夫琅禾费衍射公式的一般形式

:2002-09-02:(1962)),,,,,.2211Vol.22No.11200311COLLEGEPHYSICSNov.2003厉江帆1,2,姜宗福2,黄春佳1,贺慧勇1,朱江峰1(1.,410077;2.,410073):,.,/0,.,..:;;;:O436.1:A:1000-0712(2003)11-0009-061,,/0(),,,,,[1~5].,.,,r,,/0.2-,-.,[6]U(p)=CQ2U(x0,y0)k(H)exp(i2Pr/K)rdx0dy0(1),U(x0,y0)(x0,y0),U(p)p,r(x0,y0)p,H,k(H).,,.[7],z,/0;z,/0;/0.,,.,,,..,[8],,z[9]..(cosA0,cosB0,cosH0),OQ=x0i+y0j+0#k(0,0)(x0,y0),r0r(0,0)(x0,y0)p(,).r=r0-OQ#(icosA0+jcosB0+kcosH0)=r0-x0cosA0-y0cosB0(2)rr0,,(1),(1)rr0,r(2),rr02P/K,.p,(x0,y0)H0.k(H0)x0,y0,.(1)U(p)=k(H0)exp(i2Pr0/K)iKr0Q2U(x0,y0)#exp-i2PcosA0Kx0+cosB0Ky0dx0dy0(3).,H0,(3).,[10](4.2)[9][11]8.3.3(38),.,,z,,z,z,,(x,y),z.,r0(0,0)(x,y),r(x0,y0)(x,y),:r0=z2+x2+y2(4)r=z2+(x-x0)2+(y-y0)2(5),(x0,y0)(x,y)cosA=x-x0r,cosB=y-y0r,cosH=zr,:cosA0=xr0,cosB0=yr0,cosH0=zr0(6)rUr0-OQ#(icosA0+jcosB0+kcosH0)=r0-xx0+yy0r0(7),z,k(H)=cosH=zryk(H0)=zr0=cosH0x0,y0,k(H0).,(1)U(x,y)=k(H0)exp(i2Pr0/K)iKr0k2U(x0,y0)#exp-i2PxKr0x0+yKr0y0dx0dy0(8)x,y.3,,,-(,),C=1iKf(H,Hc)=cosH-cosHc2,,Hc.,.,(-),,C=1iK,k(H)=cosH.[12],,.1,,,,.,,,.,.k(H)=cosH,,,,,[7](,),,-,.,,1022-:U(p)=1iKQ2U(x0,y0)k(H)exp(i2Pr/K)iKrdx0dy0(9)3.1/0[12,13],z2.(9),,H,k(H)=cosHU1.(5)rr=z1+12(x-x0)2+(y-y0)2z2-12#4(x-x0)2+(y-y0)2z22+12#4#6#(x-x0)2+(y-y0)2z23-,(10)[12][13]z,(10)(9).,z:z3mP4K[(x-x0)2+(y-y0)2]2max(11)/0:U(x,y)=exp(ikz)iKzk2U(x0,y0)expik2z[(x-x0)2+(y-y0)2]dx0dy0(12)():zmk2(x20+y20)max(13)(12)x0y0,/0:U(x,y)=exp(ikz)expik2zx2+y2iKz#k2U(x0,y0)exp-i2PxKzx0+yKzy0dx0dy0(14)(12),(14).,(),[7]().[5],.,(),(3)(8),(14).,,(3)(),I=cos2H0K2r20sinc2asinH0Ksin2NPdsinH0Ksin2PdsinH0K(15)dsinH0=kK(16a)(8)(),I=k2(H0)K2r20sinc2axKr0sin2NPdxKr0sin2PdxKr0d#xr0=dsinH0=kK(16b)(14),I=1K2z2sinc2axKzsinc2axKr0sin2NPdxKzsin2PdxKzd#xz=dtanH0=kK(16c),(16c),(16a)(16b).,,:d,,:¹rmK,ºmK.,¹,º,..,.,/,.,,5,,0[14].,,(16a)(16b)d,[10].dmK,.,(k),,.,,(14)r,,(8).1111:3.2(x-x0)2+(y-y0)2z2rz,[(x-x0)2+(y-y0)2]/z2r,,(9).(10)r=z1+12x2+y2+x20+y20z2-12#4x2+y2+x20+y20z22+1#32#4#6x2+y2+x20+y20z23-,-xx0+yy0z1-12x2+y2+x20+y20z22+1#32#4x2+y2+x20+y20z23-1#3#52#4#6x2+y2+x20+y20z24+,-(xx0+yy0)22z3#1+xx0+yy0z2-32x2+y2+x20+y20z2+,(17),z,(13),(17)x0y0.(17)rUz1+12x2+y2z2-12#4x2+y2z22+1#32#4#6x2+y2z23-,-xx0+yy0z#1-12x2+y2z22+1#32#4x2+y2z23-1#3#52#4#6x2+y2z24+,(18)x0,y0,x0,y0,.(9),(9)k(H)=cosH=zr,r(18),r(18),r(18)2P/K,.(9)U(x,y)=ziKQ2U(x0,y0)exp(i2Pr/K)r2dx0dy0=expi2PKz1+12x2+y2z2-12#4x2+y2z22+1#32#4#6x2+y2z23-,#iKz1+12x2+y2z2-12#4x2+y2z22+1#32#4#6x2+y2z23-,2-1#Q2U(x0,y0)exp-i2PKxx0+yy0z#1-12x2+y2z22+1#32#4x2+y2z23-1#3#52#4#6x2+y2z24+,dx0dy0(19),(19)r01/r0,z1+12x2+y2z2-12#4x2+y2z22+1#32#4#6#x2+y2z23+,=z1+x2+y2z21/2=r0(20)1z1-12x2+y2z22+1#32#4x2+y2z23-1#3#52#4#6#x2+y2z24+,=1z1+x2+y2z2-1/2=1r0(21)(20)(21)(19)U(x,y)=zr0exp(i2Pr0/K)iKr0k2U(x0,y0)#exp-i2PxKr0x0+yKr0y0dx0dy0(22),z/r0=cosH0=k(H0)(22)(8).,(19)H[45b.H45b,tanH=(x-x0)2+(y-y0)2z1,(10),(x-x0)2+(y-y0)2z2r,(19).(19),.[5].3.3x20+y20r20-2(xx0+yy0)r20r(5)r=r01-2(xx0+yy0)r20+x20+y20r2012(23)(6)(23)r=r0-(x0cosA0+y0cosB0)+x20+y202r0-18r0#x20+y20r0-2(x0cosA0+y0cosB0)2+,(24)z(13),r0zmk2(x20+y20)max,(24),1222.rUr0-(x0cosA0+y0cosB0)(25)zy],(24).,(9),(9)rr0,r(24),k(H)k(H0)=cosH0,(8).41),,H,k(H)U1,k(H).,.[13],H18b,k(H)U15%.[2],H18b,5%.,,,,.,k(H),.,,k2(H0)=cos2H0,H00[H090b,k(H0)H0,0k(H0)[1.,k(H0)(I=0),..(15),k(H0)=cosH0(I=0);N,,(I=Imax);N(),k(H0),(),k(H0)I=0,().2)/0(8),r,;,,(8)45b.x2+y2z2=tan2Hn1,(14),,;,,,.,(8).3)(3)(8).,,(3)(8),/0(14),[1].,(3)(8)U(x0,y0)(fx=cosA0K,fy=cosB0Kfx=xKr0,fy=yKr0).,3.3(8),[5].,,(8)(14),U(x0,y0),,,,,(8),,,,.:[1].[J].,1995(2):36~39.[2].[J].,1996,16(5):215~217.[3].[J].,1995,15(6):274~276.[4].[J].,2001,27(5):472~476.[5].[J].,2000,20(1):16~20.[6],.[M].:,1978.269.[7]AW.[M].,.:,1987.132~147.[8],.[M].:,1984.209.[9]M.[M]..:,1978.501~503.[10],.[M].:1311:,1984.20,84.[11]MaxBorn,EmilWolf.PrincipleofOptics[M].Sixth(corrected)edition.London:CambridgeUniversityPress,1997.385.[12].[M].:,1985.78~102.[13]JW.[M]..:,1976.33~84.[14]MC.[M]..:,1990.199.ThegeneralexpressionsoftheFraunhoferdiffractionformulaLIJiang-fan1,2,JIANGZong-fu2,HUANGChun-jia1,HEHu-iyong1,ZHUJiang-feng1(1.DepartmentofPhysicsandCommunicationEngineering,ChangshaUniversityofElectricPower,Changsha,Hunan,410077,China;2.InstituteofScience,NationalUniversityofDefenseTechnology,Changsha,Hunan,410073,China)Abstract:ThegeneralexpressionsoftheFraunhoferdiffractionintegral,whichcanalsobetenableinthecaseofabigdiffractionangle,arederivedbyusingdifferentmethods.ItispointedoutthatthegeneralexpressionsarethebasicformulaetosolvetheproblemsofFraunhoferdiffraction.ByderivingaseriesexpressionoftheFraunhoferdiffraction,thedifferencesbetweenthegeneralexpressionsandtheFraunhoferdiffractionformulainFourieropticsisrevealed.Thewayofeliminatingthedifferenceispointedout.Inaddition,theimpactoftheinclinationfactoruponthediffractionpatternisdiscussedinsomedetail.ThisprovidesreferencesfordiffractionmeasurementandfurthercomprehensionoftheessenceoftheFraunhoferdiffraction.Keywords:Fraunhoferdiffraction;generalexpressions;seriesexpression;diffractionmeasure-ment(8)[7],.[M].:,2000.2.3,3.4,3.7Duffing.Relationbetweenperiodandenergyforperiodicmotionofo