Patent Number: 048878855
Section: description

DETAILED DESCRIPTION OF THE DRAWINGS Referring to the drawings in detail, FIG. 1 is a schematic view of a typical prior art collimator, illustrating a substantially collimated beam 10 after it has passed through three successive apertures 12 positioned in alignment along a collimation axis 14. Specifically, FIG. 1 illustrates an exaggerated view of the effects of diffraction on the beam at 16 after the beam passes through each aperture. In contrast to FIG. 1, FIG. 2 is a schematic illustration of a diffraction free aperture 18 constructed pursuant to the teachings herein, and illustrates the nondiverging output beam 20 produced thereby. Pursuant to the teachings of the present invention, a well defined traveling wave beam 20 substantially unaffected by diffractive spreading can be generated from a recognition that certain exact, non-singular solutions exist for the free space Helmholtz wave equation which represent a class of fields that are nondiffracting in the sense that the intensity pattern in a transverse plane is substantially unaltered by propagation in free space. More specifically, the only axially symmetric non-diffracting field other than a plane wave is the zero-order Bessel function of the first kind, and this beam can have an effective spatial width as small as several wavelengths. Several arrangements are disclosed herein for approximately generating J.sub.o beams, and a numerical simulation of their propagation is presented which demonstrates that they possess a remarkable depth of field. It is characteristic of the familiar wave equations of theoretical physics that they reduce to the Helmholtz equation EQU [.gradient..sup.2 +.kappa..sup.2 ].PSI.(r)=0, (1) when the time-dependence is separable. This is true, for example, of the Klein-Gordon equation, the Schrodinger equation, and various classical equations for light, sound, water, and other types of waves. A recognized feature of all previously known solutions to equation (1) is that whenever the field .PSI. is initially confined to a finite area of radius r in a transverse plane, it will be subject to diffractive spreading after propagating a distance z&gt;.kappa.r.sup.2 normal to that plane in free space. For this reason, it is commonly thought that any beam-like field (i.e., one whose intensity is maximal along the axis of propagation and tends to zero with increasing transverse coordinate) must eventually undergo diffractive spreading as it propagates. This is certainly true, for example, of Gaussian beams--Gaussian beam having spot size .theta., the root mean square width at beam waist, will -diverge at an angle inversely proportional to .kappa..theta. at distances z&gt;.kappa..theta..sup.2 from the beam waist. We present here free-space, beam-like, exact solutions of the wave equation (any of the wave equations mentioned above) that are not subject to transverse spreading (diffraction) after the plane aperture where the beam is formed. These solutions are regular and well behaved mathematical functions with finite values at all points and, like plane waves, have finite energy density rather than finite energy. Most importantly, they can have intensity distributions as small as several wavelengths in every transverse plane, independent of propagation distance. Consider the electromagnetic wave equation as a particular example. In this case .PSI. represents the complex amplitude of one component of a monochromatic electric field assumed to be polarized normal to the direction of propagation. One can easily verify that for time dependence e.sup.-i.omega.t an exact solution of equation (1) for fields propagating into the source-free region z.gtoreq.0 is EQU .PSI.(x,y,z.gtoreq.0)=e.sup.i.beta.z .PSI.(x,y,z=0), (2) with the amplitude in the z=0 plane being equal to ##EQU1## Here A(.phi.) is an arbitrary complex function of .phi., and .beta.=[.kappa..sup.2 -.alpha..sup.2 ].sup.1/2. The separable z-dependance in equation (2) is the critical property which the present invention recognizes is characteristic of non-diffracting solutions. Note that when .beta. is real it gives immediately .vertline..PSI.(x,y,z.gtoreq.0) .vertline.=.vertline..PSI.(x,y,z=0).vertline.. The transverse structure in the z=0 plane is reproduced exactly in every other plane for z&gt;0, and this recognition presents some remarkable consequences. The real time-dependent field associated with the complex amplitude .PSI. is EQU E(r,t)=1/2.PSI.(r) e.sup.-i.omega.t +c.c., (4) where c.c. denotes complex conjugate, .omega.=c.kappa., and c is the speed of light. The time-averaged intensity of this field is simply ##EQU2## For any value of .alpha. in the interval 0.ltoreq..alpha..ltoreq..kappa., a field of the form given in equation (2) will be nondiffracting in the sense that the intensity pattern in the z=0 plane is reproduced in every plane normal to the z-axis: EQU I(x,y,z.gtoreq.0) =I(x,y,z=0). (6) For values of .alpha.&gt;.kappa., the solutions are evanescent waves whose intensities decrease exponentially along the z axis. By superimposing monochromatic non-diffractive fields of amplitude V.sub.m and frequency .omega..sub.m =c[.beta..sub.m.sup.2 +.alpha..sup.2 ].sup.1/2 .gtoreq.c.alpha. one obtains a polychromatic solution of the wave equation ##EQU3## for which the time-averaged intensity is ##EQU4## Thus a field need not be monochromatic in order to be nondiffracting in the sense that we have defined. It is only necessary that all of the frequencies exceed the value .alpha.c when .PSI. is of the form given in (3). The only axially symmetric non-diffracting fields are those for which the function A(.phi.) is independent of .phi., namely, those fields whose amplitudes are proportional to ##EQU5## Here .rho..sup.2 =x.sup.2 +y.sup.2 and J.sub.o is the zero-order Bessel function of the first kind. When .alpha.=0 the solution is simply a plane wave, but for .alpha.&gt;0 we have an intensity distribution whose envelope is inversely proportional to .alpha..rho., as shown in FIG. 3. The effective width of the beam is governed by .alpha. and when .alpha. equals the maximum possible value .kappa.=2.pi..lambda. for a non-evanescent field, the central maximum has a diameter of approximately 3.lambda./4. It is easily shown that none of the nondiffracting field solutions given by equation (3) are square-integrable, but the equations and solutions are idealizations applying to infinite, empty space, and thus an infinite amount of power would be required to create a spatial mode of that form over an infinite space, and we will now examine the propagation properties of J.sub.o beams of finite aperture. FIGS. 4a through 4f are graphical comparisons of the performance of an exemplary embodiment of a diffraction free aperture pursuant to the present invention compared with a Gaussian system. In FIGS. 4a through 4f, the solid line represents the intensity distribution for a J.sub.o beam produced pursuant to the teachings of the present invention, and the dotted line represents that of a Gaussian beam, in FIG. 4a when z=0 (i.e., in the initial plane where the beams are formed), in FIG. 4b after propagating a distance z=25 cm., in FIG. 4c after propagating a distance z=50 cm., and in FIG. 4d after propagating a distance z=80 cm., with .lambda.=0.5 .mu.m.. In FIGS. 4b-d, the intensity of the Gaussian beam has been multiplied by 10 to make it visibly discernible. FIG. 5 illustrates the intensity I(.rho.=0,z) at the beam center as a function of distance of the J.sub.o and Gaussian beams, whose initial intensity distributions at z=0 are shown in FIG. 4a. The intensity distribution I(.rho.,z=0)=J.sub.o.sup.2 (.alpha..rho.) when .alpha.=.pi..times.10-4 meters .sup.-1 is shown in FIG. 4a. The central spot diameter is then 0.15 mm, and we assume that the field is zero for all .rho.&gt;2 mm. FIG. 4a also illustrates a dotted line which represents the intensity distribution of a Gaussian beam whose FWHM is 0.12 mm (the integrated energy is approximately 10 times less than that of the J.sub.o beam). FIG. 5 is a numerical simulation of the propagation of the central peak intensity (i.e., the intensity at .rho.=0) for each beam as a function of distance from the aperture when the wavelength of each field is .lambda.=0.5 .mu.m. Since the initial energy in the J.sub.o beam is substantially greater than that of the Gaussian beam, it is not remarkable that the J.sub.o beam propagates a greater distance than the Gaussian. What is remarkable is that even as the peak intensity of the J.sub.o beam oscillates (in a manner remeniscent of the intensity distribution for the diffraction pattern near a straight edge), the central maximum of the intensity profile doesn't spread along the entire range of propagation, as demonstrated in FIGS. 4b-d. Such a beam would be very useful, for example, in performing high precision autocollimation or alignment. There is a simple and accurate method for finding the range of a J.sub.o beam of finite aperture. One sees from equation (9) that the J.sub.o beam is a superposition of plane waves, all having the same amplitude and traveling at the same angle .theta.=sin.sup.-1 (.alpha./.kappa.) relative to the z-axis, but having different azimuthal angles ranging from 0 to 2 .pi.. For such a field, geometrical optics predicts that a conical shadow zone begins at the distance EQU z=r/tan.theta. =r[(.kappa./.alpha.).sup.2 -1].sup.1/2, (10) where r is the radius of the aperture in which the J.sub.o beam is formed. For the case shown in FIG. 4a one finds that .theta.=0.143.sup.o and z=80 cm, which is a point located right at the base of the sharp decline in beam intensity shown in FIG. 5. In fact, equation (10) has been found to accurately predict the effective range of J.sub.o beams of finite aperture for values of .alpha. in the range .alpha.=.kappa. (when there is no propagation) to .alpha.=2.pi./r (when the source field is essentially just a disc of radius r). One method of creating a J.sub.o beams of finite aperture is by plane wave illumination of an object whose amplitude transmission function is equal to J.sub.o (.alpha..rho.). This object would consist of a phase plate whose amplitude is +1 in those regions that J.sub.o (.alpha..rho.) &gt;0 and -1 in those regions where J.sub.o (.alpha..rho.) &gt;0, followed by a mask (e.g., photographic film) whose amplitude transmission if equal to .vertline.J.sub.o (.alpha..rho.).vertline.. Another simple method consists of uniformly illuminating a circular slit located in the focal plane of a lens. In principle, each point on the circular slit acts as a point source which produces a plane wave propagating at an angle .theta.=tan .sup.-1 (.epsilon./f), where .theta. is the radius of the slit and F is the focal length of the lens. If the incident light is of wavelength .lambda., the resulting J.sub.o beam will have a central spot diameter of .sub.( 3.lambda./4)[ 1+(f/.epsilon.).sup.2 ].sup.1/2. The embodiment of FIG. 6 generates a beam having a transverse dependence of a Bessel function by placing a circular annular source 30 of an input beam 34 in the focal plane of a lens focusing means 32, which results in the generation of a well defined beam thereby because the far field intensity pattern of an object is the Fourier transform thereof, and the Fourier transform of a Bessel function is a circular function. The arrangement of FIG. 6 forms the narrow beam 38 as predicted by the theory herein, which substantially retains its form at 38' unaffected by the normal spreading effects of diffraction. The arrangement of FIG. 6 is generally applicable to embodiments with optical components, microwave components and acoustical components because of the commercial availability of the different components of the arrangement of FIG. 6 for those types of beams. It has been shown, with reference to FIG. 6, that the sharp central spot size s is related to the radius r of the circular hole in the screen, the focal length f of the lens, and the wavelength .lambda. of the light beam by the simple formular s=(3/4) (.lambda.f/r). FIG. 7 is a schematic illustration of a second embodiment of the subject invention, analogous to the first embodiment of FIG. 6 but designed specifically for operation with acoustical waves. In this embodiment, a circular annular source 40 of an acoustical beam is placed in the focal plane of an acoustic lens 42 to produce a narrow acoustical beam 44 as predicated by the theory herein which substantially retains its form at 44' unaffected by normal spreading effects of diffraction. The annular source 40 can be formed by a circular annular diaphragm 46 reciprocally driven at a selected acoustical frequency F by an acoustic drive transducer 48. The acoustical lens 42 can take any common form such as those described in SOUND WAVES AND LIGHT WAVES, by Winston E. Kock. This reference also describes several different types of microwave lens which could operate in microwave embodiments analogous to the embodiments of FIGS. 6 and 7. The annular source of a microwave embodiment could be very similar to that illustrated in FIG. 6, with the screen 36 now being opaque to microwaves, such as by metal screen. FIG. 8 illustrates a third embodiment of the present invention, particularly applicable to operation with microwaves, and FIGS. 9, 10 and 11 illustrate respectively the phase plate transmittance, the spatial filter transmittance, and the output beam intensity of the third embodiment of FIG. 8. In microwave embodiments, the wavelength is not microscopic, but typically may be several centimeters (one inch=2.54 cm). This size allows an array of a large number of phase shifters in a phase plate 52 to be coupled with an absorption filter 54, an shown schematically is FIG. 8. The absorption filter 54 is selected of elements whose degree of absorption is tailored to produce the overall size of the required Bessel modulation, while the phase shifters generate the negative portions of J.sub.o (.alpha..rho.). In this embodiment, the beam is generated by transmitting a coherent beam sequentially through a phase modulator, having a periodic step function pattern, and a spatial filter, whose transmittance is the modulus of the Bessel function, to generate a beam having a transverse Bessel function profile. As illustrated in FIG. 9, the phase plate 52 can have a periodic step pattern which alternately transmits and blocks microwaves which is aligned with the spatial filter 54 having a microwave transmittance function as illustrated in FIG. 10. In a practical embodiment, the spatial filter 54 could be constructed by using a recording densitometer to record the function of FIG. 10. A prototype diffraction free aperture has been constructed tested with commercially available optical equipment arranged as illustrated in FIG. 6, and its operation is substantially in agreement with the mathematical conclusions drawn from the Wave Equation and expressed herein. The following detailed discussion of the five related embodiments of FIGS. 12-16 is generally generic to either Light Amplification by Stimulated Emission of Radiation (LASER's) or Microwave Amplification by Stimulated Emission of Radiation (MASER's), and the only real difference therebetween is in the selection of different components for focusing of the radiation, or different materials for reflecting or partially reflecting the particular wavelengths of radiation involved therein. The embodiments of FIGS. 12-15 are all diffraction-free mode generators, and have the common characteristic of integrating the radiation source into the diffraction-free mode generator, as opposed to directing an externally generated beam through a diffraction-free aperture. The embodiment of FIG. 16 is somewhat of a hybrid specy in this regard as a diffraction-free aperture is incorporated into one end of the resonant cavity. All of the embodiments of FIGS. 12-16 are generally expected to produce much higher output power and increased efficiency of operation. Moreover they can be used to produce intense light beams of very small diameter (60 microns or much smaller) having applications to precision pointing, microwelding, and ultra-small scale data deposition and scanning. The different disclosed embodiments of FIGS. 12-16 have several common characteristics: (a) a close relation to a known stable laser or maser cavity design, (b) a large mode volume to permit exploiting the relatively high gain of laser or maser system, and (c) little departure in principle from the design that has already led to successful observation of non-diffracting beams. FIG. 12 illustrates a first embodiment of a diffraction-free mode generator 60 having a resonant cavity with a pumped active gain medium therein. A synthesized Bessel function mask 62 is placed at one end of the resonant cavity, and is designed to achieve a required Bessel function behavior for the electric field amplitude of the radiation beam. The mask 62 is similar in principle to a combination of the phase plate 52 and spatial filter 54 illustrated in the embodiment of FIG. 8, and can be fabricated in any known manner such as holographically. This embodiment is particularly suitable for generating all of the Bessel mode beams with appropriate modifications of the mask. The so-called "higher modes" correspond to Bessel functions of index number higher than zero: J.sub.1, J.sub.2, etc. By using a collection of higher Bessel modes in conjunction with the zero-order mode, non-diffracting beams can be produced with any desired shape of beam spot-oval instead of circular, for example. The resonant cavity also includes a reflecting mirror surface 64 adjacent to the Bessel Function mask 62 and defining one end of the resonant cavity, with the other end of the resonant cavity being defined by a partially reflecting mirror surface 66. The diffraction-free Bessel mode beam 68 is formed by that portion of the radiation which is transmitted through the partially reflecting mirror surface 66. The embodiments of a diffraction-free mode generator illustrated in FIGS. 13, 14 and 15 make use of the "bright circle" Fourier principle underlying the zero-order Bessel mode corresponding to the zero-order Bessel function J.sub.o. Each of these embodiments incorporates within the resonant cavity a radiation reflective element in the shape of a narrow circle or annulus, and a lens is positioned to image the circle for transmittal outside of the cavity. In all three embodiments, the output beam draws efficiently on the gain medium, as does a laser or maser, but the optical or microwave components convert the radiation from the normal laser or maser (Gaussian) form to the non-diffracting Bessel mode beam. In the embodiments of FIGS. 13, 14 and 15, the mean diameter of the annular reflector is d(=2.rho.), the width of the annular reflector is .alpha.d, the radius of the focusing lens system is R, the focal length thereof is f, and the radiation has wavelength .lambda.. Ideally, each point along the annual reflector acts as a point source which the lens transforms into a plane wave. The set of plane waves formed in this manner has wave vectors lying on the surface of a cone, and it has shown that this can be regarded as the defining characteristic of the J.sub.o beam. The J.sub.o beam produced in this manner has a spot parameter .alpha.=(2.pi./.lambda.)sin .theta., where .theta.=tan.sup.-1 (d/2f). In practice, of course, the amplitude is modulated by the diffraction envelope of the annular reflector. That modulation is negligible within the finite output aperture R, provided that .alpha.d&lt;.lambda.f/R. The embodiment of FIG. 13 places an annular reflector or mirror (in optical embodiments) 70 on a transmitting substrate 72. The second end of the resonant cavity is defined by a partially reflecting reflector or mirror surface 74 on a focusing element 76 having the annular mirror 70 positioned in the focal plane, such that it projects a non-diffracting Bessel mode bean 78. The embodiment of FIG. 14 simply places an annular reflecting or mirror surface 80 at one end of the resonant cavity. The annular reflector or mirror 80 is placed in the focal plane of a focusing element 82 in the resonant cavity, and the output non-diffracting Bessel mode beam 86 passes through a partially reflecting output surface 84. The embodiment of FIG. 15 places an annular reflecting mirror (for optical embodiments) surface 90 on a transparent substrate 92 at one end of the laser or maser cavity. The opposite end of the resonant cavity is formed by a partially transmitting mirror (for optical embodiments) surface 94. The transmitted portion of the beam is focused by a focusing element 96 having the annular mirror 90 positioned in its focal plane to form the output non-diffracting Bessel mode beam 98. The embodiment of FIG. 16 is somewhat of a hybrid embodiment wherein a maser or laser cavity is defined by two end reflectors or mirrors 100 and 102, the latter of which has an annular aperture or slit 104 formed therein. A focusing element 106 is positioned outside of the resonant cavity to have the annular aperture 104 in its focal plane, and projects the output non-diffracting Bessel mode beam 108. In this embodiment, the width .alpha.d of the annular slit should be as narrow as possible to sustain a Gaussian mode of operation in the cavity, and preferably is of the order of one wavelength. In alternative embodiments, particularly with respect to the designs of FIGS. 13-16, other types of focusing system designs could be utilized, such as reflective-based focusing systems. Moreover, each laser cavity embodiment could be implemented in any type of laser cavity operating in the infra-red, visible or ultraviolet wavelengths of light, such as gas lasers, liquid lasers, solid lasers, laser diodes, and continuous wave or pulsed lasers. Each maser cavity embodiment could operate in any suitable portion of the microwave spectrum. During Single-Mode Operation of any of the resonant cavities illustrated in FIGS. 12-16, when the losses of the cavity are adjusted so that only a single longitudinal mode is above threshold, the output is a temporally-coherent J.sub.o beam which can be propagated as taught herein substantially unaffected by diffractive spreading. During Multimode Operation of the resonant cavity illustrated in FIG. 12, each of the longitudinal modes which are lasing or masing (i.e. those modes within the gain profile that are above threshold) will have the same transverse mode structure, namely, that of a J.sub.o beam. Although the output will now be temporarily-incoherent, the time-averaged intensity profile will be exactly the same as that obtained when only a single longitudinal mode was lasing. When the cavities of FIGS. 13-16 oscillate multimode, each longitudinal mode will be in a transverse J.sub.o mode whose spot size is proportional to the longitudinal mode frequency. The range .alpha.s in spot sizes is given by .alpha.s/s=.alpha.W.sub.G /W, where .alpha.W.sub.G is the bandwidth of the gain profile and w is the mean frequency of oscillation. In all currently known gain media, this ratio is on the order of 10.sup.-3 or less, and therefore the transverse intensity profile near the center of the beam will essentially the same as that obtained in single-mode operation as frequency w. Mode-locking (A discussion of the various methods that can be used to effect mode locking can be found, for example, in: A. Yariv, Quantum Electronics, Ch. 11, Wiley, 1975) can be used to transform the temporally-incoherent output of these laser or maser cavities into a train of pulses of width .alpha.t=2.pi./.alpha.W.sub.G (which ranges from pico to nanoseconds for typical laser media). A further advantage of mode locking is that the peak output power is increased in direct proportion to the number of modes that are lasing or masing. While several embodiments and variations of the present invention for a diffraction free arrangement are described in detail herein, it should be apparent that the disclosure and teachings of the present invention will suggest may alternative designs to those skilled in the art.