Light Ray Tracing in GRIN Lens
1:Distribution of reflective index in GRIN Lens
Imaging condition:
n(r)ds,const ………………….(1) ,s
Following equations can be got according to Fig.1(Light ray tracing in
Grin lens)
ds , dr s rBy Light reflection Law: 0
r 2a ,0 z
Fig.1
………………….(2) n(0)cos,,n(r)cos,,n(r)00
is the reflective index at axial of Grin lens. n(r) is the reflective n(0)0
index at amplitude.
drsin,,Use following equation can be calculated: ds
n(r)drds, ………………….(3) 222n(r),n(0)cos,0
If light ray tracing is sinusoid: r,rsin(Az) ………………….(4) 0
222ds,dr,dzUse ………………….(5) dr,rAcos(Az) ………………….(6) 0dz
when z=0,
1
dr ………………….(7) ,tg,,rA00dz
Using (3),(4),(5) and (7),we can get following equation: 2222 ………………….(8) n(r),n(0)(1,Acos,r)0
(1) can be calculated by (8) and (3):
r220,(1,cos)rA0(),4(0)nrdsndr ………………….(9) ,,222sin,,cos,rA0s00
Using transform integral variable,(9) can be calculated: sin,,rActg,0
,12,(1sin),20,nn4(0)2(0)222,,,n(r)ds(1sinsin)d ,,,0,,cos,AcosA,0s00
………………….(10)
If near axial light ray is considered,,(10) can be cos,,1,sin,,000
simplified:
2n(0),n(r)ds,,n(0)l ………………….(11) ,As
l is the periodic length.Formula (11) show that the rays of giving off
from a location at axial will gather together at another location of axial.
The distance between two location at axial is equal to the periodic
llength .
For near axial light ray,the distribution of reflective index can be
expressed:
11222n(r),n(0)(1,Ar),n(0)(1,Ar) ………………….(12) 2
2: the light ray tracing in GRIN Lens:
2
If ,(12) is : n(r),n,n(0),n0
12 ………………….(13) n,n(1,Ar)02
Now the near axial meridian light ray is considered only,according to
light ray equation:
,ddr(n()),,n ………………….(14) dsds
x dx2ds,dz1,(),dz ….(15) P dzs (14) can be calculated:
o z
Fig.2
2dx,n,ndxn,,, ………………….(15) 2,x,zdzdz
12n(x),n(1,Ax)and: ………………….(16) 02
1,n2,0Ax,,1Using ,and if ,we can get following simplified light ray ,z2
equation:
2dx,,Ax ………………….(17) 2dz
Light ray tracing is:
………………….(18) x,Bcos(Az),Csin(Az)
dxP,,,BAsin(Az),CAcos(Az) ………………….(19) dz
Integred constant B and C can be obtained by following beginning
x,x,Bcondition:when z=0, ,P,P,CA,we have following 00
equation:
P0 ………………….(20) x,xcos(Az),sin(Az)0A
3
………………….(21) P,,xAsin(Az),Pcos(Az)00
1,,xxcos(Az)sin(Az),,,,,,0 ………………….(22) ,,,,,A,,,,,,PP,,0,,,,Asin(Az)cos(Az),,,
Two special cases: (1):When light ray input Grin lens at axial with ,in Grin lens the ,
,angle between light ray and z axial is : so , x,B,0P,P,CA,tg,00
tg, ………………….(23) x,sin(Az)
A
………………….(24) P,tg,cos(Az)
ml,. (23) show the light ray is sinusoid.whenz,,m,m,1,2,3......,x,02A
,
,
l
Fig.3
(2):When light ray input Grin lens,the light ray is paralleled to the axial of
Grin lens:
x,BSo, , P,P,CA,000
x,xcos(Az) ………………….(25) 0
P,,xAsin(Az) ………………….(26) 0
ml,z,,m,m,1,2,3......,P,0(25) show light ray is cosine curve. when. 2A
4
P
0.75P
0.5P
0.25P
Fig.4
3: Imaging of Grin lens: (1):image: M a ’ Mb x 0 ,xxb
, ’ xa O O
LzL0
Fig.5
For a light ray in Grin-lens:
………………….(27) x,xcos(Az)a0
………………….(28) P,,xAsin(Az)a0
a Light ray output Grin lens,the slope is
,P,nP,,nxAsin(Az) ………………….(29) a0a00
a light ray output Grin lens is a beeline,the beeline equation is:
,,,x,x,Pz ………………….(30) aa
For b light ray in Grin lens:
5
xtg,0 ………………….(31) x,sin(Az),,sin(Az)bAnLA00
x0 ………………….(32) P,tg,cos(Az),,cos(Az)bnL00
b Light ray output Grin lens,the slope is
x,0 ………………….(33) P,nP,,cos(Az)0bbL0
b light ray output Grin lens is a beeline,the beeline equation is:
,,,x,x,Pz ………………….(34) bb
,,From (27)~(34), the intersection point () of a light ray (z,x),(L,x)
output Grin lens and b light ray output Grin lens can be calculated:
nLAcos(Az),sin(Az)100() ………………….(35) L,
nAnLAsin(Az),cos(Az)000
x0x,, ………………….(36)
nLAsin(Az),cos(Az)00
LIn the above expressions,is object distance, is image distance; is Lx00
object height, is image height. Transverse magnification is defined as x
ratio of image height and object by:
x,1m,, ………………….(37) xnLAsin(Az),cos(Az)000
L,0L,0m,0m,0If ,real image, ,virtual image;erect image,,inverted
m,1m,1image,,enlarged image ,,lessen image.
f (2): effective focal length f: x s
Reference to Fig.6:
x,xcos(Az) 0, x0 , x z 6 F
z Fig.6
tg, P,,xAsin(Az),,tg,,,0n0
So,the distance between focal point F and rear surface of Grin Lens s can
be calculated by:
xcos(Az)x10 ………………….(38) s,,,ctg(Az)tg,nxAsin(Az)nA000
xx100f,,, ………………….(39) tg,nxAsin(Az)nAsin(Az)000
Discussion:
(1): if input light ray is paralleled to principal axial of Grin lens:
we have formulae (38) and (39). For 0.25pitch Grin
,2*0.25Az,,lens,, ,2
s=0,
1f, ………………….(40)
nA0
It show that the focal point is at rear surface of grin lens.
L,,(2): If the image is infinite,according to (35),,So:
nLAsin(Az),cos(Az),000
cos(Az)1L,,ctg(Az),s ………………….(40) 0nAsin(Az)nA00
Object should be at front focal point.
7
本文档为【Light Ray Tracing in GRIN Lens】,请使用软件OFFICE或WPS软件打开。作品中的文字与图均可以修改和编辑,
图片更改请在作品中右键图片并更换,文字修改请直接点击文字进行修改,也可以新增和删除文档中的内容。
该文档来自用户分享,如有侵权行为请发邮件ishare@vip.sina.com联系网站客服,我们会及时删除。
[版权声明] 本站所有资料为用户分享产生,若发现您的权利被侵害,请联系客服邮件isharekefu@iask.cn,我们尽快处理。
本作品所展示的图片、画像、字体、音乐的版权可能需版权方额外授权,请谨慎使用。
网站提供的党政主题相关内容(国旗、国徽、党徽..)目的在于配合国家政策宣传,仅限个人学习分享使用,禁止用于任何广告和商用目的。