CN101290396A - Differential value feedback optimized diffraction optical element - Google Patents

Differential value feedback optimized diffraction optical element Download PDF

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CN101290396A
CN101290396A CNA2008100252562A CN200810025256A CN101290396A CN 101290396 A CN101290396 A CN 101290396A CN A2008100252562 A CNA2008100252562 A CN A2008100252562A CN 200810025256 A CN200810025256 A CN 200810025256A CN 101290396 A CN101290396 A CN 101290396A
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groove
optical element
subelement
optical
diffraction optical
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邬融
李永平
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University of Science and Technology of China USTC
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Abstract

The invention relates to a difference feedback optimized diffractive optical element, which solves the problem that the randomness of the application of SA optimization algorithm in beam shaping causes long search time, small increase amount of light spot uniformity and low cost performance. An optical substrate material of the invention is K9 glass or other optical glass, which has the refractive index between 1.5 and 1.85, the diameter between 300 and 350 mm and the thickness of 8 mm; the depth H of a subunit groove of an optical shaping groove is between 0.413 and 2.106 mu m, and the width W of the subunit groove is between 0.097 and 0.684 mm. The diffractive optical element has the advantages that: (1) in theoretical design, the design time of the element is shortened, and the uniformity index of output light spots is improved; (2) in processing and manufacture, a step type structure is suitable for the etching of ion beam mask nesting technology, and as the step type structure is uniform in the whole phase distribution, the sensitivity of the element to etching errors is reduced; (3) in the end, the output light spots on the system are well smoothened, and burrs are inhibited to the utmost extent. Therefore, the element optimized through the method can meet beam shaping requirements well.

Description

Differential value feedback optimized diffraction optical element
Technical field
The present invention relates to be used for the optimal design of the diffraction optical element of beam shaping.
Background technology
Diffraction optical element becomes the only selection in current beam shaping field with its excellent performance [1,2]Since the seventies and eighties in 20th century, along with Gerchberg Saxton (GS) [3]With input output (IO) [4], and this laboratory improved phase mixture algorithm (PMA) [5]The proposition of scheduling algorithm and development, the design of diffraction optical element are quite ripe.On the other hand, along with high power laser system is more and more stricter to the inhomogeneity requirement of hot spot, based on simulated annealing (SA:SimulateAnneal) [6]Also be applied to gradually wherein etc. optimized Algorithm, with the homogeneity of further raising hot spot.
But no matter be GS, IO, PMA, or the SA algorithm, they can't further improve the hot spot homogeneity after converging to a more excellent phase structure again.In addition, the randomness of SA makes search time very long, and the inhomogeneity raising amount of hot spot and little, cost performance is very low.Just, have to expend the plenty of time and use SA to do optimization in order to satisfy high power laser system to inhomogeneity harsh requirement.
Summary of the invention
In order to have overcome the randomness of SA, solve the problem of above-mentioned time efficiency and effectiveness of performance, the invention provides and a kind ofly can in very short time, converge to a more excellent phase structure, cost performance is greatly improved, and shortens the diffraction optical element of the differential value feedback optimized algorithm optimization design of employing in element design cycle greatly.
Concrete structural design scheme is as follows:
Differential value feedback optimized diffraction optical element comprises circular or square optical base-substrate, and a side of optical base-substrate is the light face, and the light face can plate anti-reflection film; The another side uniform the beam shaping groove of forming by deep mixed subelement groove.
Described optical base-substrate material is K9 glass or other optical glass, and its refractive index is 1.5-1.85, bore 300-350mm, thickness 8mm;
The subelement groove depth H of described optical shaping groove is 0.413-2.106 μ m, and groove width W is 0.097-0.684mm;
The groove depth computing formula of described subelement groove is: (n-1) H=λ
In the following formula, refractive index n is 1.5-1.85, and wavelength X is fundamental frequency or frequency multiplication or frequency tripling, i.e. 1.053 μ m or 0.532 μ m or 0.351 μ m,
The groove width computing formula of described subelement groove is: W=D/N
In the following formula, diffraction optical element diameter D is 300-350mm, and N is counting on dimension of 2 dimension sampling matrixs, and its number generally is taken as 512-2048.
Specify differential value feedback optimized algorithm (DFO:Difference FeedbackOptimization) below:
The differential value feedback optimized algorithm of one dimension
The phase structure that makes IO or PMA algorithm obtain is (θ 1, θ 2..., θ n), unit amplitude plane wave incident back focal plane COMPLEX AMPLITUDE is (u 1, u 2..., u n), then the latter can be calculated by Fast Fourier Transform (FFT) (FFT), but in order to introduce the difference feedback, does not directly adopt FFT, but utilize Fourier transform matrix to calculate:
u 1 . . . u n = F 11 . . . F 1 n . . . . . . . . . F n 1 . . . F nn n × n exp ( j θ 1 ) . . . exp ( jθ n ) - - - ( 5 - 1 )
Wherein, F Pq=exp (j2 π pq/n).Following formula is actual to be the Linear Equations of a n unit, by calculating (u 1, u 2..., u n) with poor (the Δ u of desirable complex amplitude 1, Δ u 2..., Δ u n), utilize the linear fit under the least square meaning to calculate (Δ θ again 1, Δ θ 2..., Δ θ n), thereby can obtain the more excellent (θ that separates 1', θ 2' ..., θ n').Because the initial bit phase structure has been an approximate optimal solution,, therefore can guarantee repeatedly to converge to more excellent separating after the iteration optimization so Δ u and Δ θ will be little values.
Introduce above-mentioned difference DELTA u and Δ θ, can be write as following system of equations:
u 1 + Δ u 1 . . . u n + Δ u n = F 11 . . . F 1 n . . . . . . . . . F n 1 . . . F nn n × n exp ( j ( θ 1 + Δ θ 1 ) ) . . . exp ( j ( θ n + Δ θ n ) ) - - - ( 5 - 2 )
In a small amount under the difference approximation relation: exp (j (θ+Δ θ)) ≈ exp (j θ)+jexp (j θ) Δ θ is arranged, is updated in (5-2) and simultaneous (5-1) has:
jexp ( j θ 1 ) F 11 . . . jexp ( j θ n ) F 1 n . . . . . . . . . jexp ( j θ 1 ) F n 1 . . . jexp ( j θ n ) F nn Δ θ 1 . . . Δ θ n = Δ u 1 . . . Δ u n - - - ( 5 - 3 )
Its matrix element note is made b Pq=jexp (j Δ θ q) F Pq, 1≤p≤n wherein, 1≤q≤n.Obviously, they are fully given.Notice Δ θ qBe real number and Δ u qIt is plural number.Therefore, system of equations (5-3) is split as:
Re ( b 11 ) . . . Re ( b 1 n ) . . . . . . . . . Re ( b n 1 ) . . . Re ( b nn ) n × n Δ θ 1 . . . Δ θ n = Re ( Δ u 1 ) . . . Re ( Δ u n ) - - - ( 5 - 4 )
Im ( b 11 ) . . . Im ( b 1 n ) . . . . . . . . . Im ( b n 1 ) . . . Im ( b nn ) n × n Δ θ 1 . . . Δ θ n = Im ( Δ u 1 ) . . . Im ( Δ u n )
Wherein, Re represents to get real part, and Im represents to get imaginary part, matrix B and Δ u that the real part on note system of equations both sides and imaginary part are combined as two 2n * n respectively.
(5-4) formula is that an equation number (2n) is the inconsistent equation group on the real number field of unknown number (n) twice.Calculating to the inconsistent equation group in numerical algebra is at square error min || AX-b|| 2 2Separating under the minimum meaning, just the inconsistent equation group under the least square meaning separates.Theorem according to linear algebra:
X is min||AX-b|| 2 2Separate, and if only if, and X satisfies A TAX=A TB.
Therefore, for above-mentioned system of equations (5-4), the difference that we can utilize this theorem to try to achieve under the least square meaning is separated (Δ θ 1, Δ θ 2..., Δ θ n).Then the flow process of 1 dimension DFO algorithm is:
(1) utilizes initial phase (θ 1, θ 2..., θ n) calculating b Pq=jexp (j Δ θ q) F Pq
(2) obtain matrix B and Δ u by the real part and the imaginary part combination of the right and left respectively by system of equations (5-4);
(3) if Δ u satisfies end condition, then algorithm stops, otherwise continues subsequent step;
(4) press the linear algebra theorem, can be written as B TB Δ θ=B TΔ u;
(5) ask and separate Δ θ=B under the least square method meaning TB B TΔ u;
(6) make new θ=θ-β Δ θ, wherein beta coefficient is being controlled the difference feedback quantity, returns (1) and proceeds iteration.
Wherein, in the step (5) under the least square meaning find the solution symbol ' ' be the account form under the Matlab.If in addition according to the actual calculation needs, can also when asking Δ u, introduce a factor alpha, be used for controlling the difference of actual output with respect to ideal output.
The differential value feedback optimized algorithm of two dimension
Make φ (x i, y i) represent that the position of adopting IO or PMA algorithm to obtain distributes I (x mutually i, y i) light distribution of expression input light, focus on the light field complex amplitude U of focal plane so o(x o, y o) can try to achieve by Fourier transform:
U o(x o,y o)=FFT2D(I(x i,y i) 1/2exp[jφ(x i,y i)]) (5-5)
Wherein FFT2D represents 2 dimension Fast Fourier Transform (FFT)s.Make I Real(x o, y o) expression focal plane actual light intensity, I Ideal(x o, y o) expression target desirable light intensity, then their poor Δ I oFor:
I real(x o,y o)=|Uo(x o,y o)| 2 (5-6)
ΔI o(x o,y o)=I ideal(x o,y o)-I real(x o,y o)
Our target is Δ I oMore little, the homogeneity of hot spot is just good more.The thinking of SA is some point of adjusting at random among the phase φ, accepts by the Metropolis probability.In theory, it can converge to certain more excellent separating, yet for being the approximate more excellent φ that separates, the optimization effect of SA is also not obvious, and randomness makes in addition needs a large amount of iterative searchs, and computing velocity is very slow.
But consider from another point of view that (5-6) formula of utilization calculates definite difference Δ I in a small amount o, another mistake is to solving the phasic difference value Δ φ that needs adjustment.So carry out repeatedly can expecting to converge to certain more excellent φ ' after iteration adjusts, make Δ I oLittler, hot spot is more even.
We know by linear algebra, the matrix A of n * n is made Two-dimensional FFT 2D just be equivalent to:
FFT2D(A)=F*A*F (5-7)
Wherein, the F implication is consistent with the front, expression Fourier coefficient matrix, ' * ' number representing matrix vector product.For a small amount of difference DELTA φ, the equation of having an appointment equally:
exp(j(φ+Δφ))≈exp(jφ)+jexp(jφ)Δφ (5-8)
Utilize (5-7) formula again, suppose that phase structure has little change Δ φ, then has following two formulas:
F*(I(x i,y i) 1/2exp[jφ(x i,y i)])*F=U o(x o,y o) (5-9)
F*(I(x i,y i) 1/2exp[j(φ(x i,y i)+Δφ(x i,y i))])*F=U′ o(x o,y o) (5-10)
(5-10)-(5-9), and utilize the approximate condition of (5-8) formula to obtain:
F*(I(x i,y i) 1/2exp[jφ(x i,y i)]Δφ(x i,y i))*F=ΔU o(x o,y o) (5-11)
Distribute owing to only approaching desirable output intensity again, so might as well make θ (x o, y o) for the initial focal plane position of IO or PMA design distributes (and all keeping this value in the process afterwards) mutually, the amplitude part is then got the Δ I in (2) formula o(x o, y o) part, then have:
ΔU o(x o,y o)=(ΔI o(x o,y o)) 1/2exp[jθ(x o,y o)] (5-12)
Simultaneously, (5-11) separating of formula can be calculated by following formula:
Δφ(x i,y i)=(F -1*ΔU o(x o,y o)*F -1)/(I(x i,y i) 1/2exp[jφ(x i,y i)]) (5-13)
Because the Δ φ that sets is the difference adjustment of argument on the real number field, and (5-13) formula is a system of equations on the complex field, so as long as get its real part as Δ φ:
φ k+1(x i,y i)=φ k(x i,y i)-β·REAL(Δφ k(x i,y i)) (5-14)
Wherein subscript k represents iteration the k time, and β is used to control the size of adjusting in a small amount, and REAL gets real part.The subsequent calculations legend shows, (5-13) to solve real part and the imaginary part of difference DELTA φ all very little for formula, satisfies the approximate condition of (5-8) formula.Finally, the flow process of 2 dimension DFO algorithms is:
(1) for given dot matrix size n, F and its inverse matrix F have been calculated in advance -1And utilize (5-7) to calculate good former φ (x in advance i, y i) be transferred to the argument θ (x of focal plane o, y o).
(2) utilize (5-7) will import an I i(x i, y i) 1/2Exp[j φ (x i, y i)] be transferred to focal plane and be normalized to I Real(x o, y o), utilize (5-6) to calculate Δ I again o(x o, y o), calculate Δ U by (5-12) then o(x o, y o).
(3) as if satisfied the imposing a condition of actual difference with target output, or enough algorithm terminations at most of cycle index.
(4) solve Δ φ (x by (5-13) i, y i), and get the order of its real part and be Δ φ (x i, y i)=β REAL (Δ φ (x i, y i)).
(5) make φ (x again i, y i)=φ (x i, y i)-Δ φ (x i, y i), as the new phase structure of next iteration, turn to step (2) to continue to carry out iteration.
The position of each subelement is worth φ (x mutually i, y i) being the degree of depth of required etching, the spacing between each subelement is groove width.Finally, the phase structure after the storage optimization, the process of whole computation optimization is finished.
Useful technique effect of the present invention is as follows: at first the subelement width is less, and processing allowed band in along with the increase meeting of sampled point N is littler, thereby make the focal plane zoning more more help the control of output light field; Secondly groove depth is controlled to be maximum 2 π bit phase delays, and has specific step, is very beneficial for the processing of ion beam mask alignment; At last, the distribution of each subelement has well homogeneity at random, does not form the distribution of any special shape, thereby has reduced the susceptibility of diffraction optical element to mismachining tolerance.
Now the index of Chang Yong evaluation output facula quality mainly contains 3: RMS (Root Mean Square) is the main lobe root-mean-square error, PV (Peak Vale) is the main lobe peak-to-valley value, the energy efficiency that DE (DiffractiveEfficiency) occupies for main lobe, their definition can be explained by following formula:
RMS = ( Σ top | I real - I ideal | 2 / I ideal 2 ) / n
PV = max { I top } - min { I top } max { I top } + min { I top } - - - ( 6 - 1 )
DE = Σ top I real / Σ total I real
Wherein descend the target implication to be respectively, top represents the hot spot main lobe, and total represents whole hot spot, and real represents actual output, and ideal represents dreamboat output.
Adopt DFO algorithm and SA algorithm that the same initial bit phase structure that is obtained by the PMA algorithm computation is optimized, the parameter that example adopted is a joint as follows, and they are as follows to the inhomogeneity raising contrast of hot spot:
The contrast of table 6.1DFO and two kinds of optimized Algorithm of SA
Sampling number and optimized Algorithm RMS reduction amount PV reduction amount The DE lifting capacity Expend time in
1 dimension, 1024 points (SA) -0.42% 5.1% -0.5% 10 minutes
1 dimension, 1024 points (DFO) 1.93% 6.7% -0.5% 8 seconds
2 dimensions, 512 * 512 points (SA) -0.58% 5.6% -0.8% 12 hours
2 dimensions, 512 * 512 points (DFO) 2.4% 7.0% -0.9% 157 seconds
Wherein, computing platform is: the 2.8G of Celeron, and the 512MB internal memory, Matlab 7.1.By last table 6.1 as seen, greatly faster than SA, especially under 2 dimensions, 512 * 512 dot matrix, DFO is consuming time less than 3 minutes on computing time for the DFO algorithm, and SA needs 12 hours; Optimizing on the efficient, the both has sacrificed a small amount of diffraction efficiency (less than 1%), and the DFO algorithm has reduced RMS and PV index simultaneously, and SA has but sacrificed the RMS index when reducing the PV index, and the PV reduction amount of DFO algorithm is greater than the PV reduction amount of SA algorithm.
Therefore, this differential value feedback optimized algorithm has fine very fast convergence, bigger raising the homogeneity of design hot spot, simultaneously shortened computing time greatly, the optimal design of later diffraction optical element is played good facilitation.In addition, for any linear system, make T, T 1For its positive inverse transformation, T*IN=OUT is arranged, T 1* OUT=IN satisfies the approximated equation of difference in a small amount as long as find, and can adopt the process step of this algorithm to do further optimization.
Description of drawings
Fig. 1 is the glass substrate cut-open view,
Fig. 2 is a diffraction optical element phase structure vertical view,
Fig. 3 is the application model figure of diffraction optical element,
Fig. 4 is the PV and the RMS change curve of execution DFO algorithm,
Fig. 5 is the rectangular light spot profile before and after the DFO optimization,
Fig. 6 crosses the long paracentral 1 dimension intensity distributions of rectangular light spot for DFO optimizes front and back.
Embodiment
Below in conjunction with accompanying drawing, the present invention is done to describe further by embodiment.
Embodiment 1:
If the circular light incident of 300 * 300mm, wavelength are 2 frequency doubled lights (0.532 μ m), focal length is 1500mm, hot spot on the target focal plane is the rectangle of 0.8 * 0.6mm, the sampling dot matrix is 512 * 512, the initial bit phase structure that elder generation obtains with the PMA algorithm, and its output facula index is:
RMS=7.20%,PV=31.6%,DE=88.6%
Result behind the DFO algorithm optimization is:
RMS=4.80%,PV=24.6%,DE=87.7%
The contrast of light beam index lifting capacity has comprised the contrast of DFO and two kinds of optimized Algorithm of SA referring to above-mentioned table 6.1.
As seen from Figure 1, every black post is represented a position phase subelement, and the required position that etches into is represented at its top.In fact, the subelement that this distance pole line points to has maximum bench height, promptly need not etching.0 reticle is represented the depth capacity of etching.The back side of the corresponding diffraction optical element of reticle " 0.2 ", less for target value for the purpose of the signal here, the depth capacity of 0 reticle in fact to the length of substrate back much larger than etching.
Fig. 2 is for tieing up vertical views through being etched in 2 of on-chip phase structure after optimizing, and substrate material is a K9 glass, and refractive index is 1.5163, thickness 8mm; (or other refractive index be the optical glass of 1.5-1.85, bore 300-350mm) chequered with black and white border circular areas is the phase structure of required etching, and wherein black represents that subelement need carve deeply, and white expression subelement should be carved shallow.The size in this zone promptly corresponding the effective aperture of incident light.Partial application is to watch clearly flute profile sub-unit structure from this direction (crossing 0 line of Y-axis, promptly horizontal center), and this promptly is the transversal profile figure of Fig. 1.As seen, the ring belt zone no longer occurs on the whole phase structure after optimizing through DFO, the requirement that processing is got up to precision has reduced a lot, and whole distribution is uniform at random, does not have other special shape appearance.
Sizing specification;
(1), the round and in position phase structure size of diffraction optical element is diameter 300mm on the substrate, the thick 8mm of substrate; Generally speaking: unitary circular or/square phase structure is big small-bore in [300,350] (mm) in the scope, substrate thickness is in [5,20] (mm) in the scope.
(2), groove depth: depth capacity is a wavelength, corresponds on the bit phase delay 2 π just, as the scale more than 0 line of last Fig. 7 .4.The computing formula of depth capacity H:
(n-1)H=λ (7-1)
This example: substitution n=1.5163, λ=0.532 μ m, H=1.03 μ m.
Generally speaking: refractive index n is different with optical glass material, in [1.5,1.85] scope; Wavelength X is generally a kind of of fundamental frequency/frequency multiplication/frequency tripling, i.e. 1.053 μ m, 0.532 μ m, 0.351 μ m are so can obtain the groove depth scope of subelement: H ∈ [0.413,2.106] (μ m).
(2), groove width: be uniform sampling on the both direction, so the groove width W of every phase subelement is the same, computing formula is:
W=D/N (7-2)
Wherein, D is the diameter of diffraction optical element, and N is counting on dimension of 2 dimension sampling matrixs, and its number generally is taken as 512-2048.
This example: substitution D=300mm, N=512, W=0.586mm.
Generally speaking: substitution D ∈ [300,350] (mm), N ∈ [512,2048] W ∈ [0.097,0.684] (mm).
As seen from Figure 4, PV and RMS value progressively restrain in the calculating, so the uniformity index of hot spot is improved; Simultaneously, in order better to investigate the actual shaping effect of diffraction optical element behind the DFO algorithm optimization, be placed in the Focused Optical system shown in Figure 3, with the homogeneity of output facula on the measurement target focal plane, corresponding mark implication is among Fig. 3:
1: the front end laser driver, in order to the laser pulse of μ J (little Jiao) high s/n ratio, high light beam quality and precise synchronization to be provided.
2: the light beam amplification system, the μ J pulse that front end is exported is amplified to a joule magnitude, and expands as required to restraint and specify caliber size.
3: the side view of this diffraction optical element.
4: the flute profile sub-unit structure signal of diffraction optical element.
5: condenser lens, be commonly referred to main lens, will focus on focal plane through the light beam of diffraction optical element modulation.
6: be placed on the receiving plane at focal plane place, just the objective plane of diffraction optical element beam shaping function.Fig. 5 has provided DFO and has optimized front and back, and the profile of the dimension of 2 on target focal plane rectangle output facula is crossed the paracentral 1 dimension intensity distributions of rectangular light spot length before and after Fig. 6 has provided and optimized.Can find that DFO has proved once more and used this algorithm optimization design back diffraction optical element to have good shaping effect the inhomogeneity raising highly significant of output facula.

Claims (1)

1, differential value feedback optimized diffraction optical element comprises circular or square optical base-substrate, and a side of optical base-substrate is the light face, and the light face can plate anti-reflection film; The another side uniform the beam shaping groove of forming by deep mixed subelement groove, it is characterized in that:
Described optical base-substrate material is K9 glass or other optical glass, and its refractive index is 1.5-1.85, bore 300-350mm, thickness 8mm;
The subelement groove depth H of described optical shaping groove is 0.413-2.106 μ m, and groove width W is 0.097-0.684mm;
The groove depth computing formula of described subelement groove is: (n-1) H=λ
In the following formula, refractive index n is 1.5-1.85, and wavelength X is fundamental frequency or frequency multiplication or frequency tripling, i.e. 1.053 μ m or 0.532 μ m or 0.351 μ m,
The groove width computing formula of described subelement groove is: W=D/N
In the following formula, diffraction optical element diameter D is 300-350mm, and N is counting on dimension of 2 dimension sampling matrixs, and its number generally is taken as 512-2048.
CNA2008100252562A 2008-05-14 2008-05-14 Differential value feedback optimized diffraction optical element Pending CN101290396A (en)

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Cited By (4)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN105629461A (en) * 2016-01-11 2016-06-01 西安交通大学 Hundred nanometer scale ultrafine light needle field focusing
WO2019148952A1 (en) * 2018-02-02 2019-08-08 北京北方华创微电子装备有限公司 Method for obtaining etching depth limit value
CN111007664A (en) * 2019-12-18 2020-04-14 中国科学院光电技术研究所 Design method of diffractive optical element with high diffraction efficiency and low speckle noise
CN113092805A (en) * 2021-04-25 2021-07-09 中国空气动力研究与发展中心设备设计与测试技术研究所 High-uniformity sheet light device for particle image speed measurement and speed measurement system

Cited By (4)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN105629461A (en) * 2016-01-11 2016-06-01 西安交通大学 Hundred nanometer scale ultrafine light needle field focusing
WO2019148952A1 (en) * 2018-02-02 2019-08-08 北京北方华创微电子装备有限公司 Method for obtaining etching depth limit value
CN111007664A (en) * 2019-12-18 2020-04-14 中国科学院光电技术研究所 Design method of diffractive optical element with high diffraction efficiency and low speckle noise
CN113092805A (en) * 2021-04-25 2021-07-09 中国空气动力研究与发展中心设备设计与测试技术研究所 High-uniformity sheet light device for particle image speed measurement and speed measurement system

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