Patent Application
20060274418
This application claims priority to Japanese Patent Application No. 2005-166337 filed Jun. 7, 2005.
This invention relates to an improvement of a homogenizer DOE (diffraction optical element) system capable of converting a Gaussian power distribution laser beam into a uniform power distribution (tophat) beam. High power lasers have been more and more applied to laser processings, for example, boring holes, cutting grooves and welding metals. The laser processings have a variety of purposes of welding, cutting, annealing, boring and so on.
Some fields of the laser processings require the conversion of a Gaussian power distribution of a laser beam into a uniform (tophat) or quasi-uniform power density distribution. There are two kinds of optical systems for converting a Gaussian beam into a uniform or quasi-uniform power distribution beam. One is an aspherical lens system. The other is a diffraction optical element (DOE) system. The optical system which converts a Gaussian power distribution beam into a uniform or quasi-uniform power beam is called a “homogenizer”. This invention relates to a DOE type homogenizer.
DOEs are endowed with high degree of freedom. High degree of freedom enables a DOE to change a Gaussian beam to a uniform or quasi-uniform beam.
There are a transmission type of DOE and a reflection type of DOE. Here the transmission type DOE is explained. The transmission type DOE is a transparent plate composed of small pixels σ with different step heights (thicknesses) aligning lengthwise and crosswise. A unit height ε of the step heights is determined to be a quotient of a wavelength light path difference λ/(n−1) divided by a constant step number w. Namely ε=λ/w(n−1), where n is a refractive index of the DOE and λ is a wavelength of light. In usual, w is an exponent of 2, for example, w is 2, 4, 8, 16, 32, 64, 256, . . . and so on. Namely w=2b: b=1, 2, 3, 4, 5, . . . and so on. The number of horizontally aligning pixels is denoted by M and the number of vertically aligning pixels is denoted by N. The total number of pixels is MN. The pixel size is denoted by d. The area of a pixel is denoted by d×d. An effective area of the DOE is Md×Nd.
A DOE varies phase of transmitting rays by the variation of thicknesses of pixels. Two dimensional coordinate (u, v) is defined on the DOE. Since pixels are discrete members, a set of integers can denote pixels. But continual variables (u, v) are allotted to pixels for designating their positions. A complex transmittance of a pixel (u,v) is denoted by T(u, v). The phase φ of a ray transmitting via (u, v) is changed by the thickness h(u, v). The phase difference (u, v) is related to the thickness h(u,v) of pixels by the following equation.
φ(u,v)=(2π/λ)(n−1)h(u,v). (1)
The complex transmittance T(u,v) is coupled with the phase difference φ(u, v) by the following relation.
T(u,v)=exp(iφ(u,v)). (2)
An incident beam is assumed to be a plane wave, which has the same phase on a plane vertical to the beam axis. Complex amplitude of light changes in two dimensions. Incident complex amplitude is denoted by a(u, v). Complex amplitude is otherwise called a wavefunction. Transmission in the DOE, which has the complex transmittance T(u,v), changes the complex amplitude to a(u,v)T(u,v). The distance from the DOE to the image plane is designated by L. Another two dimensional coordinate (x,y) is defined on the image plane. The distance from pixel (u,v) on the DOE to a point (x,y) on the image plane is designated by q. A ray from (u,v) to (x,y) changes the complex amplitude by exp(ikq). Here k is a wavenumber of the light. k=2πn/λ. Complex amplitude on the image plane is represented by I(x,y). A partial contribution dI(x,y) from pixel (u,v) to I(x,y) is
dI(x,y)=a(u,v)T(u,v)exp(ikq)dudv. (3)
Symbols dI(x,y), du and dv are differentials of I(x,y), u and v. Integration of a(u,v)T(u,v)exp(ikq) by u and v gives the image plane a whole complex amplitude (wave function).
I(x,y)∫∫=a(u,v)T(u,v)exp(ikq)dudv. (4)
∫∫dudv means summation with regard to all the pixels on the DOE. Summation is represented by double integrals. The term exp(ikq), which transcribes DOE transmittance T(u,v) to the image plane, gives the effect of diffraction. It is difficult to calculate the contribution of exp(ikq) exactly. For avoiding the difficulty, exp(ikq) shall be approximated. The approximation reduces the calculation of Eq. (4) to a Fourier transformation. The distance q between (u,v) on the DOE and (x,y) on the image plane is given by
q={(u−x)2+(v−y)2+L2}1/2. (5)
Since the DOE/image-plane distance L is far larger than u, x, v and y, q is approximately reduced to
q=L+(u2+v2+x2+y2)/2L−(ux+vy)/L. (6)
Under the approximation, the above equation means that a(u, v)T(u, v)exp {(u2+v2)/2L} is connected with I(x,y)exp {(x2+y2)/2L} by Fourier and inverse-Fourier transformations.
The image plane (object plane) requires some restrictions. The condition on the image plane is predetermined. For example, the image power distribution function I(x,y) should be constant within a definite area and be zero outside of the area on the image. I(x,y)exp {(x2+y2)/2L} is determined on the image plane. Inverse-Fourier transformation of I(x,y)exp {(x2+y2)/2L} gives a(u,v)T(u,v)exp {(u2+v2)/2L} on the DOE plane.
The complex amplitude a(u,v), which is a wavefunction of an incident Gaussian laser beam, has been determined by the incident laser beam profile. Then a DOE complex transmittance T(u,v) is obtained. DOE's restrictions are imposed upon the DOE complex transmittance T(u,v).
When the DOE is fully transparent, no absorption occurs in the DOE. The absolute value of the transmittance should be 1. Thus |T(u,v)|=1. This is a normalization condition. The inverse-Fourier transformed T(u,v) is not normalized to be 1. Thus a new DOE transmittance T(u,v) is replaced by T(u,v)/|T(u,v)|. The new transmittance T(u,v) is normalized to be 1. Intensity variation is trivial owing to transparency. Phase variation is important. Restriction-satisfying a(u,v)T(u,v)exp {(u2+v2)/2L} is obtained on the DOE. Fourier transformation of a(u,v)T(u,v)exp {(u2+V2)/2L} gives a renewal I(x,y)exp {(x2+y2)/2L} on the image. Image's restrictions should be imposed on the renewal I(x,y)exp {(x2+y2)/2L}. Maintaining the phase component, the on-image condition imposes that I(x,y) should be constant within the definite area and 0 outside of the area. A single cycle of Fourier and inverse-Fourier transformations is explained.
The same Fourier and inverse-Fourier transformations shall be repeated hundreds of times or thousands of times between the DOE plane and the image plane. The purpose of repeated calculations is to determine the transmittance T(u,v). Repetitions of Fourier/inverse-Fourier make T(u,v) converge to a definite function. A converged T(u,v) is obtained after the repeated calculations. The absolute value of T(u,v) is 1. Phase is important, as mentioned before. Eq. (2) determines phase distribution φ(u,v) of (u,v) pixels of the DOE from the calculated T(u,v). Eq. (1) allocates (u,v) pixels height distribution h(u,v) (thickness distribution) (u,v) from φ(u,v). Since the heights of all pixels are determined, a DOE for the purpose can be made by cutting all pixels to the designed heights.
The purpose of the DOE is to make a uniform power density (homogenized) beam within a definite area on the image plane 5. A righthand figure of
Heights of pixels are quantized into discrete step values in a DOE. Differences of heights of pixels are multiples of a unit height ε. Pixels are made by mechanical cutting. Some materials allow etching to make a variety of discrete pixel heights of a DOE. Cutting step of micro-sized pixels (for example, 5 μm square) is a difficult ultrafine processing. Cutting errors accompany the pixel processing. There are a variety of cutting errors. The most serious error, which gives the worst influence on the homogenizing property, is a “step height error” of pixels. The step height error means that actual step heights deviate from the calculated, designed heights of pixels.
If step height errors accompany the fabrication of a DOE, a new beam which is not fully diffracted but goes straightly along the axis is yielded. The non-diffracted straight-forward going beam is called a “zero-th order” beam. The zero-th order beam is invited by the step height errors. If the zero-th order beam happens, the zero-th order beam interferes with the diffracted beam.
Interference occurs between the zero-th order beam and the diffraction beam. The interference causes zigzag-fluctuating power distribution k′ within J on the image shown at a righthand of
The resultant beam on the image is a uniform power density beam. Big discontinuity accompanies boundaries between an irradiated area and a non-irradiated area of the resultant beam. The discontinuity invites instability of power profile on the image. Instability exaggerates the interference between the zero-th order beam and the diffraction beam. A purpose of the present invention is to provide a DOE which prevents the resultant beam from fluctuating in a big zigzag profile k′ on the image. In other words, a purpose of the present invention is to provide a homogenizer DOE which can suppress the interference between the zero-th order beam and the diffraction beam.
This invention proposes a wedged homogenizer DOE with unparallel surfaces prepared by slanting a front surface, a rear surface or both surfaces. Instead of parallel surfaces, the DOE has unparallel front and rear surfaces. The unparallel surfaced shape is called a “wedge”. The present invention gives a DOE a wedge-shape. The wedge-shaped DOE refracts the zero-th order beam in an off-axis direction but allows the diffracted beam to pass straightly on the axis. The wedge-shaped DOE can separate the zero-th order beam from the diffracted beam. The area at which the zero-th order beam attains on the image plane does not overlap with but separates from the area at which the diffracted beam attains on the image plane. No interference occurs between the zero-th order beam and the diffracted beam on the image plane.
Though the DOE has a shape of wedge, the DOE diffracts the beam in a straightforward direction. The straightforward diffraction enables the laser, the DOE and the image plane (object) to align along a straight line. Without step errors, the DOE produces only the diffracted beam 4 as shown in
Common sense would allege that a zero-th order beam would be a noise going straightforward without bending at a diffraction member. But this is wrong. Strictly speaking, the zero-th order beam should be a diffraction beam having the zero-th order of diffraction without path difference. When a DOE is not planar one but a wedged one, the zero-th order beam is not diffracted and does not go straight. The zero-th order beam is bent in the direction of the thickness increasing like refraction at a flat wedge block. Bending of the zero-th order rays is important for a wedge DOE. Since the image (plane) 5 is far separated from the DOE 3, projection of the zero-th order rays on the image 5 is distanced from the pattern 8 of the diffraction beam 4 on the image 5. The zero-th order beam 6 does not overlap with the diffraction beam 4 on the image 5. The zero-th order beam 6 (RSZ) does not interfer with the diffraction beam 4 (RST) on the image 5.
How high degrees does the separation of the zero-th order beam from the diffraction beam requires the DOE for an wedge angle? An angle of viewing the uniform power pattern on the image from the center of the DOE is denoted by . The distance between the DOE 3 and the image plane 5 is denoted by L. The diameter of the incident beam is D. The bend angle θ of the zero-th order beam must satisfy an inequality of
+(D/L)≦θ for preventing the zero-th order beam from overlapping with the diffraction beam 4 on the image plane 5. The wedge angle Θ is related to the zero-th order bending θ by an equation θ=sin−1 (n sin Θ)−θ. A condition
+D/L≦{sin−1(n sin Θ)−Θ} is imposed upon the wedge angle Θ for separating the zero-th order and diffraction beams. In brief, the spatial separation requires a more restricted condition
+(D/L)≦(n−1)Θ or {
+(D/L)}/(n−1)≦Θ for the wedge angle Θ.
The wedge DOE proposed by the present invention has a wedge shape as a whole. But microscopically every pixel has no inclining top but has a top parallel to the standard plane. The wedge is a collective shape. The wedge surface has a zigzag outline in a microscopic scale. The zero-th order beam is directed in a direction where a path difference between a ray emanating from a spot of a pixel and another ray emanating from a corresponding spot of a neighboring pixel is zero. The zero-th order beam has a definite width of D, where D is an incident beam diameter. The diffraction beam has also a width of L, where L is a distance between the DOE and the image. A sum of the widths is
+(D/L). The minimum bending power of a wedge angle Θ is (n−1)Θ, which is known in prism optics. Intuition can understand the non-overlapping requirement
+(D/L)≦(n−1)Θ. The relation will be clarified again.
This invention shapes a DOE into a wedge having inclining, non-parallel surfaces. The wedge enables the DOE to spatially separate the zero-th order beam from the diffracted beam. Even if step height errors accompany pixels, spatial separation prevents the zero-th order beam from interfering with the diffracted beam. No interference occurs. The DOE can maintain uniform power distribution of the diffracted beam on the image plane in spite of pixel step errors.
The gist of the present invention is to make a wedge-shaped DOE capable of preventing the zero-th order beam from going straight forward, bending the zero-th order beam in a slanting direction, deviating the zero-th order beam from the diffraction beam line (RST) and prohibiting the zero-th order beam from interfering with the diffraction beam. However, pixels of any DOE have flat tops parallel with the standard plane (DOE surface). A wedge-shaped DOE of the present invention has pixels with flat tops. The DOE is a wedge on the whole. But individual pixels have no slanting tops but flat tops. There is no wedge steps of pixels in a wedge DOE in a microscopic scale. Analogy of a wedge glass assumes that a wedge DOE would bend the zero-th order beam in the direction of a increasing thickness. But it is not proved yet that a Wedge DOE with flat tops of pixels would bend the zero-th order beam like refraction by a wedge glass block. Can or cannot a wedge-shaped DOE bend the zero-th order beam in the direction of increasing a thickness? Don't confuse diffraction with refraction.
Isn't the beam, which is immune from diffraction, named as zero-th order beam due to the reason that the beam is not diffracted by a DOE? Even a wedge DOE is a set of pixels. Pixels have flat tops perpendicular to the axis. If the zero-th order beam is assumed to be bent by a wedge DOE, what is the reason of the zero-th order beam bending? It would be marvelous from the standpoint of geometric optics. This is a problem to be clarified first.
The plane perpendicular to the beam axis is called a “standard plane”. In the example, the front surface is a standard plane. The rear surface bends the outgoing beam 24 at an angle θ, which is called a “bending angle”. Snell's law requires sin(θ+Θ))=n sin Θ at the refraction on the rear surface. Θ is a wedge angle, θ is a bending angle and n is a refractive index of the glass block 23. The wedge angle Θ enables the glass block 23 to bend the incident beam 22 at θ. Refraction of a glass block like a prism, gives incident light, which corresponds to the zero-th order beam, a bending angle θ.
The bending angle θ is given by
θ=sin−1(n sin Θ)−Θ. (7)
Here Θ is a wedge angle at which the rear surface inclines to the front surface of the glass block 23. The relation (7) is correct for a glass block having continual surfaces. A wedge DOE having an average inclination angle Θ between the surfaces should be considered. It is still questinable that a wedge DOE would lead the zero-th order beam in the direction of Eq. (7). Although average planes of both surfaces incline, individual pixels formed on a surface have all flat tops parallel to the other surface. Since all the pixels have microscopically flat steps which are parallel to the other surface, it would be still questionable whether the zero-th order beam is really bent by a macroscopically wedge-shaped DOE. Blunt insight is of no use.
All the steps of pixels are orthogonal to the beam axis. As long as Snell's law were taken account, the zero-th order beam which is a set of rays passing individual pixels would never bend. A sense of word “zero-th order” alleges straightforward progress. The zero-th order beam on a wedge DOE should be now considered.
The incident beam is a set of parallel rays orthogonal to the standard plane(front surface). Three rays distanced by a pixel size d are considered. A first ray passes S1 point in the DOE and goes out of the DOE via W1 point. A second ray passes S2 point in the DOE and goes out of the DOE via W2 point. A third ray passes S3 point in the DOE and goes out of the DOE via W3 point. W1, W2 and W3 are corresponding points which are separated from each other by a multiple of d (pixel size).
S1W1N1, S2W2N2 and S3W3N3 are straightforward rays (solid lines) going out via W1, W2 and W3. Are these three rays real or virtual? This is a problem. Obeying the refraction law, three rays S1W1N1, S2W2N2 and S3W3N3 advance straightforward. However, optical path differences are not zero. An optical path is defined as the sum of products nl of a refractive index n and a path length l. Σnl is a definition of an optical path length. Σ denotes summation of the following terms. The optical path length S2W2N2 is longer than the optical path length S1W1N1 by (n−1)s, where n is a refractive index of the DOE and s is a step height in
Phases of the corresponding straightforward rays increase by 2π(n−1)s/λ step by step. The number of the corresponding straightforward rays is equal to the pixel number MN. The pixel number MN is a very large number. Since the basic phase difference 2π(n−1)s/λ is not a multiple of 2π, the MN corresponding straightforward rays cancel by each other and die away. S1W1N1, S2W2N2 and S3W3N3 have no reality. Thus the corresponding straightforward rays like S1W1N1, S2W2N2 and S3W3N3 can be taken out of consideration. This point quite differs from geometric optics, which relies only upon refraction.
Light paths S1W1P1, S2W2P2 and S3W3P3, which progress in parallel in the block, go out via corresponding spots W1, W2 and W3 on the neighboring steps, bend downward and make a wavefront P1P2P3, are considered. A wavefront means a locus of the same phase or rays. If the light paths have no path difference, the rays reinforce themselves and make a real light beam in the direction. The angle of the downward beams W1P1, W2P2 and W3P3 to the axis is denoted by Φ0. A path difference between one light path and the next light path is (S2W2P2−S1W1P1)=ns−(d sin Φ0+s cos Φ0 ). If the optical path difference is a multiple of λ, the rays in the direction reinforce themselves and make a diffraction beam. In particular, if the light path difference (S2W2P2−S1W1P1) is 0, all the rays from all pixels coincide in phase with each other in the direction. The 0 difference condition is given by
ns−(d sin Φ0 )+s cos Φ0 )=0. (8)
Eq. (8) proposes a condition of the non-light path difference on a wedge DOE. Namely Eq. (8) is a zero-th order equation in a wedge DOE. The zero-th order beam is one which is determined by Eq. (8). The straightforward beam is not the zero-th order beam. Eq. (8) is the zero-th order beam in a wedge DOE. Similarly, an m-th order beam, which has a light path difference of m times as large as λ, is defined by one which satisfies the following restriction.
ns−(d sin Φm+s cos Φm)=mλ.(m=±1,±2,±3, . . . ) (9)
Eq. (8) defines the zero-th order beam. Eq. (9) defines the m-th order beam. For an arbitrary Φ,
d sin Φ+s cos Φ=(d2+s2)0.5 sin(Φ+α). (10)
Here sin α=s/(d2+s2)0.5. Substituting the relation into Eq. (8) yields
n(d2+s2)0.5 sin α=(d2+s2)0.5 sin(Φ0+α). (11)
The common factor (d2+s2)0.5 should be eliminated. Then Eq. (11) becomes
n sin α=sin(Φ0+60 ). (12)
Comparison Eq. (12) with Snell's Law sin(θ+Θ)=n sin Θ in the Θ-wedge glass block implies that replacement of α=Θ can equalizes a wedge DOE to a wedge block. In face, sin α=s/(d2+s2)1/2. tan Θ of the wedge angle Θ of the glass block is equal to the ratio s/d of height s to length d in a DOE. tan Θ=s/d.
The above explanation proves that the zero-th order beam should bend in a direction of Φ0=θ by the DOE which is prepared by quantizing the slant into pixels of a step s and a length d which satisfy tan Θ=s/d. The zero-th order beam angle Φ0 in the wedge DOE is equal to the refraction angle θ of the wedge block of
What does this result mean? The bending angle of the zero-th order beam produced by a wedge DOE is entirely equal to the bending angle of light refracted by a wedge block having the same slope. What has been proved is that the zero-th order beam should be bent in a wedge DOE as shown in
n sin Θ−sin(Φm+Θ)=mλ/(d2+s2)0.5(m=±1, ±2, . . . ). (13)
An increment of m causes a decrement of Φm. A further increase of m invites negative Φm.
(n−1)sin Θ=mλ/(d2+s2)0.5.(m=1, 2, 3, . . . ) (14)
If there were a positive integer m satisfying Eq. (14), the straightforward rays, e.g. W1N1, W2N2 and W3N3, (m=0) would exist and would form a wavefront N1N2N3. Since this meant Θ=α, sin Θ would be s/(d2+s2)1/2. Eq. (14) is reduced to
(n−1)s=mλ.(m=1, 2, 3, . . . ) (15)
λ/(n−1) is a full single wavelength step. A DOE has no step height which is higher than the full single wavelength step λ/(n−1). In any case, a step difference s is less than λ/(n−1). There is no positive integer m. The straightforward rays, e.g. W1 N1, W2N2 and W3N3, (m=0) never exist. Geometric optics is mute. Waveoptics gives us a clear solution.
The allowable range of the wedge angle Θ can be determined by the requirement that the zero-th order beam should not overlap with the diffraction beam. As mentioned before, the range of allowable Θ is represented by Θ≧(+D/L)/(n−1).
is an aperture angle glancing a pattern on an image from a DOE center, and n is a refractive index of the DOE. When Θ is determined, the preceding consideration of the zero-th order beam gives tan Θ=s/d. The pixel size d is a predetermined constant. The step height difference s is determined. The full single wavelength step height is λ/(n−1). Step height differences are conveniently quantized into w (2, 4, 8, 16, 32, 64, . . . , w=2b). A unit step ε is a quotient λ/(n−1)w of the full single wavelength difference λ/(n−1) divided by w. ε=λ/(n−1)w.
Height differences between neighboring pixels should take one of only the w steps of ε, 2ε, 3ε, 4ε, . . . , (w−1)ε. The step differences for making a wedge desired by the present invention should also be one of the w degrees of steps of ε, 2ε, 3ε, 4ε, . . . , (w−1)ε. The neighboring step height difference s for making the desired wedge is assumed to be k times as high as the unit difference ε(=λ/w(n−1)), i.e., s=kε, where k is a positive integer less than (w−1)(1≦k≦(w−1)).
s=kλ/w(n−1). (16)
This relation gives a condition of the step height difference to neighboring pixels for making the desired wedge. The step height difference is replaced by the wedge angle Θ.
tan Θ=s/d=kλ/dw(n−1). (17)
The unit of step height is ε(ε=λ/w(n−1)). By replacing λ/w(n−1) by ε, Eq. (17) is rewritten to
tan Θ=s/d=kε/d.(k; integer, 1≦k≦w−1). (18)
The tangent of the inclination angle Θ of a wedge DOE shall be a multiple of ε/d which is a quotient of the unit step ε divided by the pixel size d. The minimum of tan Θ derives from the case of putting k=1 for making a wedge.
tan Θ≧ε/d=λ/wd(n−1). (19)
The upper limit of tan Θ is determined by an inequality of k/w<1.
tan Θ<λ/d(n−1). (20)
λ/(n−1) is a full wavelength thickness difference. Eq. (20) implies that the tangent of the wedge Θ should be smaller than the quotient of one wavelength height λ/d(n−1) divided by the pixel size d.
The above requirements determine the relations between the wedge angle Θ, the neighboring step difference s and the multiple number k of the wedge DOE of the present invention. The continual slant of the wedge is replaced by the height difference s=kε between discrete neighboring pixels. The wedge has an effect of excluding the zero-th order beam. A convergence effect and a homogenizing effect together with the zero-th order exclusion effect shall constitute the DOE.
A superposition of three effects should construct the DOE. Component effects are allotted to glass blocks for clarifying the superposition for making the DOE.
In fact, the wedge DOE is an assemble of three virtual components 43, 45 and 47. Intermediate beams 44 and 46 are non-existing, imaginary rays. The components are virtual imaginary parts which facilitate to understand the function by intuition.
The rear surface (or the front surface) inclines at Θ. A beam 44 inclines. Reversing the inclining beam to a horizontal beam requires a complementary, imaginary glass block 45. The complementary glass block 45 should be quantized into a discrete stepping DOE. The wedge block 45 should have discrete steps having a common height difference s=kλ/w(n−1). The steps form a sawtooth structure on the rear surface (or front surface) of the block 45.
The third virtual component following the wedge glass block 45 is an aspherical lens 47 which has a homogenizing effect and a converging effect like a Fresnel lens. The virtual aspherical lens 47 produces a homogenized and reduced beam 48 on the image plane. The virtual aspherical lens 47 is a kind of convex lens, since the lens has convergence. The imaginary aspherical lens 47 has a flattened central part since the lens has a homogenizing function. The virtual aspherical lens should be replaced by a planar lens making use of diffraction instead of refraction. An equivalent planar lens should have a sawtooth surface which shall be produced by repeatedly subtracting a wavelength unit height λ/(n−1) from the surface height till the resultant height is reduced below λ/(n−1). The convex surface shall be replaced by a sawtooth-like surface by the repetitions of subtraction. The imaginary aspherical lens should be reduced to a sawtooth-surfaced planar element. Among three imaginary components, the wedge glass block 43 is employed as a shape of the wedge DOE. The virtual complementary wedge block 45 and the virtual aspherical lens 47 should be planar elements having sawtooth surfaces determined by the discrete step functions.
An outline shape of the DOE of the present invention shall be determined exclusively by the first glass block 43. Superficial sawtooth-like shapes of the DOE shall be determined by a superposition of the zigzag surfaces of the sawtooth and Fresnel blocks 55 and 57. A superposition of the wedge block 43 and the concentric Fresnel block 57 makes another concentric Fresnel pattern diverting the concentric center to the left. The saw-tooth component 55 gives parallel narrow grooves. The DOE pattern of
With reference to
Another upper ray passes at point 52, goes out at point 53, bends downward at Φ (dotted line) and progresses via point 63 in
ns−{(d+t−g)sin Φ+s cos Φ}=0. (21)
When (d+t−g) is equal to d (t=g), Φ0=θ=sin−1(n sin Θ)−Θ. When (d+t−g) is smaller than d (t<g), Φ is larger than Φ0 (Φ>Φ0). The minimum of (d+t−g) is 0. When (d+t−g) is 0, Φ extinguishes. The inequality Φ>Φ0 means that the zero-th order beam bends larger than Φ0. The zero-th order beam separates further from the diffraction beam. Thus Φ>Φ0 causes no problem (
When (d+t−g) is larger than d (t>g), Φ is smaller than Φ0 (Φ<Φ0). Smaller Φ brings noise rays closer to the diffraction beam formed along the beam axis. Small Φ invites the probability of interference of (zero-th order) noise rays with the diffraction beam. How far are noise rays approaching the diffraction beam? What is the minimum of Φ? This is a problem. The maximum of (d+t−g) is 2d (when t=d and g=0). When d+t−g=2d, t=d and g=0, Φ takes the minimum value Φmin. Analytical, exact calculation of the minimum Φmin is difficult. Approximate calculation yields,
Φmin=Φ0/2. (22)
An inner, axis-nearest, minimum bending angle Φmin is about a half Φ0/2 (=θ/2) of the zero-th order center angle Φ0 (=θ).
In order to prevent the zero-th order beam from overlapping with the diffraction beam, the breadth of the diffracted beam should be confined within a space of an angle smaller than a half (Φ0/2) of the bend (Φ) of the zero-th order beam. The condition ensures no overlapping of the zero-th order beam with the diffraction beam. No overlapping invites no interference between the zero-th order beam and the diffraction beam. .
and e are connected by a relation 2L tan(
/2)=e. Since the distance L from the DOE to the image is large in comparison to e, the relation is simplified to an approximate expression
L=e. (23)
The central component of the zero-th order beam is strongly bent in a direction (double dotted line) of Φ0=θ. But an upper peripheral component of the zero-th order beam is only bent by (single-dotted lines) Φmin(=θ/2). The zero-th order beam emanates from all the incident beam with a diameter D. The upper peripheral components of the minimum bending Φmin are projected all on Z1Z3 of the image (L/2 to T3=−
L/2. The position Z1 of the uppermost of the peripheral rays emanating from the upper edge of the DOE is Z1=−Lθ/2+D/2, where D is a diameter of the incident beam and θ is a bending angle of the zero-th order beam. If Z1 is below T3 (Z1<T3: −Lθ/2+D/2≦−
L/2), the zero-th order beam does not overlap with the diffracted beam and interference does not occur. The condition of non-overlapping is denoted by
L/2≦Lθ/2−D/2. (24)
Namely, exclusion of the zero-th order beam out of the diffraction beam requires the following inequality
L+D≦Lθ. (25)
Substitution of θ=sin−1 (n sin Θ)−Θ into (25) yields,
L+D≦L{sin−1 (n sin Θ)−Θ}. (26)
Dividing (26) by L leads to +(D/L)≦{sin−1(n sin Θ)−Θ}. (27)
This is a precise expression of the condition of excluding the zero-th order beam. When Θ is small enough,
(n−1)Θ<{sin−1(n sin Θ)−Θ)}. (28)
Inequality (27) can be reduced to a simplified approximate expression +(D/L)≦(n−1)Θ. (29)
Eq. (29) is a sufficient condition of excluding the zero-th order beam for a wedge DOE. Otherwise, non-interference condition can be written by
Θ≧{+(D/L)}/(n−1). (30)
This inequality teaches us that the minimum of the wedge angle Θ is {+(D/L)}/(n−1), where L is the distance between the DOE and the image, D is an incident beam diameter,
is an aperture angle of the diffraction pattern on the image viewing from the DOE center and n is a refractive index of the DOE.
L+D≦Lθ, because the divergence of rays starting from non-equivalent spots is within θ/2 and
is the aperture of rays diffracted from the DOE.
The light source is a YAG-SHG (Second Harmonic Generation) laser with a beam diameter 2φ=2 mm having a Gaussian power distribution. The purpose of the wedge DOE system is to produce a uniform-power distribution beam of a rectangle section of 0.5 mm×1 mm on an image plane distanced by 200 mm from the DOE. Main properties of the optical system are;
Wavelength: 532 mm
Beam diameter: φ2 mm (at 1/e2)
Wavefront: Flat (=Plane Wave)
Focal length (DOE/Image distance) L=200 mm
L=1 mm and D=2 mm. The condition of separation of the zero-th order beam and the diffraction beam is
L+D<Lθ. The critical condition is expressed by
L+D=Lθ. The critical condition is considered now. In this example,
L+D=1 mm+2 mm=3 mm. Lθ=3 mm gives the critical condition for θ.
In
θ=tan−1(3/200)=0.014999 rad=0.859372°
A wedge angle of the DOE is denoted by Θ. The front surface of the DOE is perpendicular to the beam axis RS. The rear surface of the DOE inclines to the axis at (90−Θ). At the rear surface, the diffraction angle θ and wedge angle Θ satisfy the following relation determined by Snell's Law.
sin(θ+Θ)=n sin Θ.
Since n and θ have been determined above) the minimum wedge angle Θ is calculated to
Θ=0.032536 rad=1.864153°.
This is the critical (minimum) wedge angle Θc of the DOE. This is a small angle. Endowment of a wedge angle more than Θc to the DOE eliminates overlapping of the diffraction beam with the zero-th order beam on the image plane. In the example, the beam breadth is 2 mm, the broadness of the diffraction beam is 1 mm and the deviation of the zero-th order beam is 3 mm on the image plane. There is no interference between the diffraction beam and the zero-th order beam. In practice, the DOE of the present invention should be assigned with a wedge angle more than the critical value Θc (in this example, 1.864153 degrees).
number of step heights: 16 steps
pixel size: 5 μm×5 μm
pixel number: 2000 pixels×2000 pixels
The pixel size is d=5 μm and the pixel number in x- and y-directions is M=N=2000 and MN=4000000. The effective area of the DOE is Md×Nd=10 mm×10 mm.
φ(u,v)=2πh(u,v)(n−1)/λ.
The phase distribution φ(u,v) shown in
Next, another wedged DOE with step errors having properties similar to the above DOE is produced for examining degradation of power uniformity. A plan view of the step-error allotted DOE quite resembles to the above non-step-error. Human eye-sight cannot discriminate the difference between the non-step-error wedge DOE (
In Embodiment 2, the step-error-allotted DOE converts the Gaussian power distribution laser beam into a rectangle uniform power distribution beam on the image plane on the same condition as the non-step-error DOE.
The small drops of power are not caused by the interference between the zero-th order beam and the diffracted beam. The on-y-axis power distribution of the right of
The wedge-type DOE of the present invention has an advantage of minimizing the degradation of the power uniformity induced by manufacturing errors.
A parallel planar DOE (non-error Comparison Example:
Wavelength: 532 nm
Beam diameter: φ2 mm (at 1/e2)
Wavefront: Flat (=Plane Wave)
Focal length (DOE/Image distance): L=200 mm
number of steps: 16 steps
pixel size: 5 μm×5 μm
pixel number: 2000 pixels×2000 pixels
Next, another parallel planar DOE with step errors having the properties similar to the above DOE is produced. A plan view of the parallel planar step-error allotted DOE resembles to the above parallel planar non-step-error DOE (
The power density on the y-axis fluctuates in a range of 0.85 to 1.0 between x=−500 μm and x=+500 μm. Power fall and power fluctuation on the y-axis are larger than the non-step-error parallel planar DOE of
The wedge DOE can be replaced by a couple of a wedge glass block and a parallel planar DOE in the present invention. The couple of the wedge glass block and the parallel planar DOE is equivalent to a wedge DOE.
Wavelength: 532 nm
Beam diameter: φ2 mm (at 1/e2)
Wavefront: Flat (=Plane Wave)
Focal length (DOE/Image distance): L=200 mm
Beam profile on image: 0.5 mm×1 mm (rectangle section; uniform power)
L=1 mm and D=2 mm gives a critical condition Lθ=3 mm. The condition of separation of the zero-th order beam and the diffraction beam is
L+D<Lθ. The critical condition is expressed by
L+D=Lθ. The critical condition is considered now. In this example,
L+D=1 mm+2 mm=3 mm. Lθ=3 mm gives the critical condition for θ.
In
α=tan−1(3/200)=0.014999 rad=0.859372°
A wedge angle of the wedge glass block 84 is denoted by Θ. The above parameters prevent the zero-th order beam from overlapping the diffracted beam on the image plane. A beam emanating slantingly at a from the DOE 83 goes at an angle θ′ in the glass block 84. Snell law requires sin α=n sin θ′ at an input boundary. The beam goes out of the glass block 84 at an angle Θ−θ′) from the rear surface. The diffracted beam should be parallel to the axis. Since the wedge angle is Θ, the diffraction beam inclines at Θ to a normal on the rear surface. Snell law requires sin Θ=n sin Θ−θ′).
θ′=0.5883°=0.01027 rad.
Θ=0.0325 rad=1.864°.
This is the critical wedge angle Θc of the glass block 84. This is a small angle. Endowment of the wedge angle to the glass block eliminates overlapping of the diffraction beam with the zero-th order beam on the image plane. The beam breadth is 2 mm, broadness of the diffraction beam is 1 mm and the deviation of the zero-th order beam is 3 mm on the image plane. There is no interference between the diffraction beam and the zero-th order beam. In practice, the glass block of Embodiment 2 should be assigned with a wedge angle more than the critical value Θc (in this example, 1.864153 degrees).
number of steps: 16 steps
pixel size: 5 μm×5 μm
pixel number: 2000 pixels×2000 pixels
The pixel size is d=5 μm and the pixel number in x- and y-directions is M=N=2000 and MN=4000000. The effective area of the DOE is Md×Nd=10 mm×10 mm.
Next, another set of a parallel planar DOE and a wedge block with step errors having properties similar to the above DOE/block is produced for examining degradation of power uniformity. A plan view of the step-error allotted DOE/block resembles to the above non-step-error DOE/block. Thus a figure of the step-error-allotted DOE/block is omitted.
In Embodiment 2, the step-error-allotted DOE/block converts the Gaussian power distribution laser beam into a rectangle uniform power distribution beam on the image plane on the same condition as the non-step-error DOE/block.
The set of the DOE and the wedge glass block of Embodiment 2 of the present invention has also an advantage of reducing the degradation of the power uniformity induced by manufacturing errors. Both a wedged DOE and a set of a parallel planar DOE and a wedge block enable the present invention to alleviate step-error caused degradation by bending the zero-th order beam in a slanting direction and exclude the zero-th order beam out of the image plane (work plane).
| Number | Date | Country | Kind |
|---|---|---|---|
| 2005-166337 | Jun 2005 | JP | national |
