JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 10, NO. 4, APRIL 1992 527

Correspondence

Comment on Article by Sennaroglu and Pollock

M. Oskar van Deventer and Jos. J. G. M. van der To1

AbstractSennaroglu and Pollock [l] claim nonreciprocal operation of a completely reciprocal device: an “unidirectional” ring laser, which is in conflict with the reciprocity principle. We show that their work is flawed and conclude that their device will operate as a reciprocal bidirectional ring laser.

Sennaroglu and Pollock [I] propose a ring laser with unidirectional operation. They claim to obtain the unidirectional operation by tilting one facet of the reciprocal waveguide ring laser (Fig. 1). As the angle of incidence is now different for clockwise (CW) and counterclockwise (CCW) running waves, there is a difference in CW and CCW reflectivities, which can be large for TM waves at angles between the Brewster angle and the total-reflection angle, as they claim. As the tilt angle is small, the mode-coupling losses are small. Therefore, the roundtrip loss for CCW waves is smaller than for CW waves and the ring laser operation would become unidirectional, instead of reciprocal bidirectional operation.

The claimed operation is obviously in conflict with the principle of reciprocity as it behaves nonreciprocal while containing only reciprocal elements. The flaw is in the assumption that the reflectivity of facets is the same for plane (TEM) waves and guided (E;,) waves. This assumption is not true for normal incidence [2 ] and it is most certainly not true for incidence angles near the Brewster angle or total reflection. Guided waves have a large numerical aperture (NA) so instead of using one angle of incidence we should consider the integral over the complete NA to calculate the effective reflectivity. Because of this integral the effective reflectivity is only a weak function of incidence angle. It is not the strongly increasing function between the Brewster angle and total reflection that is valid for plane waves. Furthermore, it should be stressed that calculations of the reflectivity of these guided waves should not be done on the basis of channel waveguide modes. These have no physical meaning near the branching point. Instead, local normal modes should be used. These complications are completely neglected in [I].

We shall not go into the elaborate calculations on the reflectivity to prove that the real operation is exactly reciprocal. Instead, we show the conflict of the claimed operation with the law of energy conservation. For this we assume ideal behavior, that is, the CW wave is incident at total reflection and the CCW wave at the Brewster angle. As the CCW wave is incident at the Brewster angle, 100% of the light is transmitted through the facet. Conversely, light can he injected through the facet. Light of the CW wave is totally reflected at the facet and added to the injected wave. We take an injected wave and CW wave that have the same wavelength and phase:

E, = E, COS d. Ecw = Ecw c o s d . - ”

The waves add coherently to E, = ( E t +Ecw C O 5 d t ) = E, COS d. For the output power, we find that

Manuscript received September 16, 1991. The authors are with PTT Research Laboratories, 2260 AK kidschendam,

IEEE Log Number 9105887. The Netherlands.

Fig. 1. The waveguide ring laser with tilted output facet favoring unidi- rectional operation. The tilt angle between AB and A’B‘ is 6 and ON is the normal to the facet A‘B‘. The arrows show the incident CW and CCW waves.

The output power Po is larger than the sum of the input powers P, and P ~ w , which is in conflict with the law of energy conservation.

We conclude that the claimed nonreciprocal of the ringlaser by Sennaroglu and Pollock is in conflict with the reciprocity principle and the energy conservation law. The flaw in their work is the assumption that facet reflectivity is the same for guided waves as for plane waves. Elaborate calculations on this reflectivity will undoubtedly reveal the exact reciprocal bidirectional operation of the proposed ring laser.

REFERENCES

A. Sennaroglu and C. R. Pollock, “Unidirectional operation of rectangular dielectric single-mode ring waveguide lasers,” J. Lightwave Technol. , vol. 9, no. 9, pp. 1094-1098, 1991. T. Ikegami, “Reflectivity of mode at facet and oscillation mode in double- heterostructure injection lasers,” J. Quantum Electron., vol. QE-8, no. 6, pp. 470-476, 1972.

Reply to the Comments Written by van Deventer and van der To1

Alphan Sennaroglu and Clifford R. Pollock

In their comments regarding the unidirectional operation of the ring waveguide laser we analyzed [l 1, van Deventer and van der To1 claim that energy would not be conserved. To show this they consider a specific case where the electric fields are assumed perpendicular to the plane containing the device and attempt to show that the output power will exceed the input power. We note, however, that they have completely ignored the boundary conditions on the tangential

Manuscript received November 13, 1991. The authors are with the School of Electrical Engineering, Cornel1 Univer-

IEEE Log Number 9105888. sity, Ithaca, NY 14853.

0733-8724/92$03.00 0 1992 IEEE