WO1991010273A1

WO1991010273A1 - Unidirectional, monolithic planar ring laser with birefringence

Info

Publication number: WO1991010273A1
Application number: PCT/US1990/007452
Authority: WO
Inventors: Alan C. Nilsson; Robert L. Byer
Original assignee: The Board Of Trustees Of The Leland Stanford Junior University
Priority date: 1989-12-28
Filing date: 1990-12-17
Publication date: 1991-07-11

Abstract

The present invention provides a means of inducing unidirectional oscillation in monolithic ring lasers in which the light path is planar. The intracavity optical diode that enforces unidirectional oscillation in the planar ring oscillator (100) is achieved by a combination of the nonreciprocal Faraday effect, a linear birefringence effect in which the principal axes of the birefringence are not parallel and perpendicular to the plane of propagation of the ring light path, and one or more partial polarizer effects. The present invention enables experimental optimization of polarization transformations within a monolithic planar ring oscillator (100) and also provides a means of tuning the frequency of the planar ring oscillator. The present invention makes it possible to vary both the reciprocal and nonreciprocal polarization transformations used to produce unidirectional oscillation.

Description

UNIDIRECTIONAL, MONOLITHIC PLANAR RING LASER WITH

BIREFRINGENCE

INTRODUCTION

ACKNOWLEDGEMENTS

This invention was supported in part by grants from NASA (Grant No. NAG-1-839). The U.S. government may have rights in this invention.

Field of the Invention

The present invention relates to solid-state ring lasers that are longitudinally pumped by one or more laser pump sources. More specifically, the present invention relates to longitudinally-pumped solid-state ring lasers in which unidirectional oscillation can be achieved. The present invention provides a means of achieving unidirectional oscillation in a monolithic resonator in which the light path is planar. The present invention also provides a means of tuning the frequency of the unidirectional ring laser.

Definitions

Ring light path: a closed polygonal path defined by the central ray of a beam of light. If all of the legs of the polygon lie within a single plane, the ring light path is planar. If the legs of the polygon do not all lie within a single plane, the ring light path is nonplanar. Ring laser: a laser in which the light path internal to the laser resonator is a ring light path.

Monolithic ring laser: a laser in which the resonator is made from a single piece of solid-state laser material, shaped such that laser oscillation occurs along an internal ring light path. The solid-state laser material may have one or more optical coatings applied to its external surfaces.

Birefringence: a lossless optical medium exhibiting optical phase anisotropy for propagation in a certain direction within the medium is said to have pure birefringence. The birefringence associated with propagation in a certain direction through an optically anisotropic medium is classifed as elliptical, circular, or linear according to the eigenstates of polarization for propagation in that direction. The most general pure birefringence is elliptical, which means that the eigenstates of polarization are orthogonal, elliptical states of polarization. Two special cases of elliptical birefringence are circular and linear birefringence. The eigenstates of polarization for circular birefringence are right and left circular polarization states. The eigenstates of polarization for linear birefringence are orthogonal, linearly polarized states. Birefringence can be intrinsic to an optical medium, or it may be induced in a medium by the action of some external influence such as a magnetic field, an electric field, or an applied stress.

Magneto-optic effects: polarization influencing effects that depend on the presence of a magnetic field within a medium. The magneto-optic polarization effects of interest here include effects associated with magnetic circular birefringence (Faraday rotation), and effects associated with reflection from magnetic materials (magnetic Kerr effects). Magnetic circular birefringence can be intrinsic to a material or may be induced in an initially optically isotropic medium (an amorphous material such as glass, or a cubic crystal such as YAG) by application of a magnetic field. Magnetic Kerr effects of interest include the polar Kerr effect, which is the reflection analog of Faraday rotation, and the transverse Kerr effect, for which the eigenstates of polarization are linear polarization states parallel and perpendicular to the plane of incidence. Note: The polarization effects associated with the simultaneous presence of a linear birefringence and magnetic circular birefringence in a medium are treated within the text of this application.

Partial polarizer: an optical component that exhibits pure amplitude anisotropy for its eigenstates. Again, the classification of partial polarizers proceeds according to their eigenstates. Thus, an elliptical partial polarizer has as its eigenstates two orthogonal, elliptical states of polarization that suffer different attenuations in the interaction with the elliptical partial polarizer. A circular partial polarizer is a special case of an elliptical partial polarizer having circular polarization states as its eigenpolarizations. A linear partial polarizer has orthogonal, linear polarization eigenstates that suffer different attenuations in the interaction with the linear partial polarizer. A common realization of a linear partial polarizer, particularly relevant for ring lasers, is oblique reflection from or transmission through an interface between different optically isotropic media.

Reciprocal polarization transformation: a polarization influence (in a fixed system) that does not rely on a magnetic field to produce the effect is said to induce a reciprocal polarization transformation. Examples of reciprocal polarization transformations include optical activity, propagation in media exhibiting birefringence that does not involve magneto-optic effects, and interaction with partial polarizers that do not involve magneto-optic effects.

Nonreciprocal polarization transformation: a polarization influence (in a fixed system) that involves a magnetic field to produce the effect. Examples of nonreciprocal polarization transformations include Faraday rotation (magnetic circular birefringence) and other magneto-optic effects.

(Intracavity) optical diode: an optical system that has different minimal attenuations for the two directions of a propagation through it is called an optical diode. For use internal to a ring laser cavity, an optical diode is created by a proper combination of three effects: a reciprocal polarization transformation, a nonreciprocal polarization transformation, and a partial polarizer.

Unidirectional ring laser: a ring laser in which a biasing influence is used to cause the laser to oscillate in only one of the two possible directions of propagation around the ring. For the purpose of discussion here, the biasing influence is taken to be an intracavity optical diode. The laser will oscillate in the direction of the intracavity optical diode having the smaller minimal attenuation.

Description of the Prior Art The prior art has presented the advantages of longitudinal-pumping of solid-state lasers. Longitudinal pumping of rare-earth-doped glasses and crystals by ion lasers, dye lasers, and diode lasers has been described in Fan et al., "Diode laser-pumped solid-state lasers," IEEE J. Quantum Electron., QE-24, 895-912 (1988); Kozlovsky et al., "Diode-pumped continuous-wave Nd:glass laser," Opt. Lett. 11, 788-790 (1986); Xing et al., "Thermal shifts of the spectral lines in the ⁴F_3/2 to ⁴I_11/2 manifold of an Nd:YAG laser," IEEE J. Quantum Electron., QE-24, 1829-1832 (1988); and Rea, Jr. et al., "Single Frequency, Unidirectional, Monolithic Nd: glass nonplanar ring laser," Conference on Lasers and Electro-optics, 1989 Technical Digest Series, vol. 11 (Optical Society of America, Washington, D.C. 1989) paper WH4, pp. 222-225. Argon-ion laser pumping of Ti:sapphire has also been discussed in Alfrey, "Modeling of longitudinally pumped CW Ti: sapphire laser oscillators," IEEE J. Quantum Electron., QE-25, 760-766 (1989). Particularly when the pump source is a properly selected semiconductor diode laser or diode laser array, longitudinal pumping of solid-state lasers is efficient and can yield narrow linewidth operation of the solid-state laser. See Zhou et al., "Efficient, frequency-stable diode laser-pumpedNd:YAG laser," Opt. Lett. 10, 62-64 (1985) and Kane, Nilsson, and Byer, "Frequency stability and offset locking of a laser-diode-pumped Nd:YAG monolithic nonplanar ring oscillator," Opt. Lett., 12:175-177 (1987).

The advantages of unidirectional ring lasers for overcoming spatial hole burning and permitting single-frequency operation at high output power levels in homogeneously broadened laser media have long been appreciated ( see, for example, A. R. Clobes, and M. J. Brienza, "Single-frequency traveling-wave Nd:YAG laser," Appl. Phys. Lett., 21:265-267 (1972). Recent publications such as T. J. Kane and R. L. Byer,

"Monolithic unidirectional singla-mode Nd:YAG ring laser," Opt. Lett., 10: 65-67 (1985), Kane, Nilsson, and Byer (cited above); W. R. Trutna, Jr., D. K. Donald, and M. Nazarathy, "Unidirectional diode-laser-pumped Nd:YAG ring laser with a small magnetic field," Opt. Lett., 12:248-250 (1987); Nilsson, Gustafson, and Byer, "Eigenpolarization theory of monolithic nonplanar ring oscillators," IEEE J. Quantum Electron. 25, 767-790 (1989); and Rea, Jr. et al. have presented the advantages of diode-laser-pumped monolithic nonplanar ring oscillators made from rare-earth-doped garnet crystals and rare-earth-doped laser glasses with regard to efficiency, single-mode operation, frequency stability, output power, and resistance to the deleterious effects of optical feedback. These publications describe the means for achieving unidirectional oscillation in monolithic nonplanar ring oscillators, but contain no mention of unidirectional, planar ring oscillators. Recent product literature from Electro-optics Technology, Inc. shows a monolithic, unidirectional planar ring laser but provides no description of the means of inducing unidirectional oscillation.

SUMMARY OF THE INVENTION The present invention provides a means of inducing unidirectional oscillation in monolithic ring lasers in which the light path is planar. The intracavity optical diode that enforces unidirectional oscillation in the planar ring oscillator is achieved by a combination of the nonreciprocal Faraday effect, a linear birefringence effect in which the principal axes of the birefringence are not parallel and perpendicular to the plane of propagation of the ring light path, and one or more partial polarizer effects. The present invention enables experimental optimization of polarization transformations within a monolithic planar ring oscillator and also provides a means of tuning the frequency of the planar ring oscillator. Applications of the present invention involving monolithic resonators are also taught.

The present invention provides an attractive alternative to the use of nonplanar ring light path geometry in monolithic resonator structures in which unidirectional oscillation is desired. The present invention makes it possible to vary both the reciprocal and nonreciprocal polarization transformations used to produce unidirectional oscillation. Such eigenpolarization-tunability is particularly advantageous in monolithic planar ring oscillators. In contrast, the reciprocal polarization transformations used to establish unidirectional oscillation in monolithic nonplanar ring oscillators of the prior art are fixed by the choice of the resonator geometry and laser medium.

Other advantages of the present invention pertaining to frequency tunability and optimization of unidirectional oscillation will be presented in connection with preferred embodiments below.

BRIEF DESCRIPTION OF THE DRAWINGS Figure 1 shows a general, asymmetric, three- reflection, monolithic planar ring oscillator of the present invention.

Figure 2 depicts a symmetric, three-reflection monolithic planar ring oscillator similar to that of Fig. 1.

Figure 3 shows an alternative embodiment of a three-reflection, monolithic planar ring oscillator. Figure 4 shows an alternative embodiment of the monolithic planar ring oscillator of Fig. 2.

Figure 5 shows an alternative embodiment of the monolithic planar ring oscillator of Fig. 2 incorporating two different applied stresses.

Figure 6 shows two embodiments of a monolithic planar ring oscillator in which no total internal reflections occur and in which dichroic mirror coatings are used in order to reflect ring laser radiation and to transmit pump laser radiation.

Figure 7 is an embodiment of a monolithic planar ring oscillator in which the ring light path is defined by four reflections.

Figure 8 is an embodiment of a monolithic planar ring oscillator in which the ring light path is defined by five reflections. Figure 9 shows three-, four-, and five- reflection rihg light paths that have a plane of mirror symmetry perpendicular to the plane of the ring light path.

Figure 10 shows general optical systems that are equivalent in their eigenpolarization characteristics to any planar ring oscillator in which the only amplitude anisotropy (partial polarizer behavior) occurs at reflection Vertex A.

Figure 11 shows the optical systems that are equivalent in their eigenpolarization characteristics to any planar ring oscillator in which the ring light path has a plane of mirror symmetry as illustrated in Fig. 9 and in which the amplitude anisotropy (partial polarizer behavior) occurs at reflection Vertex A. Figure 12 shows a mirror symmetric, three- reflection planar ring light path in which the Jones matrices corresponding to the polarization transformations for counterclockwise propagation are indicated.

TABLE OF CONTENTS

1. Principles of Operation

1.1 Optical Diode

1.2 Reciprocal Polarization Transformations in Optical Diodes

1.3 Birefringence

1.4 Simultaneous, Distributed Reciprocal and

Nonreciprocal Polarization Transformations

1.5 Frequency Tuning

1.6 Pumping

1.7 Advantages of the Invention for Monolithic Ring Lasers

2. Monolithic Unidirectional Planar Ring Oscillators

2.1 Monolithic Three-Reflection Planar Ring Oscillators

2.1.1 General, Asymmetric Case

2.1.2 Symmetric, Faceted Rod

2.1.3 Prism

2.1.4 Electro-Optic Version

2.1.5 Multiple Actuators

2.1.6 Multiport Pumping

2.2 Monolithic Unidirectional Planar Ring Oscillators Defined By Four or More Reflections 2.2.1 Symmetric, Four-Reflection, Faceted Rod

2.2.2 Symmetric, Five-Reflection, Faceted

Rod

3. Appendix A: Geometries of Light Paths

3.1 Total Internal Reflection (TIR) at B and C, No TIR at A

3.2 Multiport Pumping at A, B, and C: No TIRs

4. Appendix B: Eigenpolarization Analysis

4.1 Optical Equivalence Theorem

4.2 Synthesis of an Optimized Optical Diode

DETAILED DESCRIPTION

1. Principles of operation

1.1 Optical Diode

Unidirectional oscillation in a ring laser occurs when there is a sufficiently large difference between the net round trip gains for the two possible directions of propagation around the ring. Unidirectional oscillation is often achieved by using combinations of polarization transformations that depend on the direction of propagation to create an effective optical diode: a device that causes different directions of propagation to have eigenpolarizations with different losses. Recall that for each of the two directions of propagation around a ring laser there are two possible eigenpolarizations (see, for example, Statz et al., "The Multioscillator Ring Laser Gyroscope," in Laser Handbook, vol. 4, M. L. Stitch and M. Bass, eds., North-Holland, Amsterdam, the Netherlands, 1985). In general, the eigenpolarizations corresponding to a given direction of propagation have different round trip losses. For a homogeneously broadened gain medium, we need only consider the low- loss eigenpolarization for each direction of propagation. If the low-loss eigenpolarizations for the two directions of propagation have sufficiently different round trip losses, then the laser will oscillate in the direction corresponding to the lowest loss eigenpolarization. These arguments are more fully developed in, for example, Nilsson, Gustafson, and Byer.

Discrete-element optical diodes for dye ring lasers are described by Johnston, Jr. et al., "Design and performance of a broad-band optical diode to enforce one-direction traveling-wave operation of a ring laser," IEEE J. Quantum Electron, QE-16,483-488 (1980), and optimization of discrete-element optical diodes for use in Nd:YAG ring lasers is discussed by Kruzhalov, Parfenov, Pakhomov, and Petrun'kin, "Optical Isolators in YAG:Nd laser cavities," Sov. Phys. Tech. Phys., 30 :1145-1147 (1985). From the analysis of discrete-element optical diodes and the analysis of monolithic nonplanar ring lasers, we arrive at a crucial conclusion about the minimal requirements for creating an intracavity optical diode. Fundamentally, the operation of an intracavity optical diode requires a combination of three effects: 1) nonreciprocal polarization transformations such as Faraday rotation, 2) reciprocal polarization transformations (induced, for example, by using linear retarders or optically active media), and 3) a partial polarizer. 1.2 Reciprocal Polarization Transformations in Optical Diodes

Discrete element optical diodes typically use

Faraday rotation for the nonreciprocal polarization transformation, and either optical activity or a rotated halfwave linear retarder for the reciprocal polarization transformation. In cases involving only optical activity and Faraday rotation, the net polarization transformation resulting from optical activity and Faraday rotation is a simple rotation of the polarization state, and the magnitude of the rotation depends on the direction of propagation. For example, the Faraday rotation and the rotation due to optical activity can be made equal in magnitude. Then for one direction of propagation, the two rotations exactly cancel, and for the other direction of propagation the two rotations add. A linear partial polarizer (for example, one or more Brewster-angle interfaces) in the ring cavity then makes the eigenpolarizations for the two directions of propagation have different minimal losses, enabling unidirectional oscillation. (See Johnston, Jr. et al., Kruzhalov, Parfenov, Pakhomov, and Petrun'kin; Nilsson, Gustafson, and Byer previously cited.)

J. A. Arnaud, "Degenerate Optical Cavities," Appl. Opt. 8:189-195 (1969) pointed out that a nonplanar ring cavity exhibits reciprocal polarization rotation analogous to optical activity. F. Biraben, "Efficacite des systemes unidirectionnels utilisables dan les lasers en anneau, " Opt. Commun., 29:353-356 (1979) proposed combining a nonplanar ring cavity, a Faraday rotator, and a linear partial polarizer to produce unidirectional oscillation of a ring dye laser. Kane and Byer showed that the use of a nonplanar ring resonator could enable unidirectional oscillation in a monolithic solid-state laser.

Accordingly, in monolithic solid-state ring lasers of the prior art, the intracavity optical diode that enables unidirectional operation has comprised a combination of Faraday rotation, reciprocal polarization transformations induced by the nonplanar ring geometry of the light path and the phase changes induced at reflections, and a linear partial polarizer effect associated with reflection from a multilayer dielectric coated output coupler at oblique incidence. A complete analytical description of the eigenpolarization analysis of monolithic nonplanar ring lasers of the prior art is given by Nilsson, Gustafson, and Byer. Nilsson, Gustafson, and Byer have proved that it is impossible to create a stable optical diode effect in monolithic ring lasers made from optically isotropic media unless two conditions are satisfied: 1) there must be an applied magnetic field to induce Faraday rotation in the laser medium (the laser medium is assumed to have a nonzero Verdet constant), and 2) the light path must be nonplanar. The applied magnetic field is required to produce the nonreciprocal polarization transformation part of the optical diode; the nonplanar ring light path produces the reciprocal polarization transformation.

Note that the above discussion excludes the possibility of using one or more magneto-optic mirrors to achieve the nonreciprocal polarization transformation. We consider the use of magneto-optic mirrors in connection with certain embodiments of the invention to be described later.

The invention described here presents a novel means of creating an optical diode in monolithic ring lasers. The novel technique of this invention is distinguished from the prior art by two essential features: 1) the ring light path is planar, and 2) the light path contains a linear birefringence effect, the principal axes of which are rotated with respect to at least one set of the traveling s and p linear polarization basis vectors associated with propagation along each leg of the planar ring light path. Note: in analyzing the propagation of light in a ring laser it is convenient to introduce traveling coordinate systems in which the local direction of propagation along a leg of the ring light path is always taken to be the positive z direction. The unit basis vectors transverse to the direction of propagation are the unit vector perpendicular to the plane of propagation, denoted by s, and a unit vector in the plane propagation, denoted by p, defined such that p x s = z. The rotated linear birefringence affect is used to create a reciprocal polarization transformation analogous to that caused by the nonplanar ring light path of the prior art. In concert with the Faraday rotation and linear partial polarizer effects common to both the prior art and the current invention, an optical diode is created within the planar ring light path that causes a difference in the round trip losses for the two directions of propagation. Unidirectional oscillation is thereby possible, facilitating single- frequency operation even at high output powers, and improving resistance to the destabilizing influence of optical feedback. (See, Nilsson, Kane and Byer, "Monolithic Nonplanar Ring Lasers: Resistance To Optical Feedback," in Pulsed Single-Frequency Lasers: Technology and Applications (1988) SPIE Vol. 912.

1.3 Birefringence

The important new element of this invention is the use of a rotated, linear birefringence effect to produce reciprocal polarization transformations analogous to those caused by nonplanar geometry in the prior monolithic ring laser art. The many possible ways to implement the necessary effect in accordance with the teachings of this invention may be grouped into two categories: 1) induced linear birefringence, and 2) natural linear birefringence. Of these two categories, the examples of this invention described herein emphasize the use of induced linear birefringence, particularly for monolithic planar ring oscillators, for reasons that are explained below.

Induced linear birefringence refers to the use of some externally applied effect to break the initial optical isotropy of the laser medium. Examples of optically isotropic laser media include all of the crystalline laser hosts with cubic symmetry and the many laser glasses. Among the crystalline laser hosts with cubic symmetry, the most important are the garnets, especially YAG, GGG, GSGG, YSAG, YSGG. The most important means of inducing linear birefringence in initially optically isotropic media include the photoelastic effect and the linear electrooptic effect, as shown in J. F. NYE, "Physical Properties of Crystals," (Oxford University Press, New York, 1986). The photoelastic effect refers to the changes in the magnitudes of the principal indices of refraction and the directions of the corresponding principal axes that occur when stress is applied to a medium. The photoelastic effect occurs in all media, regardless of symmetry. The linear electrooptic effect is the analogous collection of changes induced in a medium when an external electric field is applied. The linear electrooptic effect occurs only for media whose point group symmetry excludes inversion.

Natural linear birefringence refers to media that exhibit optical anisotropy with linear eigenpolarization states in the absence of external influences. All crystals with non-cubic point group symmetry are characterized by natural birefringence, and most of these exhibit linear birefringence. Thus, many useful crystalline laser hosts are intrinsically birefringent, including YLF, YALO (also called YAP), YVO₄, AI₂O₃, Mg:F₂, and many others.

The polarization transformation associated with a rotated, linear-birefringent section of the light path is the same effect as occurs in a linear retarder (waveplate). It is essential that the principal axes of the birefringence effect be rotated with respect to at least one set of the s and p vectors of the propagating coordinate system. This requirement produces a reciprocal polarization transformation that can partially or fully cancel the Faraday rotation for one direction of propagation and can add to the Faraday rotation for the other direction of propagation. In this way, the rotated birefringence effect invoked here is analogous to the use of, for example, a rotated haIfwave plate to cancel the Faraday rotation in a discrete-element optical diode (Globes and Brienza, and Kruhalov, Parfenov, Pakhomov, and Petrun'kin, previously cited). The novelty of this feature of the present invention is that such an effect can be combined with Faraday rotation in a laser gain medium to produce an optical diode in a monolithic structure, and the light path in such a structure can be planar, as is now more fully described.

1.4 Simultaneous, Distributed Reciprocal and Nonreciprocal Polarization Transformations

Discrete-element optical diodes often use a Faraday rotator for the nonreciprocal polarizafciur. transformer and a rotated linear retarder (often a halfwave plate) for the reciprocal polarization transformer. For monolithic planar ring oscillators both the reciprocal and nonreciprocal polarization transformations must coexist in the same optical medium. Separation of these two effects is possible in some cases (as described below), but it is essential to understand how to treat the simultaneous presence of a distributed linear birefringence and a distributed Faraday effect.

Consider an optical medium that exhibits pure linear birefringence in the absence of an applied magnetic field. Application of a magnetic field to this medium will modify the way in which the polarization state of light evolves while propagating through the medium. This problem has been treated by W. J. Tabor and F. S. Chen, "Electromagnetic Propagation Through Materials Possessing Both Faraday Rotation and Birefringence: Experiments with ytterbium Orthoferrite," J. Appl. Phys., 40:2760-2765 (1969). The most important results of their analysis can be summarized as follows:

1) In the absence of linear birefringence, the medium exhibits pure Faraday rotation, and the eigenstates of polarization are circularly polarized

2) In the absence of Faraday rotation, the medium exhibits pure linear birefringence, and the eigenstates of polarization are linearly polarized. 3) In the presence of both the Faraday effect and linear birefringence, the eigenstates of polarization are orthogonal, elliptically polarized states.

4) The net polarization transformation depends on the direction of propagation with respect to the magnetic field. When the birefringence is large compared to the Faraday effect, the medium behaves more nearly like a linear birefringent medium. In particular, the maximum net rotation of the plane of polarization of an incident linear polarization state is small.

Detailed analysis of propagation of light in a medium with simultaneous linear birefringence and a Faraday effect can be carried out using the Jones matrices derived by Tabor and Chen. Intuitively, however, it is apparent that simultaneous linear birefringence and a Faraday effect represents a combination of reciprocal and a nonreciprocal polarization transformation. Since the magnitudes of the effects will depend on the direction of propagation with respect to the magnetic field, such a combined reciprocal and nonreciprocal effect can be used as a component of an optical diode. The only other necessary component for the optical diode is a partial polarizer.

Both the monolithic nonplanar ring oscillators of the prior art and the novel monolithic planar ring oscillators of this invention rely on the Faraday effect of the laser medium in the presence of an external magnetic field. The Faraday effect in laser media is typically extremely weak. Accordingly, to cancel the weak nonreciprocal Faraday effect for one direction of propagation around a ring light path requires a weak reciprocal polarization transformation. The important consequences of this idea in the context of monolithic nonplanar ring oscillators have been described by Trutna, Jr., Donald, and Nazarathy, and Nilsson, Gustafson, and Byer. The conclusion of these analyses is that optimal performance of the optical diode requires careful balancing of the Faraday rotations by weak reciprocal polarization transformations. For nonplanar ring oscillators, the implication is that the geometry of the ring light path should be only slightly nonplanar if best performance is to be achieved. By analogy, the reciprocal polarization effects in a planar ring oscillator should also be weak in order to achieve best performance. This is the advantage of using induced birefringence rather than intrinsic birefringence. The magnitude and sense of the reciprocal polarization effect in an initially optically isotropic medium can be controlled by changing the magnitude and direction of the externally applied influence that induces the birefringence. The induced effect can thus be a small perturbation on the initial isotropy of the laser medium, as is the induced Faraday rotation, and the tunability of the birefringence effect makes it possible to optimize performance. In a naturally birefringent medium, the weak Faraday effect is typically much smaller than the linear birefringence of the medium. This does not imply that unidirectional operation is impossible to achieve in a naturally birefringent medium, but optimal operation is undoubtedly more difficult to achieve.

A pertinent analogy arises in the case of monolithic nonplanar ring oscillators. The original monolithic nonplanar ring oscillators of Kane and Byer had light paths in which the two principal planes of propagation were orthogonal to each other. In such a light path the reciprocal polarization effects of the nonplanar geometry are large compared to the Faraday rotation. As described by Trutna Jr., Donald, and Nazarathy, and Nilsson, Gustafson, and Byer this orthogonal geometry greatly reduces the strength of the optical diode, but it by no means prevents unidirectional oscillation, as the experiments of Kane and Byer have shown.

An additional complication of using naturally birefringent media with large inherent birefringences is the presence of significant anisotropic optical effects such as Poynting vector walkoff, bireflectance at total internal reflection interfaces, and temperature dependences of birefringences. These effects complicate the design and analysis of workable ring lasers . Therefore , the majority of the assertions made in this application pertain to ring lasers in which the gain medium is initially optically isotropic, and in which the optical diode depends for its success upon an induced birefringence in the laser medium to cancel an induced Faraday rotation.

1.5 Frequency Tuning

For many applications, it is important to be able to tune the frequency of the ring laser radiation. It should be apparent that, in addition to providing unidirectional oscillation, the magnitudes and directions of the applied magnetic field and either stress and/or electric field, as appropriate, will also determine the allowed frequency of oscillation of the laser. By changing the applied fields, the laser frequency can be tuned. Prior art use of the photoelastic effect to tune the output frequency of a monolithic solid-state laser with a linear cavity is described in Owyoung et al., "Stress-induced tuning of a diode-laser-excited monolithic Nd:YAG laser," Opt. Lett. 12, 999-1001 (1987), J. J. Zayhowski and A. Mooradian, "Frequency-modulated Nd:YAG microchip lasers," Opt. Lett., 14:618-620 (1989), and J. J. Zayhowski and A. Mooradian, "Single-Frequency Microchip Nd Lasers," Opt. Lett., 14:24-26 (1989). The use of the photoelastic effect to tune the frequency of a monolithic nonplanar ring oscillator is described in Kane and Cheng, "Fast Frequency Tuning and Phase Locking of Diode-Pumped Nd:YAG Ring Lasers," Opt. Lett. 13, 970-972 (1988). The theory of tuning the frequency of a monolithic nonplanar ring oscillator by changing the magnitude of the applied magnetic field is given by Nilsson, Gustafson, and Byer.

To achieve continuous tuning over a large range, it is desirable for the laser to have a large separation between adjacent axial modes. To achieve this effect, the round trip path length of the cavity should be made as short as possible. Clearly, monolithic linear cavities can be made smaller than ring laser cavities. Standing wave cavities do not, however, offer the resistance to feedback inherent in the unidirectional ring lasers. Among ring lasers, a three-reflection planar ring oscillator concept makes possible the shortest round trip path lengths. Accordingly, the three-reflection planar ring oscillator should have the largest continuous tuning range available among monolithic ring oscillators. By combining temperature tuning (a slow effect) and rapid tuning via the photoelastic effect (and/or the electrooptic effect), the monolithic planar ring oscillator offers all of the performance features of the monolithic nonplanar ring oscillator with respect to the linewidth of the free-running laser and the ability to lock the laser tightly to an external standard. (See Kane, Nilsson, and Byer; Day et al., "30-Hz-linewidth, diode-laser-pumped, Nd:GGG nonplanar ring oscillators by active frequency stabilization," Elect. Lett. 25, 810-811 (1989).)

1.6 Pumping

The advantages of longitudinal pumping of linear and ring lasers have been emphasized many times in the prior art. Each reflection vertex defining the ring light path represents a potential pumping port . In all demonstrated prior art monolithic nonplanar ring lasers, all reflections except for one have been total internal reflections. Total internal reflection vertices preclude mode-matched longitudinal pumping. Proposals for using dichroic mirror coatings instead of total internal reflections have been presented in Nilsson et al., U.S. patent application Serial No.

332,232. This concept is used in certain embodiments of this invention in connection with longitudinal pumping of monolithic planar ring oscillators. Several of the embodiments presented below explicitly illustrate the idea of longitudinal pumping at several reflection vertices. The principal advantage of multiport pumping is that higher power operation of the unidirectional ring oscillator can be achieved for those cases in which the single-port pump power is limited. Diode laser pumping, for example, often limits the total output power available from current devices.

The product literature of Electro-Optics Technology, Inc., may refer to the use of the concept of multiport longitudinal pumping. The product literature states that "An important feature of the PRO is that it can be pumped by multiple diodes." It is not stated whether the multiple diodes pump the resonator through a single pump port or are used at multiple pump ports. Of interest, the concept of multiport pumping appears in the patent applications of Trutna, Jr., and Berger and Nazarathy.

1.7 Advantages of the Invention for Monolithic Ring Lasers

The advantages of the novel approach to inducing unidirectional oscillation in accordance with the teachings of this invention relate to efficiency of operation, simplicity and precision of fabrication, optimization of differential loss within the resonator, and frequency tunability.

1) Efficiency. In general, propagation in an optical medium and reflection from optical surfaces involves some loss associated with scattering and absorption. These effects reduce the efficiency of any laser. Additionally, the scattering causes power coupling between the two directions of propagation in a ring laser, as is well known from ring laser gyroscope studies. In general, then, both the efficiency of laser operation and the ease of inducing unidirectional oscillation are improved by reducing the number of reflections defining the ring light path to a minimum. The smallest number of reflections that can define a ring light path is three, and the ring light path that results from the use of three reflections is planar in a monolithic structure. Monolithic nonplanar ring oscillators require at least four reflections to define the nonplanar ring light path that makes unidirectional oscillation possible in optically isotropic media. The present invention presents the advantage of making unidirectional oscillation possible in a planar ring defined by three reflections, which leads to improved efficiency and reduced coupling between counterpropagating directions.

2) Ease of fabrication. Because fewer optically polished surfaces are required in the three- reflection planar ring device, and because the normals to the reflection surfaces are coplanar, the fabrication of a three-reflection planar ring oscillator is simpler than the fabrication of any nonplanar ring oscillator. Reduced fabrication costs and improved accuracy of production are significant benefits of the monolithic planar ring oscillator constructed in accordance with this. invention.

3) Optimization of differential loss. A significant advantage of the present invention relates to the tunability of the induced birefringence. The magnitude and direction of the reciprocal polarization effects of the induced birefringence can be altered experimentally to achieve optimal laser performance. In contrast with monolithic nonplanar ring oscillators of the prior art, the reciprocal polarization effects are determined by the geometry of the fabricated laser crystal, which cannot easily be altered once fabrication is completed.

4) Frequency tunability. The magnitude and direction of the induced birefringence and the amount of Faraday rotation can be changed in order to tune the frequency of the laser. The changes in the amount of birefringence and Faraday rotation in the resonator result in changes in the optical path length of the ring light path, thereby shifting the frequency of the laser.

2. Monolithic Unidirectional Planar Ring Oscillators

2.1 Monolithic Three-Reflection Planar Ring Oscillators

The monolithic planar ring oscillators decribed herein are laser pumped solid-state lasers in which the light path is planar. The number of reflections in a round trip may be any integer greater than or equal to three. For three-reflection monolithic planar ring oscillators, the light path is necessarily planar, whereas rings with four or more reflections may be planar or nonplanar. The embodiments presented herein are all defined to be planar. In each embodiment shown, the optical diode is produced by a combination of the Faraday effect, birefringence with principal axes rotated with respect to axes parallel and perpendicular to the plane of propagation on at least one leg of the planar ring light path, and a linear partial polarizer effect associated with reflection from a multilayer dielectric coated output coupler. The birefringence and Faraday effect may occur simultaneously in a distributed fashion, or in certain special cases the birefringence effect may be substantially separated from the Faraday effect.

2.1.1 General, Asymmetric Case

Figures 1a, 1b, and 1c show top, side, and end views of a general, unidirectional monolithic planar ring oscillator 100 in which the light path ABC is defined by three reflections. As shown here, monolithic resonator 100 is fabricated from a right circular cylinder of an optically isotropic laser gain medium with a nonzero Verdet constant. End-cap surface 110 can be spherical, aspherical, or planar. Surface 111 is the remaining cylindrical surface after fabrication of end-cap surface 110 and reflecting facets 112 and 113. Surface 110 is a multilayer dielectric coated surface that acts both to transmit the pump radiation 115 from the external laser pump source 114 and to couple out the unidirectional ring laser radiation 116 from the monolithic resonator 100. Surfaces 112 and 113 are taken to be flat facets at which either total internal reflection or approximately total reflection from multilayer dielectric mirror coatings occurs.

A magnetic field H is applied to laser resonator 100 such that the field is not perpendicular to all of the legs of the light path. Therefore, this magnetic field produces Faraday rotation in two or more legs of the light path. In Fig. 1 a linear birefringence is induced in the monolithic resonator 100 by applying a stress across the diameter of the rod. Line contacts along the rod surface 111 are shown in the end view of Fig. 1c. The principal axes of the applied stress are rotated by an angle β with respect to the plane of propagation of the light. This angle β is chosen such that the resulting linear birefringence has its principal axes also rotated with respect to the plane of propagation of the light. In this way a distributed linear birefringence with principal axes rotated with respect to the plane of propagation of the light is established in monolithic resonator 100. The nonreciprocal Faraday effect and the rotated linear birefringence serve as the nonreciprocal and reciprocal polarization transformations required to make an optical diode. The amplitude anisotropy of reflection from the output coupler at vertex A serves as the linear partial polarizer of the optical diode. By properly selecting the magnitudes and directions of the applied magnetic field H and the applied stress, unidirectional oscillation is established in monolithic planar ring oscillator 100.

More generally, at least one of the reflections defining the light path is presumed to act as a linear partial polarizer, and at least one of the reflections occurs at an interface that permits laser end-pumping of the laser medium. The remaining reflections could, for example, be high reflectors for the ring laser radiation or could be total internal reflections. No claims of special symmetries are made about this most general of three-reflection planar rings. In fact, however, preferred embodiments will generally take advantage of imposed symmetries as described below in order to simplify analysis and design, improve performance, and reduce manufacturing complexity and cost. In what follows, all of the ring light paths and the resonator structures that produce these ring light paths are considered to have a plane of mirror symmetry that is perpendicular to the plane of propagation of the light and that intersects the plane of propagation in a line that bisects the apex angle at vertex A.

2.1.2 Symmetric, Faceted Rod Figures 2a, 2b, and 2c show top, side, and end views of a preferred embodiment of a monolithic, unidirectional planar ring oscillator. Box 214 represents an external source of pump radiation 215 used to excite ring laser resonator 200 by longitudinal pumping through vertex A on dichroic output coupler 210. Ring laser 200 oscillates unidirectionally to produce output radiation 216 as shown. Monolithic resonator 200 is fabricated from a right circular cylinder of an optically isotropic, solid-state laser medium with a nonzero Verdet constant. The right circular cylinder is modified by grinding and polishing end-cap 210 and facets 213 and 212. Surface 211 is the remaining, unmodified portion of the cylinder wall. The ring light path within monolithic resonator 200 is defined by reflections at vertices labeled A, B, and C. In one embodiment, triangle ABC is an isosceles triangle. The auxiliary point labeled Q is the midpoint of leg BC. An external magnetic field is applied parallel or antiparallel to AQ in Fig. 2, but the minimal requirement on the external magnetic field is that it not be perpendicular to the plane of propagation. Surface 210 producing the reflection at A is a flat or curved, multilayer dielectric coated surface that serves as the output coupler for ring laser 200 and also admits laser pump light 215 used for end-pumping. The shape of surface 210 is chosen in accordance with the requirement that laser resonator 200 be a stable resonator with respect to its Gaussian beam modes. If surface 210 containing point A is taken to be a curved surface, it may either be spherical or alternatively may be chosen to have two different principal radii of curvature parallel and perpendicular to plane ABC in order to reduce the astigmatism of laser resonator 200. Surfaces 212 and 213 producing the reflections at vertices B and C, respectively, are taken to be flat, polished facets. In this embodiment, the angles of the isosceles triangle ABC are chosen such that the reflections at vertices B and C are total internal reflections. Analysis of the geometry of light paths required to produce total internal reflection at vertices B and C and to avoid total internal reflection at A is included as Appendix A of this specification.

The optical diode for this embodiment is created by a combination of induced linear birefringence in the laser medium, Faraday rotation in the laser medium, and a linear partial polarizer effect associated with oblique reflection from multilayer dielectric coated output coupler 210 at vertex A. The light path of the monolithic laser of Figure 2 is defined by three reflections, and hence is necessarily planar. A compressive stress is applied along the diameter of the rod from which monolithic laser 200 is made, as shown in the end view of Fig. 2c. This stress is applied by line contacts running parallel to the rod's axis, the contact lines being diametrically opposite. The length of the contact lines is variable. The axis of application of the compressive stress makes an angle β with the perpendicular to the plane of propagation ABC. Principal axes of the induced birefringence are parallel and perpendicular to the applied stress for isotropic materials such as laser glasses and for garnet crystals such as Nd:YAG if the cylinder axis corresponds to the usual [111] crystal direction.

As shown in Fig. 2a, the external magnetic field is applied parallel to AQ. The Faraday effect thus occurs along legs AB and AC but not along leg BC, which is perpendicular to the applied magnetic field. In this embodiment the polarization transformation associated with propagation along legs AB and AC involves a combination of distributed birefringence and simultaneous Faraday rotation. The amplitude anisotropy of reflection from the output coupler at vertex A and the combined reciprocal and nonreciprocal transformations along legs AB and AC leads to different eigenpolarizations with different losses for the two directions of propagation.

2.1.3 Prism

Figures 3a, 3b, and 3c show an embodiment of a monolithic, unidirectional, planar ring oscillator in which resonator 400 has a different geometry than that shown in the embodiments of Fig. 2. Monolithic resonator 400 is fabricated from an optically isotropic, solid-state laser medium with a nonzero Verdet constant. The final shape of laser resonator 400 is a prism shape. The surfaces of resonator 400 are input coupling surface 410, reflecting facets 412 and 413, sides 411 and 414, top 419, and bottom 418. Input coupling surface 410 could be spherical, aspherical or flat as described in the previous embodiments. Box 415 represents a source of pump radiation 416 used to excite laser resonator 400, which oscillates unidirectionally to produce output radiation 417 as shown.

In this embodiment the externally applied stress can be applied locally along leg BC as shown in Fig. 3 in such a way that 1) the influence of the stress is localized to be significant only on leg BC, and 2) the principal axes of the stress are rotated with respect to the s and p basis vectors of propagation along leg BC. Such a stress can be applied with normal forces or shear forces. In the embodiment of Fig. 3, the external magnetic field is once again taken to lie along AQ, although this is not essential. Making the simplifying assumption that the stress-induced birefringence along legs AB and AC is negligible, the polarization transformations around the ring are simply described. Faraday rotation occurs along legs AB and AC, and a reciprocal transformation associated with rotated birefringence occurs along leg BC. The optical diode results from the combination of the amplitude anisotropy of the mirror at vertex A, the nonreciprocal rotations on legs AB and AC, and the reciprocal transformation on leg BC. The analysis of the optical diode resulting from this system is the same as that performed previously for the four-reflection monolithic nonplanar ring oscillator by Nilsson, Gustafson, and Byer. The analysis for the current system and its relation to the analysis by Nilsson, Gustafson, and Byer is included as Appendix B in this specification.

2.1.4 Electro-Optic Version

Figures 4a, 4b, and 4c show an embodiment of a monolithic, unidirectional, planar ring oscillator 500 that could be implemented in a noncentrosymmetric cubic laser medium exhibiting the linear electrooptic effect. In this embodiment a uniform electric field is applied to the laser medium in such a way that the principal axes of the resulting induced linear birefringence are rotated by an angle β with respect to the axes parallel and perpendicular to the plane of propagation of the light. The resulting polarization transformations are in the same form as those described previously using the photoelastic effect.

Of importance, in accordance with the teachings of this invention, all of the various embodiments of Figs. 1 to 4 described above for the photoelastic effect now can be repeated by substituting the electrooptic effect for the photoelastic effect. In the interest of brevity, these variations are not explicitly repeated here.

2.1.5 Multiple Actuators

To this point the embodiments shown have only contained a single external influence for inducing linear birefringence. For purposes of modulation (for example, switching directions of oscillation of the ring laser, or providing a small frequency dither or tuning about some initial setpoint) it may be useful to apply more than one such external influence to the laser cavity. Figures 5a, 5b, and 5c show an embodiment similar to that of Fig. 2 except that two externally applied stresses P₁ and P₂ are shown. For cases in which the electrooptic effect is available, one could consider two different electric field structures, or any combination of stresses and electric fields.

2.1.6 Multiport Pumping

Figure 6 shows an embodiment of a monolithic, unidirectional planar ring oscillator similar to that of Fig. 1, where the angles of isosceles triangle ABC have been chosen such that total internal reflection does not occur at any of the reflection vertices. Instead, the reflections at vertices B and C involve reflection from dichroic high reflectors chosen to have minimal amplitude anisotropy for the ring laser radiation and to be maximally transmitting for pump radiation. The advantage of this embodiment is that longitudinal pumping can occur at all of the reflection vertices. Figure 6a shows longitudinal pumping along all of the legs of the light path. Especially when diode laser pump sources are used, the ability to increase the total pump power and maintain the advantages of longitudinal pumping is attractive for producing high power ring laser radiation with good spatial and temporal properties. A special case of the isosceles triangle light path is the equilateral triangle light path. For materials with index of refraction less than approximately 2.0 at the ring laser wavelength, an equilateral light path is an acceptable solution for avoiding total internal reflection at each of the three reflections, as is discussed in Appendix A of this specification.

2.2 Monolithic Unidirectional Planar Ring Oscillators Defined by Four or More Reflections

The advantages of the three-reflection ring have been outlined above. With respect to ability to achieve optimal performance of an optical diode in a given monolithic laser medium with a given index of refraction, it is sometimes useful to use ring light path geometries with more reflections in a round trip. For example, the implications of the restrictions associated with total internal reflection phase shifts in a given medium have been discussed by Nilsson, Gustafson, and Byer; Nilsson and Byer (U.S. patent application Serial No. 332,232 ); and Rea, Jr. and Nilsson, U.S. Patent Application Serial

No. ___________ in connection with optimal design of optical diodes in monolithic nonplanar ring oscillators. An additional advantage of including more reflections in a round trip is that more pumping ports become available, which is important for high power applications. This advantage has been discussed previously in Nilsson and and Byer (U.S. patent application Serial No. 332,232). the choice of an even or odd number of reflections in a round trip has other implications for the image and polarization transformations in a ring laser. Some of these implications are presented in S. V. Kruzhalov et al., "Optical isolators in YAG:Nd laser cavities," Sov. Phys. Tech. Phys. (1985) 30: 1145-1147.

2.2.1 Symmetric, Four-Reflection, Faceted Rod

Figures 7a through 7c show an embodiment of a monolithic, unidirectional planar ring oscillator 800 in which the planar ring light path is defined by four reflections at vertices A, B, C, and D. The discussion of the pumping of the resonator, the application of stress (and/or an electric field when using electrooptic media), the application of the magnetic field, and the operation of the resulting optical diode are similar to what has been described above in connection with three-reflection rings constructed in accordance with the teachings of this invention. Fig. 7a together with Figs. 7d and 7e represent an alternative embodiment of the four-reflection monolithic nonplanar ring oscillator of Figs. 7a-7c, differing only in the shape of the resonator body.

2.2.2 Symmetric, Five-Reflection, Faceted Rod

Figures 8a, 8b, and 8c show a monolithic planar ring oscillator 900 in which the planar ring light path is defined by five reflections at vertices A, B, C, D, and E. The light path of Fig. 8 is best suited for use with total internal reflections at all vertices except for A, because the angles of incidence at vertices B, C, D, and E are typically larger than the critical angle for typical laser gain media. A skewed stress (or electric field in the case of electro-optic media) is applied locally to leg CD.

3. Appendix A: Geometries of Light Paths

The light paths considered here are planar light paths having a plane of mirror symmetry perpendicular to the plane of the light path such that the plane of mirror symmetry intersects the light path along a line bisecting the apex angle at vertex A. The resonator producing the light path is also taken to be mirror symmetric as described above. This symmetry simplifies the analysis of the constraint conditions. For example, three-reflection light paths are necessarily isosceles or equilateral triangles. Figure 9 shows examples of three-reflection, four-reflection, and five-reflection planar figures having such mirror symmetry. Note that, for light paths defined by an odd number of reflections, the leg opposite vertex A is perpendicular to the line bisecting the apex angle at A. For light paths defined by an even number of reflections, the bisector of apex angle A passes through another vertex.

The analysis presented here treats the case of optically isotropic media only. Generalization to optically anisotropic media is an extension of the analysis presented here.

3.1 Total Internal Reflection at B and C, no TIR at A

We require total internal reflection to occur at B and C, and we require that total internal reflection not occur at A, in order that the surface at A can serve as the output coupler for the resonator. By symmetry, we can consider only vertices A and B. The angles of incidence at A and B must satisfy

θA < ^θcritical'

θB > ^θc ritical where θ_critical is the critical angle for internal reflection in the medium with index of refraction n,

θ_critical = sin^-1(l/n)

Additionally, the angles of incidence θ_A and θ_B must satisfy the trigonometric constraint

θ_A + 2 θ_B = 90.

Since the angle of incidence at B is constrained to exceed the critical angle, and since the angle of incidence at A must be less than the critical angle, there is an upper limit on the permissible angles of incidence at A:

θ_A ^max = min{ θ_{c ritical'} 90 - 2 θ_{c ritical} }

For n < 2.0, the upper limit on θ_A is

θ_A ^max = 90 - 2 θ_critical'

corresponding to setting the angle of incidence at B to its smallest value,

θ_B ^min = θ_critical.

For n = 2.0, the upper limit on θ_A is θ_A ^max = θ_critical' and the corresponding minimal value of the angle of incidence at B is

θ_B ^min = ( 90 -θ_critical )/2.

For symmetric ring light paths defined by more than three reflections, the analysis proceeds as above. In each case we find constraints on the permissible angles of incidence at the various vertices in order to assure that total internal reflection occurs at all vertices but A and that total internal reflection does not occur at A.

3.2 Multiport Pumping at A, B, and C: No

TIRs

If instead we want to be able to pump the laser longitudinally through all of the reflection vertices, the constraints on the angles are changed. In this case we require that all of the angles of incidence be smaller than the critical angle for the medium. For the mirror-symmetric triangular case this constraint is

θ_A' θ_B < θ_critical . Of course , the same trigonometric constraint exists , θ_A + 2 θ_B = 90.

Together, these constraints result in upper and lower limits for the angles of incidence at A and B as follows. First, there is an index of refraction constraint, because there are no solutions at all for materials with index of refraction exceeding 2.0. For materials with index of refraction less than or equal to 2.0, the limits are:

θ_A ^min = 90 - 2 θ_critical θ_A ^{max = θ}critical

θ_B ^min = (90 - 2 θ_critical )/2

θ_A ^max =θ_critical. As a special case of the isosceles solutions for multiport pumping, we consider the equilateral light path. This light path can satisfy the constraints for multiport pumping for all indices of refraction less than 2.0 When the index of refraction is 2.0, the equilateral light path is the only solution . Therefore , a qualitative description of the available design space for angles of incidence at A and B is as follows. For indices of refraction less than 2.0, there are isosceles and equilateral light paths that satisfy all of the constraints. The range of admissible angles of incidence decreases as the refractive index increases, until at n = 2.0, the only solution is the equilateral light path. There are no solutions for n > 2.0.

The solutions for light paths with larger numbers of reflections in a round trip are worked out using the same constraint equations as above. A possible generalization is to mix the availability of total internal reflection and multiport pumping, which is potentially desirable in order to avoid the cost and complexity of applying too many different optical coatings to a monolithic laser component.

4. Appendix B: Eigenpolarization Analysis

The purpose of this Appendix is to provide a framework for analyzing the various embodiments described in the body of the text. The detailed mathematical formalism for performing such analyses is contained in Nilsson, Gustafson, and Byer. From that analysis we distill two crucial results. The first result pertains to finding the optical equivalent of a given ring light path. The second result pertains to synthesizing strong optical diodes. 4.1 Optical Equivalence Theorem

Hurwitz et al., "A new calculus for the treatment of optical systems, II. Proof of three general equivalence theorems," J. Opt. Soc. Am., 31, 494-499, (1941) describes a theorem regarding the Jones matrices associated with propagation through lossless optical media. The optical interpretation of this theorem is the following: any system of optical elements comprising linear retarders and rotators is equivalent, with respect to polarization transformation properties, to a system containing just one rotator and one linear retarder. This theorem can be applied directly to many of the planar ring oscillator schemes described in the body of the text by considering the polarization-transforming properties of all of the elements of the ring path except for the output coupler. The output coupler is itself equivalent to a combination of a linear retarder and a partial polarizer, the principal axes of both elements being strictly aligned with one another. Application of the optical equivalence theorem to the monolithic planar ring oscillators leads, in every case in which all reflections other than the reflection from the output coupler at A are taken to be lossless, to the optical equivalent resonators shown in Fig. 10. With regard to establishing an optical diode, the important point is that in general the retardance ana orientation angle of the equivalent linear retarder depends on the direction of propagation, as does the rotation angle of the rotator. It is the difference in one or more of these parameters that causes the eigenpolarizations for the two directions of propagation to have different losses. A special case deserves to be singled out.

Under certain conditions to be described below, the optical equivalents of the resonators for the two directions of propagation simplifies to contain no additional rotator. In that case, the two optical equivalents are shown in Fig. 11. This special case has been analyzed by Nilsson, Gustafson, and Byer. Although the analysis performed there arose from consideration of symmetric light paths in monolithic nonplanar ring oscillators, the equations and their interpretation are completely identical for planar ring oscillators, as we now show. Consider the symmetric triangular light path of Fig. 12. Assume that a localized, rotated birefringence effect occurs on leg BC, and that only Faraday rotation occurs along legs AB and AC. For this case, the Jones matrices for the round trip through the resonator are

M⁺ = M_A R(γ_AB) MB R(-θ_BC) N_BC R(θ_BC) M_B R(γ_AB),

M- = R(-γ_AB) M_B R(θ_BC) N_BC R(-θ_BC) M_B R(γ_AB) M_A.

Here + and - denote the two directions of propagation around the ring, R denotes a rotation matrix, M_B denotes a diagonal Jones matrix representing the linear retarder associated with reflection at B, M_A denotes a diagonal Jones matrix representing the linear retarder associated with reflection at A, N_BC denotes a diagonal Jones matrix representing the linear retarder associated with the birefringence between vertices B and C, γ_AB denotes the Faraday rotation along leg AB, and θ_BC is the angle representing the rotation of the principal axes of the localized birefringence along leg BC with respect to the s and p basis vectors for propagation from B to C. The reason for explicitly giving this product is to enable direct comparison with Eqs. (21) and (22) of Nilsson, Gustafson, and Byer in which the round trip Jones matrices for propagation through a four-reflection monolithic nonplanar ring oscillator are given. It is seen that the forms of these equations are identical. Therefore, the optical equivalents and their interpretations are also identical. It is shown in Appendix C of Nilsson, Gustafson, and Byer that these round trip matrices yield optical equivalents that do not involve any additional rotators; hence the optical equivalents are as shown in Fig. 11. In particular, we can use the previously derived results of Nilsson, Gustafson, and Byer to describe the synthesis of an optimized optical diode.

An optical diode is defined to be a device that produces a propagation-direction-dependent difference in the losses of the eigenpolarizations of the resonator. With regard to establishing unidirectional oscillation in a ring laser, the minimal requirement on the optical diode is that the loss difference it induces exceed the coupling between the two directions of propagation due to effects such as scattering in the bulk medium and at interfaces. Optimal performance, however, imposes much more rigorous requirements on the optical diode. We define an optimized optical diode as a device with the following properties:

1) it induces unidirectional operation of the ring laser,

2) the eigenpolarization for the selected direction of propagation is the minimal loss eigenstate for the ring,

3) the differential loss between the oscillating and non-oscillating directions of propagation is as large as possible subject to constraint (2) above.

4.2 Synthesis of an Optimized Optical Diode An algorithm for synthesizing an optimized optical diode for such a ring laser is the following. First, the equivalent linear retarder for the selected direction of propagation should have its principal axes aligned with those of the output coupler. This guarantees operation in the lowest loss eigenpolarization for the resonator, regardless of the retardance value. Second, the retardance of the equivalent retarder for the nonoscillating direction should be chosen such that the sum of the retardance of the output coupler and that of the equivalent retarder is 360 degrees. For the case in which the output coupler has a standard quarterwave dielectric stack mirror, characterized by a retardance of 180 degrees, this condition reduces to setting the retardance of the equivalent retarder of the nonoscillating direction equal to 180 degrees. Finally, in order to maximize the loss difference between the oscillating and nonoscillating directions, the parameters of the ring should be chosen such that the orientation angle of the principal axes of the nonoscillating direction's equivalent linear retarder is as large as possible. This algorithm has been used in the design of optimized optical diodes in monolithic nonplanar ring lasers as described in the U.S. patent applications of Nilsson et al. and by Rea, Jr. et al., cited previously.

It is a straightforward generalization of the above results to prove that any symmetric, planar ring light path with an odd number of reflections reduces to a system with optical equivalents as in Fig. 11. The importance of this observation is that achieving the required retardance sum for the ring can be difficult because of the existence of an index-of-refraction- determined maximum phase shift upon reflection from a total internal reflection surface. Consider the symmetric three-reflection ring, for example. The induced birefringence on the leg BC must be large enough that its retardance plus twice the retardance associated with reflection from vertex B is equal to 180 degrees, if optimized operation is to be achieved. As an example, in Nd:YAG with index of refraction 1.82 at 1064 nm, the maximum total internal reflection retardance is 64.85 degrees, occurring for an angle of incidence of 42.92 degrees. To produce an optimized optical diode, the induced retardance on leg BC would have to be 180 - 2 (64.85) = 50.30 degrees. This is a large retardance, given the weakness of the photoelastic effect. For example, in YAG the typical order of magnitude of the induced birefringence in when a stress equal to the maximum tensile strength of YAG is applied is Δn = 10^-4 . One can reduce the difficulty of the requirements on the induced birefringence by including, for example, two more total internal reflections to give additional total internal reflection retardance. An analogous scheme has been applied by Rea, Jr. et al. to establish optimized optical diodes in monolithic nonplanar ring oscillators made from low-index materials such as Nd:glass.

To summarize, the analysis of monolithic planar ring oscillators can be reduced to the treatment of simplified, optically equivalent resonators. In the most general case described here, the optical equivalents involve a linear retarder with its principal axes rotated with respect to those of the output coupler, and an additional rotator. In special preferred cases, the additional rotator does not appear in the equivalent resonators. Optical diodes sufficiently strong to induce unidirectional oscillation are possible in either case, and are even possible in the most general example described in connection with Fig. 1. An algorithm for producing optimized optical diodes has been given for the preferred case in which no additional rotators appear in the optically equivalent resonators, and systems for which such optical equivalents exist have been identified.

All publications and patent applications cited in this specification are herein incorporated by reference as if each individual publication or patent application were specifically and individually indicated to be incorporated by reference.

Although the foregoing invention has been described in some detail by way of illustration and example for purposes of clarity of understanding, it will be readily apparent to those of ordinary skill in the art in light of the teachings of this invention that certain changes and modifications may be made thereto without departing from the spirit or scope of the appending claims.

Claims

WHAT IS CLAIMED

1. A monolithic laser resonator comprising: a monolithic piece of solid-state laser material ; input means for receiving input energy to pump said laser resonator; means for defining a planar ring light path involving three or more reflections internal to said piece of solid-state laser material, in which laser oscillation occurs in response to said input energy; means for extracting an output laser beam from said laser oscillation; and means for inducing unidirectional oscillation within said planar ring light path.

2. A monolithic laser resonator as in claim 1 wherein said means for inducing unidirectional oscillation includes a birefringence effect within said solid-state laser material.

3. A monolithic laser resonator as in claim 2 wherein said birefringence effect comprises linear birefringence.

4. A laser resonator as in claim 3 wherein said means for inducing unidirectional oscillation within said planar ring light path comprises: a reciprocal polarization transformation resulting from orienting the plane of said planar ring light path such that the principal axes of said linear birefringence effect are not parallel and perpendicular to said planar ring light path; a nonreciprocal polarization transformation associated with magneto-optic rotation of polarization occurring within said solid-state laser material on one or more of the legs of said planar ring light path, said magneto-optic rotation occurring only in the presence of an externally applied magnetic field; and a partial polarizer effect associated with oblique reflection from one or more reflection facets formed on said piece of solid-state laser medium, the angle of incidence on said reflection facets being less than the critical angle of incidence for total internal reflection within said solid-state laser medium.

5. A laser resonator as in claim 4 wherein said solid-state laser medium comprises a host whose point group symmetry is not cubic, such that said linear birefringence effect is inherent to the solid- state laser medium.

6. A laser resonator as in claim 5 wherein said linear birefringence effect comprises modification of said inherent linear birefringence by additional induced birefringence owing to the action of an externally applied stress via the photoelastic effect or electric field via the electro-optic effect.

7. A laser resonator as in claim 4 wherein said solid-state laser medium is, in the absence of said externally applied stress or electric field, either an amorphous glass or a crystal with cubic point group symmetry, such that said linear birefringence effect is wholly induced in the solid-state laser medium by the action of an externally applied stress or electric field.

8. A laser resonator as in claim 7 wherein said induced linear birefringence and said nonreciprocal polarization transformation occur simultaneously on one or more legs of said planar ring light path.

9. A laser resonator as in claim 7 wherein said induced linear birefringence is localized to apply to a single leg of said planar ring light path, such that said nonreciprocal polarization transformation does not simultaneously occur on said single leg.

10. A laser resonator as in claim 3 wherein said means for inducing unidirectional oscillation within said planar ring light path comprises: a reciprocal polarization transformation resulting from orienting the plane of said planar ring light path such that the principal axes of said linear birefringence effect are not parallel and perpendicular to said planar ring light path; a nonreciprocal polarization transformation associated with magneto-optic rotation of polarization occurring within said solid-state laser material on one or more of the legs of said planar ring light path, said magneto-optic rotation being an intrinsic property of said laser material; and a partial polarizer effect associated with oblique reflection from one or more reflection facets formed on said piece of solid-state laser medium, the angle of incidence on said reflection facets being less than the critical angle of incidence for total internal reflection within said solid-state laser medium.

11. A laser resonator as in claim 10 wherein said solid-state laser medium comprises a host whose point group symmetry is not cubic, such that said linear birefringence effect is inherent to the solid- state laser medium.

12. A laser resonator as in claim 11 wherein said linear birefringence effect comprises modification of said inherent linear birefringence by additional induced biref ringence owing to the action of an externally applied stress via the photoelastic effect or electric field via the electro-optic effect.

13. A laser resonator as in claim 10 wherein said solid-state laser medium is, in the absence of said externally applied stress or electric field, either an amorphous glass or a crystal with cubic point group symmetry, such that said linear birefringence effect is wholly induced in the solid- state laser medium by the action of an externally applied stress or electric field.

14. A laser resonator as in claim 13 wherein said induced linear birefringence and said nonreciprocal polarization transformation occur simultaneously on one or more legs of said planar ring light path.

15. A laser resonator as in claim 13 wherein said induced linear birefringence is localized to apply to a single leg of said planar right light path, such that said nonreciprocal polarization transformation does not simultaneously occur on said single leg.

16. A laser resonator as in claim 3 wherein said means for inducing unidirectional oscillation within said planar ring light path comprises: a reciprocal polarization transformation resulting from orienting the plane of said planar ring light path such that the principal axes of said linear birefringence effect are not parallel and perpendicular to said planar ring light path; a nonreciprocal polarization transformation associated with reflection from one or more magnetic Kerr effect magneto-optic mirrors located on reflection facets defining said planar ring light path; and a partial polarizer effect associated with oblique reflection from one or more reflection facets formed on said piece of solid-state laser medium, the angle of incidence on said reflection facets being less than the critical angle of incidence for total internal reflection within said solid-state laser medium.

17. A laser resonator as in claim 16 wherein said solid-state laser medium comprises a host whose point group symmetry is not cubic, such that said linear birefringence effect is inherent to the solid- state laser medium.

18. A laser resonator as in claim 16 wherein said linear birefringence effect comprises modification of said inherent linear birefringence by additional induced birefringence owing to the action of an externally applied stress via the photoelastic effect or electric field via the electro-optic effect.

19. A laser resonator as in claim 16 wherein said solid-state laser medium is, in the absence of said externally applied stress or electric field, either an amorphous glass or a crystal with cubic point group symmetry, such that said linear birefringence effect is wholly induced in the solid- state laser medium by the action of an externally applied stress or electric field.

20. A monolithic laser resonator comprising: a monolithic piece of solid-state laser material; input means for receiving input energy to pump said laser resonator; means for defining a ring light path involving three or more reflections internal to said piece of solid-state laser material, in which laser oscillation occurs in response to said input energy; means for extracting an output laser beam from said laser oscillation; means for inducing unidirectional oscillation within said ring light path; and means for tuning the eigenpolarization states of said monolithic laser resonator, said means for tuning comprising a tunable birefringence effect within said solid-state laser material.

21. A monolithic laser resonator as in claim 20 wherein said means for defining a ring light path comprises means for defining a non-planar ring light path involving four or more reflections internal to said piece of solid-state laser material.

22. A monolithic laser resonator as in claim 20 wherein said birefringence effect comprises linear birefringence.

23. A laser resonator as in claim 22 wherein said means for tuning the eigenpolarization states of said monolithic laser resonator comprises: a reciprocal polarization transformation resulting from orienting the plane of said planar ring light path such that the principal axes of said linear birefringence effect are not parallel and perpendicular to said planar ring light path; a nonreciprocal polarization transformation associated with magneto-optic rotation of polarization occurring within said solid-state laser material on one or more of the legs of said planar ring light path, said magneto-optic rotation occurring only in the presence of an externally applied magnetic field; and a partial polarizer effect associated with oblique reflection from one or more reflection facets formed on said piece of solid-state laser medium, the angle of incidence on said reflection facets being less than the critical angle of incidence for total internal reflection within said solid-state laser medium.

24. A laser resonator as in claim 23 wherein said solid-state laser medium comprises a host whose point group symmetry is not cubic, such that said linear birefringence effect is inherent to the solid- state laser medium.

25. A laser resonator as in claim 24 wherein said linear birefringence effect comprises modification of said inherent linear birefringence by additional induced birefringence owing to the action of an externally applied stress via the photoelastic effect or electric field via the electro-optic effect.

26. A laser resonator as in claim 23 wherein said solid-state laser medium is, in the absence of said externally applied stress or electric field, either an amorphous glass or a crystal with cubic point group symmetry, such that said linear birefringence effect is wholly induced in the solid- state laser medium by the action of an externally applied stress or electric field.

27. A laser resonator as in claim 26 wherein said induced linear birefringence and said nonreciprocal polarization transformation occur simultaneously on one or more legs of said planar ring light path.

28. A laser resonator as in claim 26 wherein said induced linear birefringence is localized to apply to a single leg of said planar ring light path, such that said nonreciprocal polarization transformation does not simultaneously occur on said single leg.

29. A laser resonator as in claim 22 wherein said means for tuning the eigenpolarization states of said monolithic laser resonator comprises: a reciprocal polarization transformation resulting from orienting the plane of said planar ring light path such that the principal axes of said linear birefringence effect are not parallel and perpendicular to said planar ring light path; a nonreciprocal polarization transformation associated with magneto-optic rotation of polarization occurring within said solid-state laser material on one or more of the legs of said planar ring light path, said magneto-optic rotation being an intrinsic property of said laser material; and a partial polarizer effect associated with oblique reflection from one or more reflection facets formed on said piece of solid-state laser medium, the angle of incidence on said reflection facets being less than the critical angle of incidence for total internal reflection within said solid-state laser medium.

30. A laser resonator as in claim 29 wherein said solid-state laser medium comprises a host whose point group symmetry is not cubic, such that said linear birefringence effect is inherent to the solid- state laser medium.

31. A laser resonator as in claim 30 wherein said linear birefringence effect comprises modification of said inherent linear birefringence by additional induced birefringence owing to the action of an externally applied stress via the photoelastic effect or electric field via the electro-optic effect.

32. A laser resonator as in claim 30 wherein said solid-state laser medium is, in the absence of said externally applied stress or electric field, either an amorphous glass or a crystal with cubic point group symmetry, such that said linear birefringence effect is wholly induced in the solid- state laser medium by the action of an externally applied stress or electric field.

33. A laser resonator as in claim 32 wherein said induced linear birefringence and said nonreciprocal polarization transformation occur simultaneously on one or more legs of said planar ring light path.

34. A laser resonator as in claim 34 wherein said induced linear birefringence is localized to apply to a single leg of said planar right light path, such that said nonreciprocal polarization transformation does not simultaneously occur on said single leg.

35. A laser resonator as in claim 22 wherein said means for tuning the eigenpolarization states of said monolithic laser resonator comprises: a reciprocal polarization transformation resulting from orienting the plane of said planar ring light path such that the principal axes of said linear birefringence effect are not parallel and perpendicular to said planar ring light path; a nonreciprocal polarization transformation associated with reflection from one or more magnetic Kerr effect magneto-optic mirrors located on reflection facets defining said planar ring light path; and a partial polarizer effect associated with oblique reflection from one or more reflection facets formed on said piece of solid-state laser medium, the angle of incidence on said reflection facets being less than the critical angle of incidence for total internal reflection within said solid-state laser medium.

36. A laser resonator as in claim 35 wherein said solid-state laser medium comprises a host whose point group symmetry is not cubic, such that said linear birefringence effect is inherent to the solid- stace laser medium.

37. A laser resonator as in claim 35 wherein said linear birefringence effect comprises modification of said inherent linear birefringence by additional induced birefringence owing to the action of an externally applied stress via the photoelastic effect or electric field via the electro-optic effect.

38. A laser resonator as in claim 1 which further comprises: a cylindrical rod having: a first end including two planar facets; and a second end including input coupling means and output coupling means and said partial polarizer means, wherein said cylindrical rod has a plane of mirror symmetry containing the line of intersection of said two planar facets and the longitudinal axis of said cylindrical rod, wherein said planar ring light path is defined by three internal reflections in a round trip, one reflection occurring at said second end, and one reflection occurring at each of said planar facets of said first end; and means for applying stress or an electric field substantially perpendicular to the longitudinal axis of said cylindrical rod and azimuthally oriented such that the principal axes of the resulting induced linear birefringence are not parallel and perpendicular to the plane of the ring light path.

39. A laser resonator as in claim 38 wherein said two planar facets are uncoated and said reflections from said planar facets are total internal reflections.

40. A laser resonator as in claim 38 wherein one or both of said planar facets are coated and the ring light path geometry is chosen such that total internal reflection does not occur at said one or both planar facets.

41. A laser resonator as in claim 38 in which said second end comprises a spherical end cap which provides reflection which is not a total internal reflection.

42. A laser resonator as in claim 41 wherein said spherical end cap includes a multilayer dielectric coating.

43. A laser resonator as in claims 1, 20, or 21 comprising: a plurality of planar facets; a spherical end cap; and a multilayer dielectric coating on said spherical end cap, wherein said spherical end cap serves as said input means, said output means, and said partial polarizer means.

44. A laser resonator as in claim 43 wherein all reflections in said ring other than at said spherical end cap comprise total internal reflections from planar facets.

45. A laser resonator as in claim 43 wherein said ring light path includes an odd number of reflections and wherein said monolithic resonator has a plane of mirror symmetry perpendicular to the plane of the ring light path, the plane of mirror symmetry bisecting the apex angle at the reflection vertex in said spherical end cap.

46. A laser resonator as in claim 45 wherein said birefringence effect is located locally to the leg of said ring light path opposite the reflection vertex in said spherical end cap.

47. A laser resonator as in claim 45 wherein said externally applied magnetic field lies substantially along the line of intersection of the plane of mirror symmetry and the plane of the ring light path, thereby causing said magnetic field to be substantially perpendicular to the leg of said ring light path in which said linear birefringence is induced.

48. A laser resonator as in claims 7, 13, 19, 20, or 21 which comprises a plurality of means for inducing linear birefringence.

49. A laser resonator as in claim 48 wherein said plurality of means for inducing linear birefringence are configured to enable modulation of the eigenpolarization states or frequency of said laser resonator while maintaining unidirectional oscillation.

50. A laser resonator as in claim 48 wherein said plurality of means for inducing linear birefringence are configured to enable switching the direction of oscillation of said ring light path.