Design of multiplexed phase diffractive optical elements for focal depth extension

Transcription

1 Desig of multiplexed phase diffractive optical elemets for focal depth extesio Hua Liu, Zhewu Lu,,* Qiag Su ad Hu Zhag,,2 Opto_electroics techology ceter, Chagchu Istitute of Optics ad Fie Mechaics ad Physics, Chiese Academy of Scieces, Chagchu 00, Chia 2 Graduate School of the Chiese Academy of Scieces, Beijig, 0009, Chia *luzhewu55@yahoo.com.c Abstract: A more computatioally tractable method to desig a multiplexed phase diffractive optical elemet with optical desig software to exted the depth of focus is proposed, through which the itesity distributio of the output beams ca also be cotrolled with great flexibility. The desig priciple is explaied i detail. Ad the feasibility of this desig method is illustrated through a desig example followed by computer simulatio verificatio. 200 Optical Society of America OCIS codes: ( Diffractive optics; ( Les system desig; ( Multiplex holography. Refereces ad liks. C. J. R. Sheppard, ad Z. S. Hegedus, Axial behaviour of pupil-plae filters, J. Opt. Soc. Am. A 5(5, ( G. Idebetouw, ad H. Bai, Imagig with Fresel zoe pupil masks: exteded depth of field, Appl. Opt. 2(2, (984.. A. Flores, M. R. Wag, ad J. J. Yag, Achromatic hybrid refractive-diffractive les with exteded depth of focus, Appl. Opt. 4(0, ( E. Be-Eliezer, Z. Zalevsky, E. Marom, ad N. Koforti, All-optical exteded depth of field imagig system, J. Opt. A, Pure Appl. Opt. 5(5, S64 S69 ( C. Iemmi, J. Campos, J. C. Escalera, O. López-Coroado, R. Gimeo, ad M. J. Yzuel, Depth of focus icrease by multiplexig programmable diffractive leses, Opt. Express 4(22, ( N. Yoshikawa, ad T. Yatagi, Phase optimizatio of a kioform by simulated aealig, Appl. Opt. (5, ( J. Fieup, Phase-retrieval algorithms for a complicated optical system, Appl. Opt. 2(0, ( A. Flores, M. R. Wag, ad J. J. Yag, Achromatic hybrid refractive-diffractive les with exteded depth of focus, Appl. Opt. 4(0, ( H. Liu, Z. Lu, ad F. Li, Redistributio of output weightig coefficiets for complex multiplexed phasediffractive elemets, Opt. Express 2(9, ( T. Stoe, ad N. George, Hybrid diffractive-refractive leses ad achromats, Appl. Opt. 27(4, (988.. G. I. Greisukh, E. G. Ezhov, ad S. A. Stepaov, Diffractive-refractive hybrid corrector for achro- ad apochromatic correctios of optical systems, Appl. Opt. 45(24, ( Zemax Optical Desig Program User s Guide Zemax Developmet Corparatio Itroductio Diffractive optical elemets(does have bee used to exted the depth of focus, such as ouiform trasmissio filters [], Fresel zoe pupil masks [2], hybrid refractive diffractive optical systems [] ad multiplexed phase diffractive optical elemets(mpdoes [4,5]. The covetioal way to desig such log-focus-depth DOEs is to use various umerical optimizatio algorithms such as simulated aealig [6] or iterative gradiet approaches [7,8]. However, these methods have all suffered from oe sigificat deficiecy: these optimizatio approaches are computer itesive ad ot ituitive, which makes the desig of DOEs very complex ad ot compatible with the commercial optical desig software packages such as ZEMAX, CODE-V. I this paper we propose a more computatioally tractable method to desig multiplexed phase diffractive optical elemets with optical desig software to exted the depth of focus. It # $5.00 USD Received 26 Apr 200; accepted May 200; published 28 May 200 (C 200 OSA 7 Jue 200 / Vol. 8, No. 2 / OPTICS EXPRESS 2798

2 is show that the MPDOE desiged by this method caot oly correct chromatic aberratio ad spherical aberratio but also exted the focal depth. The desig priciple is described i Sectio 2. A illustrative example is give is Sectio. Ad our coclusios are put forward i Sectio Desig priciple The MPDOE is oe kid of diffractive optical elemets i which several phase fuctios with weights are multiplexed ito a sigle oe. The desig procedure of MPDOEs is as follows: First, the separate phase fuctio exp( iφ should be derived through the optimizatio of the optical system at correspodig defocusig positio t i a appropriate rage of defocusig distaces. The those separate phase fuctio ca be multiplexed ito oe sigle phase oly fuctio. Cosider a liear combiatio of N phase fuctios exp( iφ with real weights A (where ϕ is two-dimesioal fuctio ad =, 2 N. We assume that i= A =, which defies a ew fuctio M exp( ia for MPDOE, where M is the amplitude ad a is the ew phase: For M 0, the multiplexed phase fuctio is N M exp( ia = A exp( iφ ( = 2 i where N A exp( ia = exp( iφ (2 M = M = [ A + A + + A N + 2A A cos( φ φ + 2A A cos( φ φ + + 2A A cos(( φ φ 2 2 N + 2A A cos(( φ φ ( φ φ + + 2A A cos(( φ φ ( φ φ N N 2 + 2A A cos(( φ φ ( φ φ ] N N N N /2 N ( Obviously the phase a derived by Eq. (2 is a discotiuous fuctio with modular 2π. Whe the multiplexed phase diffractive optical elemet with phase a is employed i optical systems, the output is a ew liear combiatio for the origial phase fuctios with ew weights ad other spurious terms, as proved below [9]. Sice / M ca be cosidered as a periodic fuctio of ( φ φ2, ( φ φ ( φ φ N with period 2π, this leads to a Fourier series expasio: ( exp( (4 ' M ββ2 β N = = am m2 m im N β+ im2β2+ + imn β N M m mn Hereβ = ( φ φ 2, β2 = ( φ φ, β = N ( φ φn ad m, m2, mn are itegral umbers. Ad the ew weights a m m 2 mn ca be writte as 2π 2π a = exp( imβ im β dβ dβ (5 mm2 mn N N N N (2 π M 0 0 # $5.00 USD Received 26 Apr 200; accepted May 200; published 28 May 200 (C 200 OSA 7 Jue 200 / Vol. 8, No. 2 / OPTICS EXPRESS 2799

3 From Eq. (2 ad Eq. (4, ( A exp i m + + mn + φ imφ 2 imn φn + A2 exp i( m mn i( m 2 im N N.. exp( ia a + + φ φ φ = m m..(6 N m mn + + AN exp i( m + + mn φ imφ 2 i( mn - φ N Two isights ca be got from the above aalyses: First, as log as the phase fuctios φ, φ2 φn of the diffractive optical elemet are separately got from the optimizatio at differet positios withi a reasoable rage of defocusig distace, the phase fuctio of the MPDOE ca be calculated with the help of Eq. (2 ad the depth of focus of the optical system will be exteded by this multiplexed phase diffractive optical elemet. Secod, from the relatio betwee the iput weight A ad the output weight a m m 2 mn, the iput weights or itesity of the phase fuctios ca be optimized ad chose accordig to the wated output itesity distributio of the MPDOE.. Optical system ad specificatios The followig optical system is iteded to illustrate the desig of oe MPDOE with N weighted phase fuctios ad to demostrate its effectiveess i extedig depth of focus. With the F-umber (F #, extedig focal depth ( d, ad the domiat wavelegth (λ of the optical system, the umber of phase fuctio (N ca be rouded upwards to the earest iteger as follow. d N = ceil( (7 # 2 4 λ( F The specificatios of our desig example are as follows: wavelegth rage is 8-2um, the domiat wavelegth is 0 um, total Field of View is 0, Effective Focal Legth is 90mm, the F-umber is.28, ad the iteded extedig focal depth is 0.8mm.The focal legth of covetioal optical system with.28 F-umber is about 0.06mm. With Eq. (7, the umber iput weights of phase fuctios ca be determied as. Accordig to Eq. (6, whe N =, the output of the MPDOE ca be writte as a ew liear combiatio of the origial phase fuctios ad spurious terms: exp( ia = + ( a A + a A + a A exp( iφ ( a A + a A + a A exp( iφ ( a A + a A + a A exp( iφ = + a exp( iφ + a exp( iφ + a exp( iφ Where the ellipses correspods to the spurious terms itroduced by the multiplexed procedure. Give the requiremet of the itesity distributio alog the axis, the weight a, a2, a of the output phase fuctios ca be got ad the the iput weights A, A2, A of the phase fuctios ca be determied accordig to Eq. (5 ad Eq. (6. For example, whe more eergy is distributed to the outer regios of the extedig focal depth, i order to improve correspodig image quality, the iput weights A, A should be larger tha A 2. O the cotrary, whe more eergy is distributed to the cetral regios of the extedig focal depth, i order to improve correspodig image quality, the iput weights A2 should be larger tha A, A. For simplicity, we assume the iput itesity of three phase fuctio is equal, which results the iput weights: (8 # $5.00 USD Received 26 Apr 200; accepted May 200; published 28 May 200 (C 200 OSA 7 Jue 200 / Vol. 8, No. 2 / OPTICS EXPRESS 2800

4 A = A = A = 2 The layout of the optical system is show i Fig.. The optical system cosists of two positive leses made of Germaium. The diffractive surface is o the covex surface of the first les. With the help of the optical desig program ZEMAX, this system is set up with three cofiguratios: they have the same structure with the same parameters except the distace t betwee the widow ad the image plae ad the phase coefficiets of the diffractive surface. Fig.. Layout of the optical system All diffractive surfaces i ZEMAX, for example biary 2, bed rays accordig to gratig equatio: mλ 2 siθ2 siθ = = mλt (9 d Where is the idex of the material before the diffractive surface, 2 is the idex of the material after the diffractive surface, θ is the agle of icidece, θ2 is the agle of exitace, m is the diffractio order, λ is the wavelegth ad T is the gratig period (iverse of the lie spacig d. The equatio above is Sell's law for refractio, plus a additioal ray bedig term represetig diffractio. The Biary 2 surface allows the gratig period to vary as a rotatioally symmetric polyomial. Zemax uses the phase advace or delay represeted by the biary 2 surface locally to chage the directio of the propagatio of the ray. The biary 2 surface adds phase to the ray accordig to the followig equatio o matter what the wavelegth is: co 2i = i (0 i= φ ( r m B r Where φ ( r is the phase i periods at radius r, co is the cofiguratio umber, is the co umber of the polyomial coefficiets i the series, Bi is the coefficiet o the 2i th power of r, which is the ormalized radial aperture coordiate, m is the diffractio order ad the maximum value of used i this system is three. The coefficiet B of Eq. (0 is the determiat of the diffractive surface optical power whe m = : # $5.00 USD Received 26 Apr 200; accepted May 200; published 28 May 200 (C 200 OSA 7 Jue 200 / Vol. 8, No. 2 / OPTICS EXPRESS 280

5 Φ ( r D = = λ B ( f Whereλ is the wavelegth. It meas that differet wavelegths will give differet optical powers. The coefficiets dispersio of the diffractive surface is determiat with Eq. (: V D λ0 = λ λ max Where λ mi is the shortest wavelegth, λmaxis the shortest wavelegth, λ 0 is the domiat wavelegth. It is cotrary to the coefficiets of dispersio of most les materials. So diffractive-refractive hybrid ca correct chromatic aberratio of optical system [0 2]. mi (2 Fig. 2. Quatified phase of the MPDOE Fig.. Uwrapped phase of MPDOE The parameters of this optical system are show i Table. The distace t betwee the widow ad the image plae is 0mm, 0.06mm, ad 0.06mm i three cofiguratios respectively, so the depth of focus is 0.8mm. Whe the diffractive order m= is chose, optimizig the phase coefficiets B, B 2 ad B of the diffractive surface for each cofiguratio. Because oly the material Germaium is used i the optical system, the diffractive les plays a key role i correctig chromatic aberratio. Meawhile the diffractive les also correct some spherical aberratio itroduced by the two positive leses. # $5.00 USD Received 26 Apr 200; accepted May 200; published 28 May 200 (C 200 OSA 7 Jue 200 / Vol. 8, No. 2 / OPTICS EXPRESS 2802

6 Table. Parameters of optical system Surf Type Radius(mm Thikess(mm Glass Coic stadard Germaium Biary stadard Germaium stadard stadard ifiity Germaium stadard ifiity stadard ifiity t Three phase fuctios correspodig to three cofiguratios are as follows: ( r = r r r ( φ ( r = r r r (4 φ ( r = r r r (5 φ I order to icrease the depth of focus, the above three phase fuctios with equal amplitude eed to be multiplexed ito oe diffractive les. Accordig to Eq. (2, the multiplexed phase fuctio of the MPDOE is exp( ia = exp( iφ + exp( iφ2 + exp( iφ (6 M Ad the discotiuous phase fuctio of the MPDOE with modular 2π is sketched i Fig. 2. Through phase uwrappig, the cotiuous phase fuctio is give i Fig.. The miimum zoe spacig of the MPDOE is about.7mm, so this MPDOE ca be fabricated easily by diamod turig techique. # $5.00 USD Received 26 Apr 200; accepted May 200; published 28 May 200 (C 200 OSA 7 Jue 200 / Vol. 8, No. 2 / OPTICS EXPRESS 280

7 Fig. 4. the modulus trasfer fuctio of optical system usig multiplexed phase DOE (a with 0.5mm defocusig legth (bwith 0mm defocusig legth (c with 0.5mm defocusig legth Fig. 5. the poit spread fuctio of optical system usig multiplexed phase DOE (a with 0.5mm defocusig legth (bwith 0mm defocusig legth (c with 0.5mm defocusig legth # $5.00 USD Received 26 Apr 200; accepted May 200; published 28 May 200 (C 200 OSA 7 Jue 200 / Vol. 8, No. 2 / OPTICS EXPRESS 2804

8 Fig. 6. the modulus trasfer fuctio of optical system without usig multiplexed phase DOE (a with 0.5mm defocusig legth (bwith 0mm defocusig legth (c with 0.5mm defocusig legth Fig. 7. PSF of optical system without MPDOE (a with 0.5mm defocusig legth (bwith 0mm defocusig legth (c with 0.5mm defocusig legth # $5.00 USD Received 26 Apr 200; accepted May 200; published 28 May 200 (C 200 OSA 7 Jue 200 / Vol. 8, No. 2 / OPTICS EXPRESS 2805

9 For the secod cofiguratio, the diffractive les surface is replaced by the MPDOE with multiplexed phase fuctios ito the optical system i optical desig software ZEMAX. The polychromatic MTF curves ad PSF curves of the optical system with the MPDOE are show i Fig. 4 ad Fig. 5 at differet defocusig locatio, respectively. It ca be observed that MTF ad PSF are almost uchaged at the three positios alog the axis ad it ca be regarded that the depth of focus of this system is exteded to 0.8mm. For the purpose of comparisos, the polychromatic MTF curves ad PSF curves of the optical system with the traditioal diffractive les at the secod cofiguratio are also give i Fig. 6 ad Fig. 7 respectively. It is clear that good imagig performace is oly achieved at the omial focal plae ad the imagig quality is greatly deteriorated i defocusig positios. It should be oted that the peak itesity is ormalized to the peak of the uaberrated PSF i Fig. 5 ad Fig. 7. The simulated results cofirm that the MPDOE ca efficietly exted the depth of focus of the optical system. 4. Coclusio Based o the relatioship betwee the multiplexed phase fuctios ad the output phase fuctios of the MPDOE, a more computatioally tractable method of desigig multiplexed phase diffractive optical elemets is proposed to icrease the depth of focus of the optical systems. The simulatio results of the desig example with the help of the optical desig software ZEMAX cofirm that the MPDOE does exted the depth of focus by keepig a stable imagig quality withi a wide rage of the defocusig distace. Although a host of questios cocerig this ew desig method such as the umber of weighted phase fuctios, the determiatio of the optimal weights of the multiplexed phase fuctios ad maximizig the diffractive efficiecy, our desig experiece will reveal a good foregroud by usig MPDOEs i the applicatio of extedig the depth of focus of the optical systems. Ackowledgemets The work described i this paper is supported by the Natioal Natural Sciece Foudatio of Chia uder Grat The authors also thak the aoymous reviewers for their isightful commets ad suggestios which sigificatly improve this paper. # $5.00 USD Received 26 Apr 200; accepted May 200; published 28 May 200 (C 200 OSA 7 Jue 200 / Vol. 8, No. 2 / OPTICS EXPRESS 2806