Source: http://www.google.com/patents/US4743083?dq=5,490,216
Timestamp: 2013-12-05 14:28:15
Document Index: 435833556

Matched Legal Cases: ['art 44', 'arts 46', 'art 44', 'arts 46', 'art 152', 'art 153', 'art 300', 'art 300', 'art 302', 'art 302', 'art 302', 'art 300', 'art 300', 'art 300']

Patent US4743083 - Cylindrical diffraction grating couplers and distributed feedback resonators ... - Google PatentsSearch Images Maps Play YouTube News Gmail Drive More »Advanced Patent Search | Sign inAdvanced Patent SearchPatentsDevices for controlling and guiding waves, such as electromagnetic waves, acoustic waves, and the like are disclosed which include at least one wave-transmitting medium and a diffraction grating, associated with the transmitting medium, for scattering wave energy into or out of the guided waves. The...http://www.google.com/patents/US4743083?utm_source=gb-gplus-sharePatent US4743083 - Cylindrical diffraction grating couplers and distributed feedback resonators for guided wave devicesPublication numberUS4743083 APublication typeGrantApplication numberUS 06/814,612Publication dateMay 10, 1988Filing dateDec 30, 1985Priority dateDec 30, 1985Fee statusLapsedPublication number06814612, 814612, US 4743083 A, US 4743083A, US-A-4743083, US4743083 A, US4743083AInventorsRobert M. SchimpeOriginal AssigneeSchimpe Robert MPatent Citations (14), Non-Patent Citations (30), Referenced by (70), Classifications (27), Legal Events (4) External Links: USPTO, USPTO Assignment, EspacenetCylindrical diffraction grating couplers and distributed feedback resonators for guided wave devicesUS 4743083 AAbstract Devices for controlling and guiding waves, such as electromagnetic waves, acoustic waves, and the like are disclosed which include at least one wave-transmitting medium and a diffraction grating, associated with the transmitting medium, for scattering wave energy into or out of the guided waves. The device may take the form of a "sandwich"-type or thin film waveguide, in which case the transmitting medium has at least one curvi-planar boundary, or it may take the form of a "rod"-shaped or cylindrical waveguide, in which case the transmitting medium has a substantially cylindrical boundary. According to the invention, the diffraction grating comprises a large number of substantially closed loop grating lines consecutively enclosing each other from inner to outer, with all of the grating lines extending substantially parallel to the phase fronts of the guided waves.
What is claimed is: 1. In an apparatus for controlling waves including a waveguide configured to confine and guide the energy of propagating waves and comprising at least one wave-transmitting medium having a substantially curviplanar boundary and a diffraction grating formed by a plurality of grating lines configured to scatter wave energy with respect to guided waves having phase fronts extending substantially parallel to said grating lines;the improvement wherein said grating lines form substantially closed loops consecutively enclosing each other from inner to outer. 2. The improvement defined in claim 1, wherein said grating lines are formed of physical ridges and grooves in said transmitting medium at said boundary.
29. The improvement defined in claim 27, wherein said transmitting medium is selected from the group consisting of GaAs; Ga.sub.1-x Al.sub.x As; Ga.sub.x In.sub.1-x As.sub.y P.sub.1-y ; and Ga.sub.x In.sub.1-x As.sub.y Sb.sub.1-y.
DESCRIPTION OF THE PREFERRED EMBODIMENTS Both the theoretical background as well as exemplary applications of the present invention will now be described with reference to FIGS. 1-20 of the drawings. Identical elements in the various figures are designated by the same reference numerals.
Considering first the waveguide of FIGS. 1A and 1B (hereinafter referred to as "FIG. 1") there is shown a cross-section of the sandwich structure comprising a central wave transmitting layer 10, two wave transmitting "cladding" layers 12 and 14, an anti-reflection layer 16 and a reflective or mirror layer 18. The index of refraction n.sub.1 of the central layer is higher than the indices of refraction n.sub.2 and n.sub.3 of the cladding layers 12 and 14, respectively. The central and cladding layers thus serve as a waveguide for electromagnetic radiation (EMR) in the infrared, visible light or ultraviolet light spectra. The transmission of such radiation along the waveguide is indicated by the arrows 19.
Surrounding FIG. 1A are graphs of the index of refraction, n(z) of the waveguide layers; the intensity distribution I.sub.x, along the x axis within the central layer 10 of the waveguide; and the axial intensity distribution I.sub.z, at the center of the grating.
Considering first the graph at the left-hand portion of FIG. 1A, it is seen that the refractive index n varies along the z axis as indicated by the characteristic 26. The characteristic 26 reaches a maximum equal to the refractive index n.sub.1 along the portion of the z axis containing the central layer 10, and falls off in the cladding layers 12 and 14 which have indices of refraction n.sub.2 and n.sub.3, respectively. In the region 27 along the z axis of the diffraction grating, the index n depends upon the respective widths of the ridges and grooves of the grating. The shaded rectangle 28 has a length along the z axis equal to the effective width T.sub.eff of the axial intensity distribution I.sub.z and a height along the n axis equal to the effective refractive index n.sub.eff of the waveguide. The quantities T.sub.eff and n.sub.eff will be defined further below.
The axial distribution of the intensity within the waveguide formed by the sandwich structure is indicated by the graph at the right-hand portion of FIG. 1A. Shown in this graph is the intensity distribution of electromagnetic radiation for the fundamental mode within the waveguide as a function of position along the z axis. As indicated by the curve 32, the intensity reaches a maximum within the central layer 10 of the waveguide at the axial position z=z.sub.m and falls off sharply in the cladding layers 12 and 14. The width of the intensity curve 32 along the z axis at which the intensity falls to one tenth of the maximum is designated by the value T.sub.eff.
Considering now the complementary waveguide structure in FIG. 2, there is shown a rod-shaped waveguide made of wave transmitting core material 40 having an index of refraction n.sub.4. The core 40 is surrounded by air or vacuum, which acts as a cladding layer. Etched into the outer circumference are a plurality of ridges and grooves 42 forming a diffraction grating. These ridges (grooves) are coaxial and extend in planes perpendicular to the rod axis (z axis). As in the case of the sandwich structure shown in FIG. 1, the grooves are arranged in three parts: a center part 44 and outer parts 46. Within the center part 44, the grooves (ridges) are equally spaced, and within the outer parts 46 the grooves (ridges) are equally spaced. The spacing of the grooves in the center part is designed to couple EMR energy out of the waveguide, whereas the spacing of the grooves in the outer part is designed to couple wave energy travelling in the z axis direction backward.
Since the wave transmitting medium of the waveguide is made of one type of material only, the refractive index is equal to n.sub.4 across the cross section of the rod and falls abruptly to the vacuum index outside as indicated by the curve 48. Only in the areas 50 of the grating, where the waveguide material is interrupted by air or vacuum, n depends upon the respective widths of the ridges and grooves of the grating. The shaded rectangle 51 illustrates again the quantities T.sub.eff and n.sub.eff as defined further below.
Similarly, the curve 56 shows the intensity distribution along the x axis for the fundamental mode of the rod-like waveguide and has indicated the width T.sub.eff of the intensity distribution. In this case, the intensity distribution extends into the air, which acts as a cladding layer.
To define the field distribution with circular cylinder symmetry, we use the polar coordinates r, φ and z of a circular cylinder. FIG. 3 illustrates this coordinate system in which the z axis coincides with the cylinder axis, r is the distance of a point from the cylinder axis and φ is the angular position of the point. Also shown is the field vector H.sub.z of the axial directed magnetic field and the field vector E.sub.φ of the circumferentially directed electric field.
For a monochromatic field of circular cylinder symmetry, the axial directed electric field E.sub.z and the axial directed magnetic field H.sub.z can be considered as the generating field components of the total electromagnetic field, which means that all other field components can be derived from E.sub.z and H.sub.z. Maxwell's equations for these axial field components can be separated into two sets of scalar wave equations: ##EQU1## where K.sub.z and K.sub.r, the axial and radial propagation constants, respectively, are related via the equation:
K.sub.z.sup.2 +K.sub.r.sup.2 =(2&#960;n/&#955;).sup.2.     (3)
These constants K.sub.z and K.sub.r will be further specified below. In equation (3), n is the refractive index of the waveguide layer at the point where the field is evaluated. The equation (1) describes the z dependence of the axial field components and the equation (2) describes the r, φ dependence of the axial field components.
The wave action within the structure of FIG. 1 will now be described in detail. For definiteness, the guided wave in the waveguide structure of FIG. 1 is considered to be monochromatic with the free space wavelength λ and to be confined to a single waveguide mode, which has only the axial magnetic field H.sub.z as generating field component (E.sub.z .tbd.O). The solution of equation (1) for each waveguide layer yields the z-dependence of the axial magnetic field H.sub.z and therefrom derived the z-dependence of the circumferential electric field E.sub.φ α∂H.sub.z /∂r.
The z-dependence of the intensity distribution I.sub.z αE.sub.φ.sup.2 is illustrated on the righthand side of FIG. 1A. The intensity extends over the axial distance T.sub.eff , which is shown in FIG. 1A. T.sub.eff is denoted as the effective thickness of the waveguide. The solution of equation (1) yields via equation (3) also the radial propagation constant
K.sub.r =2&#960;n.sub.eff /&#955;.                         (4)
In equation (4), n.sub.eff is the effective refractive index of the waveguide, which can be interpreted as a field-weighted average of the indices of refraction of the different waveguide layers. The quantities n and T characterize an effective dielectric waveguide, which will be used for the description of the action of the grating-waveguide configuration. The concept of the effective dielectric waveguide encounters a wide class of waveguides. The model is particularly useful for the description of a waveguide with a graded refractive index as shown in FIG. 1 or of a multilayer waveguide such as a multiple quantum well waveguide. It is noted that the model also applies to a waveguide with one or several metal boundaries such as a metal hollow waveguide.
The superposition of a guided wave with circular phase fronts outgoing from the cylinder axis, and a guided wave incoming toward the cylinder axis, forms a circular cylindrical standing wave pattern, which is described by equation (2). The standing wave solutions of equation (2) for the axial magnetic field H.sub.z and the solution for the circumferentially directed electric field E.sub.φ, derived therefrom, are well known to be given by the linear superposition ##EQU2##
J.sub.m denotes the cylinder function (Bessel function) of the first kind of the integer order m=0, 1, . . . , and J.sub.m ' denotes the derivative of J.sub.m with respect to the argument K.sub.r r.
As an example, the Bessel function J.sub.1 and its derivative J.sub.1 ' are illustrated in FIG. 4A as a function of the radius r, for a range of small radii and for a range of larger radii, respectively. In this Figure, the vertical scale in the left-hand portion of the diagram is smaller than the vertical scale in the right-hand portion. Nevertheless, the magnitude of the functions J.sub.1 and J.sub.1 ' is greater near the left-hand (z) axis than toward the right.
According to the equations (5) and (6), the standing wave pattern corresponding to a pair of coefficients h.sub.m and e.sub.m is characterized, in particular, by its number of radial directed nodal lines, along which the cosine function, and therefore H.sub.z and E.sub.φ vanish. The standing wave pattern corresponding to a pair of coefficients h.sub.m and e.sub.m is usually denoted as the mode of the circumferential order m.
The r, φ dependence of field intensity E.sub.100 .sup.2 is illustrated in FIGS. 5A, 5B and 5C by the shaded areas 130, 132 and 134, respectively, for the mode of the circumferential order m=0, 1 and 3, respectively. Since E.sub.100 is the electric field component of a standing wave, the shaded areas illustrate pockets of stored wave energy. The arrows 136, 138 and 140 within the energy pockets illustrate the direction of the electric field. Generally, the circumferentially oriented field is symmetric (antisymmetric) with respect to the field center for modes with odd (even) integer order m, as illustrated by the arrows 136, 138 and 140.
Radial directed nodal lines 142 and 143 are also shown. In FIGS. 5A and 5C, a field free area 144 and 145 in the center of the standing wave pattern can be seen. The radius of this field free area extends with increasing mode order m, as prescribed by the functions J.sub.m '. It is noted that radially consecutive energy pockets contain essentially equal amounts of energy. Since the circumferential length of the pockets linearly decreases with decreasing radius r, the volume energy density increases with decreasing radius. The most inner energy pockets illustrated by the areas 146, 147 and 148 are therefore the energy pockets with the highest energy density for the mode with m=0, 1 and 3, respectively. This concentration of wave energy in the center of the standing wave pattern cannot be achieved with state of the art sandwich type waveguides containing a grating (nor with the resonator according to FIG. 2). This "piling" of wave energy is most pronounced for modes with a low circumferential order m.
Having a waveguide according to FIG. 1 provided with a nonlinear material (for example LiNbO.sub.3, ZnS or a nonlinear organic material), for example in the cladding layer 12 or the central layer 10, the cylindrical grating in the waveguide is a promising arrangement to enhance nonlinear interactions. This attractive feature of the arrangement in FIG. 1 can be used for frequency conversion (via second order nonlinearities) or for affecting the z dependence of the field distribution by intensity-induced refractive index changes (using third order nonlinearities).
Given a waveguide according to FIG. 1 with an electro-absorptive material (for example GaAs, Al.sub.p Ga.sub.1-p As, InP, Ga.sub.p In.sub.1-p As.sub.p P.sub.1-p in the central layer 10 or in a cladding layer 12 or 14, the absorption of resonator modes can be influenced by applying an electric field perpendicular to the layers. Such an electric field can be built up by reverse biasing a pn-junction formed by two of the mentioned semiconductor materials. The influence of this electric field will be particularly high, if it is applied in the area around the center of the grating. The influence of the electric field is different for waveguide modes which have E.sub.z as the generating axial field component than for the waveguide modes which have H.sub.z as the generating axial field component. It is therefore possible to select the polarization of the waveguide mode by changing the strength of the applied electric field. As a consequence, different polarization characteristics of the emitted beam 34 can be achieved. Since changes in absorption are accompanied by changes of the refractive index in these materials, the free space wavelength λ of the resonator modes can be influenced. The absorbed radiation generates electrical carriers which can be collected by external contacts (not shown in FIG. 1).
With reference to the equation (6), the wavelength λ.sub.m of the guided wave in the mode of the circumferential order m is now defined as twice the radial distance between two neighboring circumferential nodal lines 150 and 151 of the electric field component E.sub.φ. The circumferential nodal lines 150 and 151 are illustrated in FIG. 5B. This means that the difference of two consecutive zeros of the function J.sub.m ' is used as a measure for the length of the guided wave. For sufficiently large r, the functions J.sub.m and J.sub.m ' follow essentially the trigonometric functions:
J.sub.m (K.sub.r r)&#8594;&#8730;2/(&#960;K.sub.r r) cos (K.sub.r r-m
J.sub.m '(&#960;K.sub.r r)&#8594;-&#8730;2/(&#960;K.sub.r r) sin (K.sub.r r-m
&#955;.sub.m &#8771;&#955;/n.sub.eff,          (9)
which would be equal to the wavelength of a guided wave with plane wave fronts. Only in the very center of the standing wave pattern of the mode of order m, the wavelength λ.sub.m becomes noticably larger than the length given by equation (9). As an example, FIG. 4A shows the wavelength λ.sub.1, in the range of small radii, where the wavelength λ.sub.1, slightly varies with r and in the range of larger radii, where the wavelength λ.sub.1 is essentially constant.
As a first approximation, the effect of the grating on the waveguide properties can be modelled by relating the changes of the position of the boundary containing the grating to changes of the effective thickness T.sub.eff and of the effective refractive index n.sub.eff of the waveguide. FIG. 4B shows schematically the radial variation of the effective thickness of the waveguide along the x axis due to the ridges of the waveguide structure in FIG. 1. The radial variation of the effective refractive index is illustrated by the more or less shaded areas within the center part 152 and the outer part 153 of the effective dielectric waveguide. Qualitatively speaking, with reference to FIG. 1 and FIG. 4B, if the guided wave is highly confined to the higher refractive index layers of the guide by increasing the thickness of the central layer, the effective refractive index is higher. Additionally, the ray 154, which runs through the position of the maxima of consecutive axial intensity distributions 156 and 157, is bent toward the side of the ridge. Also, the effective thickness of the guide decreases. Each ridge in the central layer in FIG. 1A therefore corresponds to an effective boundary displacement 160 on the opposite side in the effective dielectric waveguide and to a region of higher effective refractive index in FIG. 4B.
The effect of the grating on an outgoing wave will now be described, beginning with the grating of the center part of FIG. 1. FIG. 6 shows a magnified portion of the center part of the effective dielectric waveguide structure. In FIG. 6 the sides of each effective boundary displacement 160, 161 or 162 are two regions 164 and 165 with a change ΔT.sub.eff of the thickness and a change Δn.sub.eff of the effective refractive index along the ray 154. Each effective boundary displacement has an average radial width W. Neighboring boundary displacements 160 and 161 are the radial distance Λ apart.
The electric field vector E.sub.φ out of the outgoing wave in the presence of a waveguide inhomogeneity in the region 164 is thought to cause a scattered wave with a magnetic field vector H.sub.rad and an electric field vector E.sub.rad. The scattered wave propagates in the direction of the ray 166.
The range of angles θ, in which scattering occurs is relatively large, as illustrated by the phase front 168, due to the small extension of the waveguide inhomogeneity in region 164 of the order or smaller than the wavelength λ.sub.m.
The superposition of the rays 178 and 179 diffracted from two neighboring regions 180 and 181 corresponding to the region 164 yields a condition for the angles θ.sub.c of emission of the grating due to multiple constructive interference. Mathematically expressed the rays 178 and 179 are in phase, if
&#923;-B=j&#955;.sub.m                                 (10)
where λ.sub.m and λ are noted, j is an integer and B is the distance shown in FIG. 6 along the ray 178 from the ray 154 at the center of the region 180 to the wavefront that passes through the center of the region 181.
B=&#923; cos &#952;.sub.c,                              (11)
cos &#952;.sub.c =1-j&#955;.sub.m /&#923;.            (12)
By varying the radial spacing Λ of the boundary displacements, a broad range of angles θ.sub.c can be achieved.
Of particular interest is emission of the grating at the angle θ.sub.c =.+-.π/2 radians, which is vertical emission out of the waveguide plane. According to equation (12) it is necessary to have Λ=jλ.sub.m, which means that the spacing Λ between the boundary displacements is equal to the wavelength in the guide at the position of the boundary displacement, or an integer multiple thereof.
2W/&#955;.sub.m +(&#916;&#966;.sub.2 -&#916;&#966;.sub.1)/(2&#960;)=k+1/2, (13)
where W and λ.sub.m are noted, k is an integer and Δφ.sub.1 and Δφ.sub.2 are the phase changes of the rays 174 and 176, respectively, after scattering in the regions 164 and 165 due to changes of the effective refractive index, respectively.
For a width W much smaller than λ.sub.m, there should be no back scattering. In order to satisfy equation (13) for this case, Δφ.sub.2 -Δφ.sub.1 must be equal to (2k+1) Δφ.sub.2 -Δφ.sub.1 =π radians.
For the central portion of the grating of the waveguide in FIG. 1, the radial spacing Λ is chosen to be equal to λ.sub.1, and the radial width W is chosen to be equal to λ.sub.1 /2 so that the coupling of power of an outgoing guided wave with the wavelength λ.sub.1 into a vertical directed beam is maximized.
Also of particular interest is the emission of a grating at an angle θ.sub.c =.+-.π radians, which is reflection of the outgoing guided wave into the incoming guided wave, thereby producing a standing wave. Then it follows from equation (12)
&#923;=j
which means that the spacing Λ between the boundary displacements is equal to one half of the wavelength in the guide at the position of the boundary displacement or an integer multiple thereof. Each wavelength, which satisfies the equation (15) is called a Bragg wavelength of the grating. A grating, which satisfies Λ=λ.sub.m /2 is called a first order grating, and a grating which satisfies Λ=λ.sub.m is called a second order grating, in particular. FIG. 4B shows the rays 190 and 191 back scattered due to changes of the effective refractive index in the regions 192 and 193, respectively.
2W/&#955;.sub.m +(&#916;&#966;.sub.2 -&#916;&#966;.sub.1)/(2&#960;)=&#961;, (16)
where Δφ.sub.2 -Δφ.sub.1 =π, as noted above, and ρ is an integer. Equation (16) solved for W yields
W=(2&#961;-1)
Equation (17) means that the back reflection of the outgoing wave can be made particularly efficient, if the radial width of the boundary displacements is equal to a quarter wavelength in the guide or an odd integer multiple thereof. For the grating in the outer part, the radial distance Λ is chosen equal to λ.sub.1 /2 and the radial width W is chosen equal to λ.sub.1 /4, as to maximize the reflection of an outgoing guided wave with the wavelength λ.sub.1 into an incoming guided wave.
It has been shown in conjunction with FIG. 4B that, in the vicinity of the Bragg wavelengths defined by equation (15), an outgoing guided wave is accompanied by an incoming guided wave, thereby causing a standing wave pattern. The incoming guided wave will superimpose on the primary outgoing wave after a round trip in the grating-waveguide configuration. The optical path length of the round trip, measured in units of the wavelength in the guide, is different for each mode, due to the different wavelengths λ.sub.m for small radii, as discussed in conjunction with the equations (6) and (9). In order to achieve constructive interference of the primary wave and the wave having undergone the round trip for a mode of the circumferential order m, the position of the boundary displacements has to be "fine tuned" with respect to the phase fronts of the standing wave pattern of this mode. With reference to FIGS. 4A and 4B, the waveguide inhomogeneities corresponding to the region 193, which produce a drop of the effective refractive index along the ray 154, have to be placed at the loci of maximum absolute values of the electric field E.sub.φ, as illustrated in the right-hand portion of FIGS. 4A and 4B for the mode with m=1. The waveguide inhomogeneities corresponding to the region 192, which produce an increase of the effective refractive index along the ray 154, have to be placed at the loci of the maximum absolute values of the magnetic field H.sub.z.
&#966;.sub.out (r+W)-&#966;.sub.in (r)=2&#960;i,               (18)
where the width W of the boundary displacement 160 is noted, i is an integer and φ.sub.out and φ.sub.in are the phase of the outgoing ray 154 at the center of the region 180 and the phase of the incoming ray 200 at the center of the region 202, respectively.
The left-hand portion of the FIGS. 4 illustrates the fine tuning of the radial position of the boundary displacements with a width W equal to one half of the wavelength of the mode with the circumferential order m=1. For example, the boundary displacement 160 is arranged symmetrically with respect to the local minimum 206 of the function J.sub.1 ', which is proportional to the electric field E.sub.φ of the mode with m=1. The regions 180 and 202 with a complementary waveguide inhomogeneity are situated at consecutive zeros of J.sub.1 ', because of the particular width W=λ.sub.1 /2.
(&#966;.sub.out -&#966;.sub.in)/(2&#960;)+&#916;&#966;.sub.m +2S.sub.2 n.sub.2 /&#955;+2S.sub.1 n.sub.1 /&#955;=k                    (19)
where λ, φ.sub.in, φ.sub.out and n.sub.1, are noted, ##EQU3## is the average along the z axis of the graded refractive index n.sub.2, S.sub.1 and S.sub.2 are the distances shown in FIG. 8, Δφ.sub.m is the phase change of the ray 228 due to the reflection at the mirror layer and k is an integer. For definiteness, Δφ.sub.m is chosen equal to π radians, which is the phase change of optical radiation at a high reflectivity gold layer.
It has been explained in conjunction with the equation (18) and the FIGS. 4A and 4B that the outgoing ray 226 and the incoming ray 230 are out of phase in the region 224 of a waveguide inhomogeneity, which means φ.sub.in -φ.sub.out =.+-.π radians, or without loss of generality φ.sub.in -φ.sub.out =π radians. With these particular choices of Δφ.sub.m and φ.sub.in -φ.sub.out, the equation (19) yields
S.sub.opt =k
denotes the optical path length between the center of the waveguide in the region 224 and the mirror layer 18. Equation (21) means that, for the particular mirror layer with Δφ.sub.m =π, the optical path length has to be equal to an integer multiple of one half of the free space wavelength λ. In general, the optimum optical path length S.sub.opt depends through φ.sub.out -φ.sub.in on the width W of the ridge 220 and on the fine tuning of the radial position of the ridges, as explained in conjunction with FIG. 6.
In dielectric waveguides the refractive indices n.sub.1 and n.sub.2 are often only slightly different so that one can set n.sub.1 ≃n.sub.2 ≃ n.sub.eff to a first approximation. Equation (21) then reduced to
S&#8771;k
where the spacing S is noted and shown in FIG. 8. Equation (23) means that the spacing S has to be approximately equal to an integer multiple of one half of the wavelength λ/n.sub.eff in the waveguide.
The righthand portion of the FIG. 8 shows the intensity distribution E.sub.rad.sup.2 of the standing wave due to the reflection of the ray 228. The electric field E.sub.rad of the emitted ray 228 is parallel to the layers as has been illustrated in FIG. 6. The intensity distribution E.sub.rad.sup.2 has therefore a node at the (metal) mirror layer 18.
The layer 14 thus stores wave energy in a standing wave pattern, thereby forming an external resonator coupled to the guided-wave resonator. By increasing the spacing S.sub.2 the amount of stored energy increases. The total quality factor of the two coupled resonators then also increases accordingly.
The grating of FIG. 9 consists of a plurality of elliptical ridges with two common foci F.sub.1 and F.sub.2, which are the distance 2c=10 Mathematically the ellipses can be expressed by:
A cylindrical wave with elliptical phase fronts can be described to a first approximation by the superposition of two in-phase circular cylindrical waves of equal amplitude, which have their axes at the foci F.sub.1 and F.sub.2, respectively. This will be explained below in greater detail. Since the distance 2c between the foci F.sub.1 and F.sub.2 in FIG. 9 is equal to a multiple of the wavelength λ/n.sub.eff, the circular cylindrical waves essentially interfere constructively along the x axis outside the connection line F.sub.1 F.sub.2 of the foci. By choosing the spacing Λ of the ridges along the x axis substantially equal to λ/n.sub.eff, the grating is operated along the x axis at one of its Bragg wavelengths according to equation (15) (second order grating). By choosing the width W of the ridges along the x axis substantially equal to λ/(4n.sub.eff) the back reflection of an outgoing wave into an incoming wave is made particularly efficient according to equation (17). With the grating of FIG. 9 we aim for a standing wave pattern with one nodal line along the y axis. This wave pattern can be produced approximately by choosing, for the two circular cylindrical waves, the mode with the circumferential order m=1.
a=c+k
which means that the distance c is equal to an integer multiple k of the wavelength λ/n.sub.eff of a guided wave with plane wave fronts at the free space wavelength λ (see equation (9)).
FIG. 10 illustrates the superposition of these two primary waves. The primary waves are represented by the respective wave vectors k.sub.1 and k.sub.2 and the vectors E.sub.100 1 and E.sub.φ2 of the respective electric fields. The wave vectors point in the directions of propagation of the primary waves. The electric field vectors E.sub.φ1 and E.sub.φ2 are perpendicular to the respective wave vectors and are oriented tangential to the respective phase fronts 250 and 252. Also shown is the vector E of the electric field of the sum wave. The vector E is oriented tangential to the phase front 254 of the sum wave. The normal vector k to the field vector E represents the wave vector of the sum wave. The wave vector k is oriented tangential to the ray 256 of the sum wave.
For further illustration, we return to the simple superposition of only two in-phase circular cylindrical waves. We examine the curvature of a phase front 254. FIG. 11 shows the superposition of the two primary circular cylindrical waves in greater detail. If one goes a small distance along the phase front 254 from the point P.sub.1 to the point P.sub.2, the phase front 250 has to go forward (in the direction of k.sub.1) by the length dr.sub.1 and the phase front 252 has to go backward (opposite to the direction of k.sub.2) by the length dr.sub.2. Mathematically, it is
dr.sub.2 /dr.sub.1 =-sin &#946;.sub.2 /sin &#946;.sub.1    (28)
where the angles β.sub.1 and β.sub.2 are shown in FIG. 11. The negative sign occurs, since the phase fronts 252 is going backward. From the diagram of the electric field vectors in FIG. 11 one obtains
E.sub.&#966;1 /E.sub.&#966;2 =sin &#946;.sub.2 /sin &#946;.sub.1, (29)
where E.sub.φ1 and E.sub.φ2 are the amplitudes of the electric field of the primary waves at the point P.sub.1. Equation (28) and (29) yield
E.sub.&#966;2 dr.sub.2 =-E.sub.&#966;1 dr.sub.1.             (30)
Equation (30) is the differential equation of a phase front in terms of the coordinates r.sub.1 and r.sub.2.
For a circular cylindrical wave, the field amplitudes follow for greater radii r essentially cos (mφ)/√r as has been shown in conjunction with the equations (6) and (8) for a standing wave. For radii r.sub.1 and r.sub.2 much greater than 2c, and both circular cylindrical waves being in a circumferential mode of the same order m, the ratio ε=E.sub.φ1 /E.sub.φ2 is approximately constant.
&#949;r.sub.1 +r.sub.2 =d,                              (31)
where d is a constant for each phase front. The constant d can be expressed for example by distances measured along the axis running through the two centers of the primary waves. In FIG. 10 this axis coincides with the x-axis. The phase front 254 intersects the x axis at the distance a from the point 0 in the middle between the two centers of the primary waves. Along the x axis it is r.sub.1 =a+c and r.sub.2 =a-c so that equation (31) yields
&#949;r.sub.1 +r.sub.2 =(1+&#949;)
where all distances r.sub.1, r.sub.2, a and c are noted and shown in FIG. 10. It is understood that equation (32) has been obtained under the assumption r.sub.1, r.sub.2 &gt;&gt;2c.
We now consider briefly the results of equation (32) for different values of ε. For ε=0 we find circular phase fronts with radii r.sub.2 =a-c, as expected. For ε=1 we find elliptical phase fronts with r.sub.1 +r.sub.2 =2a. This shows that the superposition of two in-phase circular cylindrical waves of equal amplitude yields a first approximation for the construction of an elliptical wave, as stated above in conjunction with the discussion following the equations (24) and (25).
The grating 304 of the center part consists of a plurality of concentric circular boundary displacements forming ridges and grooves in the lower confinement layer 322. The ridges have vertical walls of 0.22 micrometer height. The bottoms of the grooves closest to the central layer 308 are shown in FIG. 12B in solid black. The ridges have constant radial width W.sub.c and spacing Λ.sub.c. The radial spacing Λ.sub.c of the ridges of 0.46 micrometer is made equal to the wavelength in the guide (2nd order grating) at the free space operating wavelength λ. The radial width W.sub.c is made equal to one half of the wavelength in the guide to maximize the vertical emission for the ridges with rectangular cross-section. The positions of the ridges are radially fine tuned to maximize the coupling of power from a circular standing wave pattern with one nodal line to vertical emitted radiation, as explained in conjunction with FIG. 6. (The centers of the ridges are situated along circles with radii equal to (k+5/8)λ/n.sub.eff, where λ is noted, k is an integer and n.sub.eff is the effective refractive index of the waveguide of the inner part).
The grating 310 of the outer part (outer radius: 250 micrometer) consists of a plurality of concentric circular ridges and grooves in the central layer 326 in the boundary common with the lower cladding layer 324. The bottoms of the grooves in the central layer are shown in FIG. 12B in solid black. The ridges have vertical walls of 0.11 micrometer height and constant radial width W and spacing Λ.sub.0. The radial spacing of the ridges Λ.sub.0 of 0.23 micrometer is made equal to one half of the wavelength in the guide (1st order grating) and the radial width W.sub.0 is made substantially equal to one quarter of the wavelength in the guide to maximize the back reflection of a guided wave as has been outlined in conjunction with FIG. 4. The concentric cross-section of the grating-waveguide configuration is different for two axes perpendicular to each other, with smooth transitions in between to provide further means for circumferential mode selection. This nonrotational symmetric cross-section is represented in FIG. 12B by a width W.sub.0 of the ridges somewhat smaller than a quarter wavelength in the guide along the y axis. The radial position of the ridges is fine tuned such that they provide maximum distributed feedback for the resonator mode with one nodal line running parallel to the y axis. This fine tuning has been explained in conjunction with the FIG. 4. Care should be taken to take into account a residual difference of the effective refractive indices of the waveguides, which would make the wavelengths in the waveguide of the center part and outer part slightly different.
To provide a low electrical resistance path for the electrical current, the upper cladding layer 320 of the center part 300 contains a highly P-type doped region 332 and the lower cladding layer 324 contains a highly N-type doped region 334. The region 334 (radius: 70 micrometer) extends over the center part 300 and partly into the outer part 302. On the top and on the bottom of the layers of the outer part 302 are ring shaped SiO.sub.2 isolation layers 336 and 338. The lower isolation layer 338 only covers the area adjacent to the highly doped region 334. In the center of the ring contact 316 (inner radius: 5 micrometer) on the resonator surface is formed a monolithically integrated lens 340. On the resonator surface opposite to the mirror layer is deposited an anti-reflection layer 342, to enhance the output coupling efficiency. The laser diode is mounted with the mirror layer 316 downward on a heat sink 344. The heat sink 344 has a conical hole 346 to allow radiation to be coupled in and out of the resonator. The laser diode is biased on one hand by a direct current I and has additionally a modulation current Δ I(t) flowing through it superimposed on the direct current I.
With the structure of FIG. 12 it has been shown that a thin film waveguide with a circular cylindrical grating can be used to produce laser action. Ga.sub.p In.sub.1-p As.sub.q P.sub.1-q double heterostructures lattice-matched to InP produce laser emission with a relatively low pump threshold in the free space wavelength range of about 1.1 to 1.65 micrometers. Laser operation in the near infrared (0.7-0.9 micrometer) can be achieved by Al.sub.p Ga.sub.1-p As heterostructures lattice-matched to GaAs.
FIG. 13 shows a multiple quantum well waveguide formed by a plurality 350 of quantum well layers, which are alternatingly made of Al.sub.p Ga.sub.1-p As and GaAs, embedded between two Al.sub.q Ga.sub.1-q As cladding layers 352 and 354. The thickness of each of the quantum well layers is of the order of 20 nm or smaller. In the outer boundary of the cladding layer 352 is a circular grating 356 with rectangular ridges. The radial width W of the ridges is one quarter of the radial spacing Λ of the ridges. The spacing Λ of about 0.2 micrometers corresponds to the wavelength of the guided wave in the wavelength range, where the multiple quantum well structure can provide optical gain due to pumping with an electron beam.
Laser operation in the mid-infrared (λ=1.8-4 micrometers) can be achieved with a GaInAsSb/AlGaAsSb double heterostructure lattice-matched to GaSb and provided with a cylindrical grating. FIG. 14 shows a double heterostructure formed by a Ga.sub.p In.sub.1-p As.sub.q Sb.sub.1-31 q central layer 360 embedded between two Al.sub.r Ga.sub.1-r As.sub.s Sb.sub.1-s cladding layers 362 and 364. The GaInAsSb central layer is pumped by a short wavelength optical beam. The spacing Λ of the ridges is equal to the wavelength of amplified guided waves so that laser action occurs at sufficiently strong pumping.
FIGS. 16A and 16B show an arrangement similar to FIG. 12, but without means for circumferential mode selection. In particular, the ridges of the grating 377 of the outer part 302 have substantially constant width W.sub.o =Λ.sub.o /2 and the sector shaped regions 312 of higher optical absorption are omitted. The grating 378 of the center part 300 remains unchanged. Care is taken to account for a residual difference between the effective refractive indices along the x and y axis. The curvature of the grating lines therefore rather follows the curvature of the phase fronts of the standing wave patterns than the curvature of exact circles. The means 340 and 346 for rear output coupling are also omitted. Instead, the contact/mirror layer 316 covers the whole area of the center part 300. The beam 306 is then coupled via a transmission medium 380 into a cladded-core multimode fiber 382 with a core radius substantially equal to the radius of the inner part 300.
It has been pointed out in conjunction with the discussion of the standing wave patterns in FIG. 5 that a concentration of radiation energy is produced at the center of a cylindrical grating. This effect considerably enhances nonlinear interactions within the waveguide. FIG. 17 shows a grating-waveguide configuration similar to FIG. 1 but with a waveguide for surface polaritons. The waveguide includes a metal layer which is capable of second harmonic generation, if energized by a high intensity wave. The waveguide supports the fundamental pump wave and the second harmonic wave. These fundamental and second harmonic waves are phase matched, which means that the effective refractive index n.sub.eff of the waveguide is substantially equal for both waves.
FIG. 17 shows a cross section of the waveguide structure. On a fused quartz substrate 400 is deposited a glass film 402 (Corning 7059) with a higher refractive index than the substrate. The thickness D of the film (about 1 μm) is adjusted such that the difference of the effective refractive indices for the fundamental wave and the second harmonic wave is as minimal as possible. In the glass film is formed a circular grating 404. On top of the grating is deposited a thin Ag layer 406 (5 nm thick). The radial spacing Λ of two consecutive grating lines is equal to one half of the wavelength of the guided wave at the free space emission wavelength λ=1.064 μm of a neodym:yttrium aluminum garnet (Nd:YAG) laser, which means that Λ is about 0.35 μm. The radial width W of the ridges is equal to one half of the radial spacing Λ. The radial position of the ridges is fine tuned such that the grating lines provide distributed feedback for the fundamental wave for the mode with one radial nodal line and with the magnetic field oriented parallel to the layers of the waveguide. This means that the centers of the ridges are situated along circles with radii substantially equal to (k+1/2)λ/(2n.sub.eff), where λ and n.sub.eff are noted k is an integer. The pump ray 408 is coupled as a guide wave radially into the resonator. The second-harmonic wave generated by the silver layer 406 is coupled into a beam 410, which is linearly polarized perpendicular to the direction of the pump ray 408.
A vertical beam 432 with a conical phase front 434 is irradiated along the symmetry axis 436 of the device. The irradiated beam 432 has a free space wavelength λ of about 1 micrometer, at which the metal film 428 is normally almost totally reflecting. If the angle of incidence θ.sub.i is such that the interface between the metal 428 and the air can guide a surface plasma wave, the quantum efficiency for the extraction of the photocurrent I is strongly enhanced. The axial intensity distribution for this case is illustrated by the diagram 438. This photodetector configuration is particularly good in attaining pico-second response speed with enhanced quantum efficiency.
If a millimeter wave is fed through the coaxial waveguide it is coupled into a guided wave with concentric circular wave fronts. The ridges scatter the radiation into a beam 460 with generally coaxial wave fronts. If the wavelength λ/n.sub.eff of the guided wave is substantially equal to the spacing Λ of the ridges, the emission occurs into a vertical beam 460. The emitted beam can be made linearly polarized, if the wave fed through the coaxial guide has one radial nodal line. The beam has a low side lobe level due to the smooth exponential decay in the radial direction of the near field of the antenna.
The standing wave pattern due to the two counter-propagating waves is illustrated by the curve 488. The curve 488 shows the square H.sub.φ.sup.2 of the circumferentially oriented magnetic field due to the two axially counter-propagating waves along the cylinder axis. The intensity decreases exponentially with increasing axial distance from the center plane 484. Therefore, all of the energy is axially trapped if the rod resonator is sufficiently long. Since we have a second order grating, radiation is emitted vertically. This is illustrated by the rays 490 which lie in planes parallel to the center plane 484. The radiation pattern, as a function of the angular offset from the vertical direction, is symmetric, since the axial nearfield H.sub.φ is symmetric with respect to the center plane 484. The emission is essentially equally intense in all radial directions in the plane 484, since the antenna is fed by a waveguide mode without a radial nodal line. The emission characteristic as a function of the angular offset from the vertical direction has a low side lobe level, due to the smooth exponential decay of the axial antenna nearfield.
FIG. 11 is an enlarged view of a portion of FIG. 10 showing the area in the region of the point P.sub.1.
SUMMARY OF THE INVENTION It is an object of the present invention to provide apparatus for controlling and guiding waves which includes at least one wave-transmitting medium and a diffraction grating for producing standing waves within the wave transmitting medium.
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2009Jul 12, 2011Steyphi Services De LlcReconfigurable optical add-drop multiplexer incorporating sets of diffractive elementsUSRE43226May 29, 2009Mar 6, 2012Steyphi Services De LlcOptical multiplexing deviceWO2001053718A1Jan 16, 2001Jul 26, 2001Christopher Alan WickliffFront derailleur for a bicycle with annular chain guideWO2002004999A2 *Jun 27, 2001Jan 17, 2002Massachusetts Inst TechnologyGraded index waveguideWO2010114834A1 *Mar 30, 2010Oct 7, 2010The Trustees Of The University Of PennsylvaniaCloaked sensor* Cited by examinerClassifications U.S. Classification385/37, 385/130, 257/E31.65, 385/147, 257/E31.128International ClassificationH01S5/125, H01S5/12, G02B6/124, H01L31/0232, H01S5/187, H01L31/108, H01S5/183, G02B6/42Cooperative ClassificationG02B6/124, H01S5/18319, H01L31/108, H01S5/187, H01S5/125, H01S5/1215, H01L31/0232, G02B6/4202European ClassificationH01L31/0232, G02B6/42C2, H01S5/125, H01L31/108, G02B6/124, H01S5/187Legal EventsDateCodeEventDescriptionJul 23, 1996FPExpired 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