作者kramnik (progressive)
看板optical
标题[心得] 6mm瞳孔直径下..4阶球差对眼睛度数的影响..
时间Mon Sep 28 21:58:43 2009
以下论述说明在瞳孔直径在 6mm内时..
仅考虑2阶和4阶球差影响前提下..
最小模糊圆约坐落於於最大2阶屈光误差的3/4处..
当瞳孔直径6mm时..mpmva会比理想值多出 -0.195 ± 0.24 (D)
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对於4阶球差致使之2阶屈光误差..
#像差,像点位移误差,屈光度误差展开式互换推导
http://www.wretch.cc/blog/kramnik1/13622399
我们令2阶屈光误差为ΔF..2阶焦距误差为Δf..
r为光束与折射面交点与光学中心垂直之间距..
f(eye)为眼轴长..F(eye)为眼球总屈光度..a,b为比例常数..
则
ΔF = a*r^2
Δf = -(a*r^2)*[f(eye)/F(eye)]
= -b*r^2
设R为模糊圆半径..模糊圆撷取处为与0阶屈光误差为0处距离为ΔS..
http://www.wretch.cc/album/show.php?i=kramnik1&b=23&f=1606078602&p=3
http://www.wretch.cc/album/show.php?i=kramnik1&b=23&f=1606078603&p=4
R = r*[f(eye)-Δf]^(-1)*(Δf-ΔS) ….when ΔS <Δf
R = r*[f(eye)-Δf]^(-1)*(ΔS-Δf) ….when ΔS >Δf
==================================================================
R = r*[f(eye)-Δf]^(-1)*(Δf-ΔS) ….when ΔS <Δf
即ΔS < -b*r^2 => 0 < r <(-ΔS/b)^(1/2)
我们令ΔS为不变量..对上式之r微分..
dR/dr = [f(eye)-Δf]^(-1)*(Δf-ΔS)
+ r*(-1)* [f(eye)-Δf] ^(-2)*(2*b*r)*(Δf-ΔS)
+ r*[f(eye)-Δf]^(-1)*(-2*b*r)
= [f(eye)-Δf]^(-1)
*{(Δf-ΔS) – 2*b*r^2* [f(eye)-Δf] ^(-1)* (Δf-ΔS) -2*b*r^2} = 0
= [f(eye)-Δf]^(-1)
*{(Δf-ΔS) – 2*b*r^2* [1/f(eye)]* (Δf-ΔS) -2*b*r^2} = 0
= [f(eye)-Δf]^(-1)
*{(Δf-ΔS) + 2*Δf* [1/f(eye)]*(Δf-ΔS) + 2*Δf}
≒ [f(eye)-Δf]^(-1)* (3*Δf-ΔS )
当dR/dr = 0 时..即r =[ΔS/(-3*b)]^(1/2)
则r = [ΔS/(-3*b)]^(1/2)有极大值
Rmax = [ΔS/(-3*b)]^(1/2)*[f(eye)]^(-1)*(-2/3)*ΔS .......(a)
==========================================================================
R = r*{1/[f(eye)-Δf]}*(ΔS-Δf) ….when ΔS >Δf
即ΔS > -b*r^2 => h > r > (-ΔS/b)^(1/2)
Rmax = h*[f(eye)]^(-1)*( -b*h^2-ΔS) ......................(b)
===========================================================================
由上述推导可知
最小模糊圆(the circle of least confusion )出现在条件 (a) = (b)
[ΔS/(-3*b)]^(1/2)*[f(eye)]^(-1)*(-2/3)*ΔS
= h*[f(eye)]^(-1)*( -b*h^2-ΔS)
=> (-4/27*b)*ΔS^3 = h^2*( b^2*h^4 + 2*b*h^2*ΔS +ΔS^2 )
我们用graphmatica软体可以跑出上式解出现在 ΔS ≒ -0.75*b*h^2 处 ........(c)
http://www.wretch.cc/album/show.php?i=kramnik1&b=23&f=1606078599&p=0
http://www.wretch.cc/album/show.php?i=kramnik1&b=23&f=1606078600&p=1
==============================================================================
telescope＠ptics.net
4. INTRINSIC TELESCOPE ABERRATIONS
http://www.telescope-optics.net/spherical1.htm
此网页给予相同的答案..
即最小模糊圆约坐落於於最大2阶屈光误差的3/4处..
最小模糊圆边缘光线由 0.866h 处入射的光线所提供..
============================================================================
根据WAVEFRONT ABERRATION AND ITS ASSOCIATION WITH INTRAOCULAR PRESSURE ，
CENTRAL CORNEAL THICKNESS AND AXIAL LENGTH IN MYOPIC EYES by林楠
人眼瞳孔6mm内zernike分析数据
http://www.wretch.cc/album/show.php?i=kramnik1&b=23&f=1606078601&p=2
我们忽略掉其余像差..仅考虑4th spherical aberration(Z12)..
Z12 = 0.1 ±0.12 (μm)
============================================================================
又 W(SA) = 6*(r/rmax)^4*Z12
根据像差,屈光度误差互换式
△F(max) = -2*[δW/δ(r^2)] | r = rmax
= -24*Z12*(1/rmax)^2
= -24* (0.1±0.12)*10^(-6) * (1/0.003)^2
= -0.26 ±0.32 (D) ......................................(d)
=============================================================================
将(c)代入(d)式..
△F(the least confused) = 0.75*(-0.26 ±0.32)
= -0.195 ± 0.24 (D)
可知在瞳孔直径6mm时..mpmva会比理想值多出 -0.195 ± 0.24 (D)
=============================================================================
由於人眼在暗室瞳孔直径有可能比6mm还要大..
需要瞳孔大於6mm的zernike分析数据..
然而当瞳孔大於6mm时..其余高阶像差的影响比例可能会大幅增加..
单单计算4th-order spherical aberration的影响可能不够贴近实际值..
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※ 发信站: 批踢踢实业坊(ptt.cc)
◆ From: 118.168.83.105
※ 编辑: kramnik 来自: 118.168.82.200 (09/29 13:27)
1F：推 myhole:推! 09/30 12:17
2F：→ sazana:最近做了一堆数据(>30个) 暗室下红绿的值平均比MPMVA多上 10/01 16:48
3F：→ sazana:0.25~0.75D 供参考 10/01 16:49
4F：→ sazana:指近视度数 10/01 16:49
5F：→ kramnik:近视度数区间取高值较真实度数高-0.25 ~ 0D 10/01 18:13
6F：→ kramnik:近视度数区间取低值较真实度数高 0 ~ +0.25D 10/01 18:15
7F：→ kramnik:您的mpmva是取高值还是低值? 10/01 18:15
※ 编辑: kramnik 来自: 118.168.83.236 (10/01 18:16)
8F：→ sazana:高值低值是指? 一般就是最高视力值 10/02 01:10
9F：→ sazana:也许是暗室下瞳孔大於6mm 这部分未测量 10/02 01:15