Waveguide structures in anisotropic lasing and nonlinear optical media

Abstract

A laser or nonlinear optical waveguide is presented that is formed from a core anisotropic crystal sandwiched by a cladding of anisotropic crystals of the same material but slightly rotated optical axes. The core and cladding crystals can be cut from the same crystal boule and bonded without adhesives between them. Because the crystals are anisotropic, the core and slightly skewed cladding crystals exhibit different refractive indexes to a propagating light beam. The difference in refractive indexes should be 1.210.sup.6 for mode confinement and 2d/*Sqrt(n.sub.core.sup.2n.sub.clad.sup.2)1.37 to achieve single mode operation in a square cross section, 1 for a planar cross section. Alternative embodiments use slightly different doping amounts in crystals to achieve the difference in refractive indexes between the core and cladding.

Claims

1. A single-mode waveguide apparatus comprising: an anisotropic core crystal having a core optical axis, at least two flat surfaces opposite one another, a thickness d between the at least two flat surfaces, and a light beam propagation axis that is parallel to two of the at least two flat surfaces; and a pair of anisotropic cladding crystals having flat surfaces that are intimately joined with two of the at least two flat surfaces of the core crystal and sandwiching the core crystal, the cladding crystals having a cladding optical axis, the cladding optical axis and the core optical axis being rotated with respect to each other in a direction of, or direction opposite to, the light beam propagation axis, the anisotropic core crystal and anisotropic cladding crystals being comprised of a same material, wherein the core optical axis and the cladding optical axis are aligned such that: $\frac{2 d}{} \sqrt{n_{core}^{2} - n_{clad}^{2}} 1.37$ where n.sub.core is a refractive index of the core crystal for a light beam with wavelength propagating parallel to the light beam propagation axis; n.sub.clad is a refractive index of the cladding crystals for the light beam with wavelength propagating parallel to the light beam propagation axis.

2. The apparatus of claim 1 wherein the core crystal has a square cross section and the at least two flat surfaces include four flat surfaces, the apparatus further comprising: a second pair of anisotropic cladding crystals having flat surfaces that are intimately joined with a remaining two of the four flat surfaces.

3. The apparatus of claim 1 wherein the core crystal is planar and the core optical axis and the cladding optical axis are aligned such that: $\frac{2 d}{} \sqrt{n_{core}^{2} - n_{clad}^{2}} 1.$

4. The apparatus of claim 1 wherein the wavelength is within a visible light wavelength between 390 nanometers (nm) and 700 nm.

5. The apparatus of claim 1 wherein the wavelength is within an infrared wavelength between 700 nanometers (nm) and 1 millimeter (mm).

6. The apparatus of claim 1 wherein: the material of the anisotropic core crystal and cladding crystal is a positive uniaxial nonlinear crystal material consisting essentially of: ZGP (ZnGeP.sub.2).

7. The apparatus of claim 1 wherein: the material of the anisotropic core crystal and cladding crystal is a negative uniaxial nonlinear crystal material selected from the group consisting essentially of: BaB.sub.2O.sub.4, CsLiB.sub.6O.sub.10, LiNbO.sub.3, MgO:LiNbO.sub.3, AgGaS.sub.2, or AgGaSe.sub.2.

8. The apparatus of claim 1 wherein: the material of the anisotropic core crystal and cladding crystal is a biaxial nonlinear crystal material selected from the group consisting essentially of: KTP (KTiPO.sub.4), LiB.sub.3O.sub.5, KNbO.sub.3, CsB.sub.3O.sub.5, BiB.sub.3O.sub.6, CsTiAsO.sub.4, and RbTiOAsO.sub.4.

9. The apparatus of claim 1 wherein the core crystal and cladding crystals are joined without adhesive.

10. The apparatus of claim 1 wherein a cross section of the core crystal and cross sections of the cladding crystals are constant throughout the waveguide apparatus.

11. The apparatus of claim 1 wherein a cross section of the core crystal is wedged along the light beam propagation axis.

12. The apparatus of claim 1 further comprising: an outer cladding joined with the cladding crystals.

13. A laser apparatus comprising: the waveguide apparatus of claim 1; and a pump laser.

14. A method of manufacturing a single-mode waveguide, the method comprising: providing a first anisotropic crystal with a flat surface; polishing the flat surface; measuring, using X-rays, a first angle of the flat surface with respect to an optical axis of the first crystal; determining a refractive index n.sub.1 of the first crystal for a light beam with wavelength propagating in a direction parallel to the flat surface; gauging, using X-rays, an optical axis orientation of a second anisotropic crystal, the first and second crystals being comprised of a same anisotropic material; calculating a second angle with respect to the optical axis orientation of the second crystal such that a refractive index n.sub.2 of the second crystal for a light beam with wavelength propagating parallel to the second angle is such that: $\frac{2 d}{} \sqrt{n_{1}^{2} - n_{2}^{2}} 1.37$ where d is a nominal lateral core width for the waveguide; cutting and polishing the second anisotropic crystal to form a flat surface on the second anisotropic crystal at the second angle to the optical axis of the second anisotropic crystal; and joining the flat surfaces of the first and second crystals together.

15. The method of claim 14 wherein the joining of the flat surfaces is achieved without adhesive between the first and second anisotropic crystals.

16. The method of claim 14 further comprising: cutting the first anisotropic crystal from a single crystal boule, wherein the cutting of the second anisotropic crystal is from the single crystal boule.

17. The apparatus of claim 1 wherein: the core optical axis is perpendicular to the light beam propagation axis; and the cladding optical axis is tilted with respect to the light beam propagation axis.

18. The apparatus of claim 1 wherein: the core optical axis is tilted with respect to the light beam propagation axis; and the cladding optical axis is perpendicular with respect to the light beam propagation axis.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) FIG. 1 illustrates a light beam propagating in a uniaxial nonlinear crystal or anisotropic laser crystal.

(2) FIG. 2 illustrates a refractive index ellipsoid of rotation in a positive (n.sub.e>n>n.sub.o) uniaxial nonlinear crystal or anisotropic laser crystal.

(3) FIG. 3 illustrates a refractive index ellipsoid of rotation in a negative (n.sub.o>n>n.sub.e) uniaxial nonlinear crystal or anisotropic laser crystal. The ellipse has a long axis of n.sub.o and a short axis of n.sub.e.

(4) FIG. 4A is an isometric view of a planar waveguide apparatus in accordance with an embodiment.

(5) FIG. 4B is an isometric exploded view of the waveguide apparatus of FIG. 4A.

(6) FIG. 4C is an end view of a waveguide apparatus of FIG. 4A.

(7) FIG. 4D is a side view of a waveguide apparatus of FIG. 4A with a positive crystal.

(8) FIG. 4E is a side view of a waveguide apparatus of FIG. 4A with a negative crystal.

(9) FIG. 5A is an isometric view of a square cross-section waveguide apparatus in accordance with an embodiment.

(10) FIG. 5B is an end view of a waveguide apparatus of FIG. 5A.

(11) FIG. 5C is a side view of a waveguide apparatus of FIG. 5A.

(12) FIG. 5D is a close up view of the middle center of end view of FIG. 5B.

(13) FIG. 6A shows a periodically poled waveguide with a uniaxial material in accordance with an embodiment.

(14) FIG. 6B is an end view of a waveguide apparatus of FIG. 6A.

(15) FIG. 6C is a side view of a waveguide apparatus of FIG. 6A.

(16) FIG. 7 shows a refractive index ellipsoid of a biaxial material, with axes n.sub.x, n.sub.y, and n.sub.z, where n.sub.x<n.sub.y<n.sub.z.

(17) FIG. 8 illustrates a core using biaxial nonlinear materials where light propagates on the x-axis in accordance with an embodiment. The core employs on-axis material, while the inner cladding employs slightly off-axis material to form the waveguide structure;

(18) FIG. 9 illustrates an inner cladding to be used with the core of FIG. 8.

(19) FIG. 10 illustrates a core using biaxial nonlinear materials where light propagates on the z-axis. The inner cladding employs on-axis material, while the core employs slightly off-axis material;

(20) FIG. 11 illustrates an inner cladding to be used with the core of FIG. 10.

(21) FIG. 12 illustrates a core using biaxial nonlinear materials where light propagates on the y-axis. No proper orientation of material can be found to form a waveguide structure.

(22) FIG. 13 illustrates a cladding using biaxial nonlinear materials where light propagates on the y-axis, conceptually to be used with that in FIG. 12.

(23) FIG. 14A shows a legend for a double-clad laser waveguide fabrication process in accordance with an embodiment.

(24) FIG. 14B shows a step in a double-clad laser waveguide fabrication process in accordance with an embodiment.

(25) FIG. 14C shows a step in a double-clad laser waveguide fabrication process in accordance with an embodiment.

(26) FIG. 14D shows a step in a double-clad laser waveguide fabrication process in accordance with an embodiment.

(27) FIG. 14E shows a step in a double-clad laser waveguide fabrication process in accordance with an embodiment.

(28) FIG. 14F shows a step in a double-clad laser waveguide fabrication process in accordance with an embodiment.

(29) FIG. 14G shows a step in a double-clad laser waveguide fabrication process in accordance with an embodiment.

(30) FIG. 14H shows a step in a double-clad laser waveguide fabrication process in accordance with an embodiment.

(31) FIG. 14I shows a step in a double-clad laser waveguide fabrication process in accordance with an embodiment.

DETAILED DESCRIPTION

(32) Considerations for Establishing Desired Single-Mode or Multimode Operation

(33) Forming an intrinsic single-mode waveguide structure often requires selecting the core and cladding materials with an appropriate refractive index difference n=|n.sub.coren.sub.clad|.

(34) For planar waveguides, intrinsic single-mode in the lateral direction largely requires Eq. (2) to be fulfilled:

(35) $\begin{matrix} \frac{2 d}{} \sqrt{n_{core}^{2} - n_{clad}^{2}} < 1 & (2) \end{matrix}$
where d is the waveguide core width in the guiding direction and is the wavelength of light.

(36) In comparison, for a circular cross section channel waveguide, i.e., conventional glass fiber, intrinsic single-mode operation largely requires Eq. (3) to be fulfilled:

(37) $\begin{matrix} \frac{2 d}{} \sqrt{n_{core}^{2} - n_{clad}^{2}} < 1.53 & (3) \end{matrix}$
where d is the core diameter and is the wavelength of light.

(38) For square cross section channel waveguides, e.g., an adhesive-free bonded crystalline fiber waveguide, intrinsic single mode largely requires Eq. (4) to be fulfilled:

(39) $\begin{matrix} \frac{2 d}{} \sqrt{n_{core}^{2} - n_{clad}^{2}} < 1.37 & (4) \end{matrix}$
where d is the core width and is the wavelength of light.

(40) The quantity on the left hand side of Eq. (2)-(4) is twice the characteristic core dimension divided by the wavelength times the numerical aperture. The quantity on the right side of the equations is a threshold for the different representative waveguide geometries.

(41) When the value of (2d/)*NA is less than the threshold value, only the lowest-order mode is allowed in the waveguide. In this case, some of the energy in the incident beam would be coupled into the mode, with the remainder lost to mode mismatch.

(42) When the value of (2d/)*NA is greater than the threshold value, there exists a design region where the number of actually observed modes depends on the actual experimental conditions. That is, with such waveguide design, multimode operation becomes a possibility rather than something certain to happen. Single-mode output with a multimode design can still occur with more constrained experimental conditions of input beam quality, such as an incident angle parallel to the input end normal, and diffraction-limited beam quality with M.sup.2 value close to 1. The more mode numbers that are to be allowed by the design, the better the input beam quality must be to achieve actual single-mode output.

(43) Consider the use of a waveguide as an amplifier in a laser-pumped system, as opposed to diode-pumped system. A laser pump typically has better beam quality than a diode pump, and can implement actual single-mode output more easily than a diode pump, even though the waveguide is designed for low-order multimode with (2d/)*NA larger than the corresponding threshold. The better the beam quality of the pump, the larger (2d/)*NA can be while still resulting in single-mode output.

(44) Achieving Index Differences (Different Dopant Concentration Vs. Optical Axes at an Angle)

(45) The core size is typically few tens of thick microns, and the operation wavelength is typically between 0.5 and 2 microns. For this case, the refractive index difference between the core and inner cladding should be within a few 10.sup.4 to fulfill the above inequalities. Operation wavelengths between 390 nm and 700 nm for visible light, between 700 nm and 1 mm for infrared light, and other wavelengths are envisioned. The present disclosure demonstrates the methodology of selecting and manipulating laser and nonlinear materials to implement intrinsic single-mode operation.

(46) The suitable cladding material should meet the following criteria: 1) The refractive index should be slightly smaller (by 10.sup.4) than the core material, for both fundamental and harmonic conversion waves; 2) The crystal structure should be similar or the same as the core material in order to implement adhesive-free-bonding; and 3) The refractive index should be selectable in a certain range to fulfill single transverse-mode operation.

(47) One way to achieve a small refractive index difference n is to use material with a higher dopant concentration in the core and lower dopant concentration in the inner cladding, with the same host material. An example of a waveguide made of isotropic crystal materials, a planar waveguide, with a lateral width of 25 m, has core material of 3% Yb:YAG and inner cladding of 2.5% Yb:YAG. The outer cladding is sapphire, which has higher thermal conductivity and mechanically supports the waveguide structure. This technique of obtaining n between the core and the cladding can also be used for anisotropic laser crystals.

(48) A constraint of using a dopant concentration difference to achieve an index difference is that there may be a limited number of material dopant concentrations available. That is, the proper combination may or may not exist due to a limited selection of available crystals. Further, reproducibility, in terms of refractive index difference from one batch of crystals to the next and of the uniformity of dopant concentration from one end of a crystal to the other, may be an issue. To achieve proper doping concentrations, the refractive index difference would likely need to be measured using an interferometric method. Once that is established, the waveguide core size may need to be adjusted to maintain single-mode operation.

(49) Within the constraints of anisotropic crystals with different refractive indices, it may or may not be possible to achieve a waveguide with desired geometrical parameters. There is a dearth of crystal materials that have a matching coefficient of thermal expansion (CTE), lattice spacings, and other important parameters.

(50) Yet, for anisotropic laser materials such as YLF and nonlinear crystals, the refractive index of light changes with incidence angle. The inventors recognized that crystals can be cut at precise angles to very tight tolerances and then joined with little-to-no interface between their flat surfaces. Therefore it is possible to either increase or decrease a refractive index by slightly tilting the material to reach desired index difference value.

(51) Technical advantages of this angle-tilting method are: a) only one type of crystal or crystal boule is needed; b) the CTE is well compatible between the core and inner cladding because the tilt angle (e.g., 1) is small enough to make an immaterial difference in CTE; and c) a continuously tunable refractive index difference as function of tilt angle is available for design purposes. Therefore, the core cross section of the core can be designed simply based on the tilt angle and a predictable refractive index difference. This is done without the need of trying to match the core and the inner cladding with two different crystal materials or rely upon a crystal manufacturer's doping precision. Different boules may have slightly different impurities and thus slightly different refractive indexes. Using the same boule for both the core and cladding can eliminate reliance on the manufacturer's tolerances such that an in-house process can concentrate on measuring and cutting at the proper angle to give a desired difference in refractive indexes.

(52) Various Types of Anisotropic Materials

(53) FIG. 1 illustrates a light beam propagating in a uniaxial nonlinear crystal or anisotropic laser crystal. Light with different polarizations experiences differences in refractive indexes when propagating in uniaxial material, due to birefringence. The refractive index equals n.sub.e when the polar angle between the polarization and the optical axis is 0 (parallel), n.sub.o when the polar angle between the polarization and optical axis is /2 (perpendicular), and n, min(n.sub.e, n.sub.o)<n<max(n.sub.e, n.sub.o) when the polar angle between the light polarization and optical axis follows the relationship 0<</2.

(54) Refractive index n is a function of polarization angle and quantified by Eq. (5):

(55) $\begin{matrix} n^{e} () = n_{e} \sqrt{\frac{1 + \tan^{2}}{1 + {(n_{e} / n_{o})}^{2} \tan^{2}}} & (5) \end{matrix}$

(56) n can also be viewed as the distance between any point on the refractive index ellipse and its center with polar angle , where the refractive index ellipse has a long axis of max(n.sub.e, n.sub.o) and a short axis of min(n.sub.e, n.sub.o). Therefore, by adjusting the polar angle , one may continuously tune the refractive index difference between 0 and |n.sub.en.sub.o|.

(57) FIG. 2 illustrates a refractive index ellipsoid of rotation in a positive (n.sub.e>n>n.sub.o) uniaxial nonlinear crystal or anisotropic laser crystal. The ellipse has a long axis of n.sub.e and a short axis of n.sub.o. The actual refractive index with an arbitrary polar angle is the distance between the point with polar angle on the ellipse and its centroid.

(58) For positive uniaxial material, n.sub.e>n.sub.o, the optical axis of the core material should be parallel to the light polarization, while the optical axis of the cladding should be slightly tilted.

(59) FIG. 3 illustrates a refractive index ellipsoid of rotation in a negative (n.sub.o>n>n.sub.e) uniaxial nonlinear crystal or anisotropic laser crystal. The ellipse has a long axis of n.sub.o and a short axis of n.sub.e. The actual refractive index with an arbitrary polar angle is the distance between the point with polar angle on the ellipse and its centroid.

(60) For negative uniaxial material, n.sub.o>n.sub.e, the optical axis of the cladding material should be parallel to the light polarization, while the optical axis of the core should be slightly tilted.

(61) Therefore, it is almost always is possible to design a n in a uniaxial laser crystal that will result in waveguiding, single-mode or multimode operation, depending on the magnitude of n and the core cross section.

(62) In neither the positive nor negative uniaxial nonlinear optical crystals can a waveguide structure be achieved if two or more interaction beams have orthogonal polarization. This is because the ordinary beam and the extraordinary beam experience an opposite refractive index change in an optical-axis-tilted crystal with respect to an on-axis crystal. Although a nonlinear waveguide for a normal type I or type II conversion is not achievable with such a technique due to the fact that interaction beams with orthogonal polarizations are common in nonlinear conversions, nonlinear frequency conversion still takes place as long as an electric field distribution is present in the nonlinear crystal.

(63) FIGS. 4A-4E illustrate a planar waveguide apparatus in accordance with an embodiment. In assembly 400, waveguide system 401 is constructed of five layers.

(64) Core and inner cladding waveguide 402 are sandwiched by outer cladding 403, which provides mechanical support and a thermal conduit. Outer cladding 403 can be a wide variety of materials, such as silica glass, yet it may be sapphire for greater thermal conductivity. Electromagnetic radiation propagates in the direction of light beam propagation axis 410.

(65) FIG. 4B shows an exploded view of the five layers. Anisotropic core crystal 405 has two flat planar surfaces 409 and 411 that are parallel and opposite to one another. Its thickness d is a constant between the two surfaces. The core is configured such that it is parallel with light beam propagation axis 410.

(66) For core and inner cladding waveguide 402, a pair of anisotropic cladding crystals 406 and 407 with flat surfaces 408 and 410, respectively, sandwiches core crystal 405. Bottom flat surface 408 of cladding crystal 406 is directly and intimately joined with top flat surface 409 of core crystal 405. Top flat surface 410 of cladding crystal 407 is directly and intimately joined with bottom flat surface 411 of core crystal 405.

(67) Outer cladding 403 is joined on the top and bottom of waveguide 402 to sandwich it.

(68) FIGS. 4C-4E illustrate crystal alignments in core crystal 405 and inner cladding crystals 406 and 407. Viewed end on (in FIG. 4C), core optical axis 415 appears straight up and down, like cladding optical axes 416 and 417 in upper and lower inner cladding crystals 406 and 407.

(69) Viewed to the side in a positive crystal (in FIG. 4D), core optical axis 415 is straight up and down but cladding optical axes 416 and 417 are tilted. This results in a light beam seeing refractive indexes n.sub.core=n.sub.e and n.sub.clad=n.

(70) Viewed to the side in a negative crystal (in FIG. 4E), core optical axis 415 is tilted but cladding optical axes 416 and 417 are straight up and down. This results in a light beam seeing refractive indexes n.sub.core=n and n.sub.clad=n.sub.e.

(71) In the embodiment of the figures, width d is contact throughout the cross section. In some embodiments, d may change linearly along the axis of light beam propagation to form wedge.

(72) FIGS. 5A-5D illustrate a square cross-section waveguide apparatus in accordance with an embodiment. In assembly 400, waveguide system 501 is constructed of multiple layers in which a square waveguide is sandwiched from the top and bottom and from the sides by inner cladding crystals.

(73) Core and inner cladding waveguide 502 are sandwiched by outer cladding 503, which provides mechanical support and a thermal conduit. Electromagnetic radiation propagates in the direction of light beam propagation axis 510.

(74) FIG. 5D is a close up view of the small rectangular region of 502. Core crystal 505 is sandwiched by inner cladding 506 and 507 on the top and bottom as well as inner cladding 512 and 513 from the left and right sides.

(75) Either the core optical axis is straight and the cladding optical axis is tilted or the core optical axis is tilted and the cladding optical axis is straight. The difference in optical axis orientations causes a difference in refractive indexes of the materials as seen by a beam of propagating light.

(76) The above technique can be combined with a periodically poling technique to fabricate a periodically poled nonlinear waveguide to enhance frequency conversion efficiency by fulfilling a quasi-phase matching condition.

(77) FIGS. 6A-6C illustrate a periodically poled waveguide with a uniaxial material, such as LiNbO.sub.3. In system 600, all interaction beams are polarized along the z-axis and can be waveguided in waveguide 602. A periodically poled LiNbO.sub.3 waveguide is fabricated by an adhesive-free bonding technique where an optical axis 605 of core crystal 605 is slightly tilted while an optical axis 616 and 617 of cladding crystals 606 and 607 is not. In such a waveguide, all beams are polarized in the same direction so that waveguiding is possible.

(78) Although no critical phase matching can be achieved with all interaction beams polarized in the same direction, after a certain propagation distance where the phase mismatch accumulates to integer multiples of , the domain flips so as to change the sign of the nonlinear coefficient. In this way, the electric field of converted radiation keeps growing, with all beams confined and well overlapped in the waveguide structure.

(79) Biaxial Crystals

(80) While conventional type I or type II nonlinear frequency conversion may not be implemented in a waveguide employing uniaxial crystals, they are feasible in biaxial crystals.

(81) FIG. 7 shows a refractive index ellipsoid of a biaxial material, with axes n.sub.x, n.sub.y, and n, where n.sub.x<n.sub.y<n.sub.z. The actual refractive index with arbitrary polar angle and azimuth angle is defined as the distance between the point with a corresponding polar angle and azimuth angle on the ellipsoid surface to its centroid.

(82) Biaxial materials demonstrate a refractive index ellipsoid instead of a refractive index ellipse as in uniaxial materials, with axes n.sub.x, n.sub.y, and n, where n.sub.x<n.sub.y<n.sub.z. The actual refractive index with arbitrary polar angle and azimuth angle is defined as the distance between the point with the corresponding polar angle and azimuth angle on the ellipsoid surface to its centroid.

(83) FIGS. 8-9 illustrate a light beam propagating along the x-axis and two orthogonal polarizations in the y-z plane. For simplicity, we assume they are polarized along the y-axis with refractive index n.sub.y and the z-axis with refractive index n.sub.z, respectively. If we tilt the crystal first with angle with respect to y-axis, and then angle with respect to the original z-axis (see FIG. 9), i.e., in the original x-y plane, the beam originally polarized along they-axis will now experience refractive index n.sub.1 with n.sub.x<n.sub.1<n.sub.y, and the beam originally polarized along the z-axis will now experience a refractive index n.sub.2 with n.sub.x<n.sub.2<n.sub.z. Both refractive indices decrease. Therefore, the tilted biaxial material may be used as cladding bonded to the same un-tilted material as core. It is evident that the situation for obtaining a n in a biaxial laser crystal is similar to a uniaxial laser crystal where no interacting beams need to be considered.

(84) FIGS. 10-11 illustrate a light beam propagating along the z-axis and the two orthogonal polarizations will be in the y-x plane. For simplicity, we assume they are polarized along the y-axis with refractive index n.sub.y and the x-axis with refractive index n.sub.x, respectively. If we tilt the crystal first with angle with respect to x-axis, and then with angle with respect to the original y-axis, i.e., in the original x-z plane, the beam originally polarized along they-axis will now experience refractive index n.sub.3 with n.sub.y<n.sub.3<n.sub.z, and the beam originally polarized along the x-axis will now experience refractive index n.sub.4 with n.sub.x<n.sub.4<n.sub.z. We find that both refractive indices increase. Therefore, the tilted biaxial material may be used as core bonded to the same un-tilted material as cladding.

(85) FIGS. 12-13 illustrate a light beam propagating along they-axis and the two orthogonal polarizations will be in the z-x plane.

(86) If we assume the beams are polarized along the z-axis and the x-axis respectively, we find the refractive index change will always go the opposite way no matter how we tilt the crystal, since n.sub.z is the largest refractive index possible and it always decreases, while n.sub.x is the smallest refractive index possible and it always increases. In that case, as long as the phase-matching condition does not require strictly that the two orthogonal polarizations be polarized along the x-axis and the z-axis, it is still possible that the sign of refractive index change of the two beams is the same, i.e., they both increase or both decrease, so that a waveguide structure is feasible.

(87) FIGS. 14A-14I illustrate a fabrication process of a double-clad laser waveguide such as that shown in FIG. 5A. The process consists of at least eight major steps. Each step includes cutting, polishing and heat treatment.

(88) FIG. 14B illustrates an adhesive free bonding of one of the inner cladding crystals to the core crystal. It also shows the adhesive free bonding of left and right outer cladding crystals to left and right inner cladding crystals.

(89) FIG. 14C illustrates an adhesive free bonding of the bottom inner cladding crystal to the core crystal in order to create a top cladding, core, bottom cladding sandwich.

(90) FIG. 14D illustrates a cutting of the left cladding and then an adhesive free bonding of the cut left cladding to the top cladding, core, bottom cladding sandwich.

(91) FIG. 14E illustrates a cutting of the top cladding, core, bottom cladding sandwich.

(92) FIG. 14F illustrates a cutting of the right cladding and then an adhesive free bonding of the cut right cladding to the top cladding, core, bottom cladding sandwich to form a middle portion.

(93) FIG. 14G illustrates a horizontal cutting of the middle portion and adhesive free bonding of a top outer cladding to the cut middle portion.

(94) FIG. 14H illustrates a horizontal cutting of the middle portion and adhesive free bonding of a bottom outer cladding to the cut middle portion.

(95) FIG. 14I illustrates an end view of the final product, like that in FIGS. 5A-5D.

(96) During manufacture, the orientation of the anisotropic laser materials and/or nonlinear optical crystals, can be determined by a D2 CRYSO X-ray Diffraction machine manufactured by Bruker AXS GmbH of Karlsruhe, Germany. The crystals are mounted on a goniometer, and can be tuned with an accuracy of better than 0.1.

(97) The actual refractive index difference between on-axis and slightly tilted crystals can be measured and confirmed using a VeriFire interferometer manufactured by Zygo Corporation of Middlefield, Conn., U.S.A. For example, at a 1.55 m operating wavelength, the interferometer can measure refractive index difference to the accuracy of 110.sup.6.

Example 1: LBO Nonlinear Waveguide

(98) An embodiment of the present invention is a lithium triborate LiB.sub.3O.sub.5 (LBO) waveguide. LBO is a biaxial optics crystal with a wide transparency range. In the preferred embodiment, a slightly off-axis LBO is used as cladding material to sandwich an on-axis LBO core.

(99) The purpose of the waveguide is to achieve second-harmonic generation from a fundamental beam of 1070 nm to its second harmonic of 535 nm. The fundamental beam is polarized on the z-axis, while the second-harmonic beam is polarized on they-axis. Both beams propagate along the x-axis.

(100) Technical advantages of using LBO include but are not limited to: 1) The refractive index is slightly smaller than the core material, for both fundamental and second harmonic waves; 2) Since they are the same material, the coefficients of thermal expansion are compatible, given that the core and the cladding LBO crystals only have a small angular deviation between them; and 3) The refractive index is continuously tunable by accurately rotating the offset angle on both the y-axis and the z-axis.

(101) The crystal should to be tilted first at an angle with respect to the y-axis, and then at an angle with respect to the original z-axis, i.e., in the original x-y plane to achieve the required refractive index difference for both the fundamental and second harmonic beams. The wave vector of the two beams can be described by Eq. (6):
{right arrow over (k)}={right arrow over (k)}.sub.0(cos cos {right arrow over (X)}+cos sin {right arrow over (Y)}+sin {right arrow over (Z)})(6)

(102) The refractive index for the fundamental beam in the cladding would be the long axis (LA) of the refractive index ellipse, determined by Eq. (7):

(103) $\begin{matrix} {\begin{matrix} X \cos \cos + Y \cos \sin + Z \sin = 0 \\ \frac{X^{2}}{{(n_{X}^{1070 n m})}^{2}} + \frac{Y^{2}}{{(n_{Y}^{1070 n m})}^{2}} + \frac{Z^{2}}{{(n_{Z}^{1070 n m})}^{2}} = 1 \end{matrix} & (7)) \end{matrix}$

(104) The refractive index of the fundamental beam n.sub.f is then found in Eq. (8):

(105) $\begin{matrix} n_{f} = LA = \frac{1}{\sqrt{\frac{\sin^{2} \cos^{2}}{{(n_{X}^{1070 n m})}^{2}} + \frac{\sin^{2} \sin^{2}}{{(n_{Y}^{1070 n m})}^{2}} + \frac{\cos^{2}}{{(n_{Z}^{1070 n m})}^{2}}}} & (8) \end{matrix}$

(106) Similarly, the refractive index for the second harmonic beam in the cladding would be the short axis (SA) of the refractive index ellipse, determined by Eq. (9):

(107) $\begin{matrix} {\begin{matrix} X \cos \cos + Y \cos \sin + Z \sin = 0 \\ \frac{X^{2}}{{(n_{X}^{535 n m})}^{2}} + \frac{Y^{2}}{{(n_{Y}^{535 n m})}^{2}} + \frac{Z^{2}}{{(n_{Z}^{535 n m})}^{2}} = 1 \end{matrix} & (9) \end{matrix}$

(108) And the refractive index of second harmonic beam n.sub.s is found in Eq. (10):

(109) 0 $\begin{matrix} n_{s} = SA = \frac{1}{\sqrt{\frac{\sin^{2}}{{(n_{X}^{535 n m})}^{2}} + \frac{\cos^{2}}{{(n_{Y}^{535 n m})}^{2}}}} & (10) \end{matrix}$

(110) Given n.sub.X.sup.1070 nm=1.565, n.sub.Y.sup.1070 nm=1.591, n.sub.Z.sup.1070 nm=1.605, n.sub.X.sup.535 nm=1.578, n.sub.Y.sup.535 nm=1.606, we calculate the refractive index difference for both fundamental and second harmonic beam for off-axis angle , <1, and obtain the following table. For a core width of 50 m, single-mode output would require an index difference less than 2.2410.sup.6. The index difference is small enough to support a good beam quality in the waveguide if the azimuth angle deviation is less than 0.5.

(111) A table below lists the refractive index difference when tilting the crystal by and :

(112) TABLE-US-00001 () () (Fund. Index) (10.sup.4) (SH. Index) (10.sup.5) 0 0.0 0 0 0.1 0 0.0088 0.2 0 0.035 0.3 0 0.0788 0.4 0 0.1401 0.5 0 0.2189 0.6 0 0.3153 0.7 0 0.4291 0.8 0 0.5604 0.9 0 0.7093 1.0 0 0.8757 0.1 0.0 0.0013 0 0.1 0.0013 0.0088 0.2 0.0013 0.035 0.3 0.0013 0.0788 0.4 0.0013 0.1401 0.5 0.0013 0.2189 0.6 0.0013 0.3153 0.7 0.0013 0.4291 0.8 0.0013 0.5604 0.9 0.0013 0.7093 1.0 0.0013 0.8757 0.2 0.0 0.0051 0 0.1 0.0051 0.0088 0.2 0.0051 0.035 0.3 0.0051 0.0788 0.4 0.0051 0.1401 0.5 0.0051 0.2189 0.6 0.0051 0.3153 0.7 0.0051 0.4291 0.8 0.0051 0.5604 0.9 0.0051 0.7093 1.0 0.0051 0.8757 0.3 0.0 0.0114 0 0.1 0.0114 0.0088 0.2 0.0114 0.035 0.3 0.0114 0.0788 0.4 0.0114 0.1401 0.5 0.0114 0.2189 0.6 0.0114 0.3153 0.7 0.0114 0.4291 0.8 0.0114 0.5604 0.9 0.0114 0.7093 1.0 0.0114 0.8757 0.4 0.0 0.0202 0 0.1 0.0202 0.0088 0.2 0.0202 0.035 0.3 0.0202 0.0788 0.4 0.0202 0.1401 0.5 0.0202 0.2189 0.6 0.0202 0.3153 0.7 0.0202 0.4291 0.8 0.0202 0.5604 0.9 0.0202 0.7093 1.0 0.0202 0.8757 0.5 0.0 0.0316 0 0.1 0.0316 0.0088 0.2 0.0316 0.035 0.3 0.0316 0.0788 0.4 0.0316 0.1401 0.5 0.0316 0.2189 0.6 0.0316 0.3153 0.7 0.0316 0.4291 0.8 0.0316 0.5604 0.9 0.0316 0.7093 1.0 0.0316 0.8757 0.6 0.0 0.0456 0 0.1 0.0456 0.0088 0.2 0.0456 0.035 0.3 0.0456 0.0788 0.4 0.0456 0.1401 0.5 0.0456 0.2189 0.6 0.0456 0.3153 0.7 0.0456 0.4291 0.8 0.0456 0.5604 0.9 0.0456 0.7093 1.0 0.0456 0.8757 0.7 0.0 0.062 0 0.1 0.062 0.0088 0.2 0.062 0.035 0.3 0.062 0.0788 0.4 0.062 0.1401 0.5 0.062 0.2189 0.6 0.062 0.3153 0.7 0.062 0.4291 0.8 0.062 0.5604 0.9 0.062 0.7093 1.0 0.062 0.8757 0.8 0.0 0.081 0 0.1 0.081 0.0088 0.2 0.081 0.035 0.3 0.081 0.0788 0.4 0.081 0.1401 0.5 0.081 0.2189 0.6 0.081 0.3153 0.7 0.081 0.4291 0.8 0.081 0.5604 0.9 0.081 0.7093 1.0 0.081 0.8757 0.9 0.0 0.1025 0 0.1 0.1025 0.0088 0.2 0.1025 0.035 0.3 0.1025 0.0788 0.4 0.1025 0.1401 0.5 0.1025 0.2189 0.6 0.1025 0.3153 0.7 0.1025 0.4291 0.8 0.1025 0.5604 0.9 0.1025 0.7093 1.0 0.1025 0.8757

(113) At such a small off-axis angle for the cladding material, the coefficient of thermal expansion (CTE) along the three axes will be minimally affected. The CTE change percentage, as influenced by the deviation angle, is derived as in Eq. (11):

(114) $\begin{matrix} {\begin{matrix} _{x}^{} =_{x} \cos \cos -_{y} \sin -_{z} \sin \cos \\ _{y}^{} =_{x} \cos \sin +_{y} \cos -_{z} \sin \sin \\ _{z}^{} =_{x} \sin +_{z} \cos \end{matrix} & (11) \end{matrix}$

(115) As a numerical example, at , =0.5, the difference of CTE at dielectric X,Y,Z directions will be 0.33%, 1.1%, 2.12% respectively.

(116) To conclude, LBO with a slightly off-axis tilt angle is a good candidate as cladding material for LBO oriented along the x-axis. This method can be applied to fabricate waveguides for other biaxial crystals, such as KTP, BiBO, RTA, KNbO.sub.3, etc.

Example 2: LiNbO3 Nonlinear Waveguide

(117) An embodiment of the present disclosure includes a planar, periodically poled waveguide (such as that in FIGS. 6A-6C) with a lateral core width d=40 m. The core material is slightly tilted periodically poled lithium niobate, while the cladding material is bulk, un-tilted lithium niobate. The planar waveguide is designed for second harmonic generation from 1.064 m to 0.532 m. The poling period is designed to be 6.78 m in order to fulfill quasi-phase matching condition for the intended second order nonlinear frequency conversion process.

(118) Both 1.064 m and 0.532 m beams are polarized along optic axis of lithium niobate. If the shortest wavelength (0.532 m) is intrinsically single mode in the waveguide, the other longer wavelength(s) will be intrinsically single mode as well. Therefore, the waveguide should be designed such that it can support only the fundamental mode of the 0.532 m beam.

(119) Eq. (2) gives the criteria for designing intrinsically single-mode planar waveguides. The refractive index difference between the core and the cladding cannot exceed (n).sub.max=9.8910.sup.6. Since the polarization of both beams is along the optic axis, a 0.532 m beam will experience refractive n.sub.e in the cladding. The core is tilted to have a slightly higher refractive index, which is limited by (n).sub.max. The maximum tilting angle calculated by Eq. (5) is 0.62 degrees. The maximum tilting angle is preferred because it provides the maximum refractive index contrast allowable and consequently the optimal confinement of an electric field in the core region.

Example 3: Yb:YLF Laser Waveguide

(120) Consider a 40 m thick planar laser waveguide employing Yb:YLF as core material, such as that in FIGS. 4A-4E, for the purpose of emitting at 1030 nm.

(121) YLF is positive uniaxial crystal whose extraordinary refractive index n.sub.e=1.4704 and ordinary refractive index n.sub.o=1.4482. For Yb:YLF laser, the emission line at 995 nm is polarized along the c-axis (extraordinary). Therefore, the inner cladding should be slightly tilted (see FIG. 4E) such that a single-mode waveguide along the lateral direction can be formed. According to Eq. (2), the index difference is calculated to be n=5.8910.sup.5. According to Eq. (5), tilt angle <0.051. If single-mode operation is not a requirement, the tilting angle can be larger.

Example 4: Nd:YVO4 Laser Waveguide

(122) A Nd:YVO.sub.4 waveguide is formed using a higher concentration Nd:YVO.sub.4 material as the core and lower concentration Nd:YVO.sub.4 material as the inner cladding. The refractive index difference needed to form a single-mode or multimode waveguide is provided by different Nd.sup.3+ doping levels in the vanadate host. The higher the doping level, the higher the refractive index.

Example 5: Inactive Waveguide Capped Yb:YLF Laser Waveguide

(123) Consider an active gain medium of Yb:YLF, the same type as in Example 4. On the two ends, a short Nd:YLF waveguide, whose length is in the order of 5 mm, is bonded to the Yb:YLF waveguide. The absorption peak for Nd:YLF is at 808 nm, while the absorption peak for Yb:YLF is at 960 nm. When pumped at 960 nm, the two short Nd:YLF waveguides will not lase. The mode profile of the Nd:YLF waveguide is designed in such a way that it matches the mode profile of the Yb:YLF to the best extent by slightly tilting the inner cladding Nd:YLF, as described before, such that the laser radiation can propagate in the entire waveguide structure with minimal loss.

(124) Technical advantages of such a capped laser waveguide design include: 1) protecting the optical coating; and 2) mitigating thermal-induced fracture, by moving the focus point of pump beam to the inactive end rather than directly focusing on the active gain medium.

(125) Other Notes

(126) A difference in refractive index for anisotropic laser crystals also is obtainable at a propagation direction that is not along a crystal axis and a small deviation from it, but also at an arbitrary orientation between the propagation direction and the crystal axis. As example, there may be a benefit to lasing behavior to propagate at an angle of e.g., 5-15 degrees off the c-axis of rare earth doped YLF. The same rules for selecting an index difference apply to this situation.

(127) In general, using differential doping of the core and the cladding can be applied to anisotropic crystals, but it is most applicable to laser crystals. Some nonlinear crystals also are co-doped or can be co-doped (e.g., H.sup.+ can be exchanged with Li.sup.+ in LiNbO.sub.3 or He.sup.+ can be implanted into KTP) to change the refractive index. There is, however, a possible issue that doping a non-linear crystal could change the nonlinear coefficient of the material.

(128) If categorized by crystal structure types, cubic is isotropic; trigonal, tetragonal, and hexagonal belong to the uniaxial, anisotropic group, and triclinic, monoclinic, and orthogonal belong to biaxial, anisotropic crystals.

(129) Conclusion

(130) In conclusion it can be seen that embodiments of the invention can provide the ability to design waveguide structures in anisotropic lasing and nonlinear optical media with desired properties such as intrinsic single-mode operation. Precise control over the index difference provides design flexibility. The invention offers a new way to make nonlinear optical crystal waveguides of specific orientations for specific harmonic conversions.

(131) A further understanding of the nature and advantages of the present invention may be realized by reference to the remaining portions of the specification and the drawings.

Waveguide structures in anisotropic lasing and nonlinear optical media

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