entire region.

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Unstable resonator cavity semiconductor lasers

Coupled-mode rate equations for coupled-cavity lasers are derived using a novel approach. The method, based on the Mittag-Leffler theorem, is exact. The coupling coefficients are compared to those derived by several different approximations. >

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An exact formulation of coupled-mode theory for coupled-cavity lasers

We analyze the threshold behavior of a pair of laterally coupled semiconductor lasers of different lengths. The predictions include longitudinal mode selectivity leading to single longitudinal mode operation with a periodicity determined by the length mismatch, and ripples in the equipower curves in the current plane due to carrier-induced index shifts. We present experimental measurements that confirm these predictions.

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Robert J. Lang

Papers

Wavelength-division multiplexed optical integrated circuit with vertical diffraction grating

Self-stabilized nonlinear lateral modes of broad area lasers

Unstable resonator cavity semiconductor lasers

An exact formulation of coupled-mode theory for coupled-cavity lasers

Laterally coupled-cavity semiconductor lasers