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Chapter 18.Matrix Methods in paraxial optics
Chapter 18.Matrix Methods in paraxial optics
< Cardinal points (planes) >Focal (F) points (planes)
• EFL: Effective Focal Length (or simply “focal length”)• FFL: Front Focal Length• BFL: Back Focal Length
Principal (H) surface (planes)
Nodal (N) points (planes)
< Matrix methods to find the cardinal points in paraxial optics >Ray-transfer matrix
• Translation Matrix• Reflection Matrix• Refraction Matrix• Lens Matrix• Mirror Matrix
Complex optical systemsComplex optical systems
Thick lenses, combinations of lenses etc..Thick lenses, combinations of lenses etc..
tt
nnLL
nn nn’’
Consider case where Consider case where tt is not negligible. is not negligible.
We would like to maintain our Gaussian We would like to maintain our Gaussian imaging relationimaging relation
PRR
nnsn
sn
L
21
11)'(''
But where do we measure But where do we measure s, ss, s’’ ; f, f; f, f’’ from? How do we determine from? How do we determine PP??
We try to develop a formalism that can be used with any system!!We try to develop a formalism that can be used with any system!!
We need to define cardinal points (planes) : focal (F), principal (H), and nodal (N) points (planes)
Example: Focal Lengths & Principal Planes
generalized optical system(e.g. thick lens,
multi-element system)
EFL: Effective Focal Length (or simply “focal length”)FFL: Front Focal LengthBFL: Back Focal LengthFP: Focal Point/Plane PS: Principal Surface/Plane
The significance of principal planes
object
multi-elementoptical system
lateral hold, where f= (EFL)
Utility of principal planesUtility of principal planes
HH22
ƒ’ƒ’
FF22
HH11
ƒƒ
FF11
s s’
nnLLnn nn’’
hh
hh’’
Suppose Suppose s, ss, s’’, f, f, f, f’’ all measured from Hall measured from H11 and Hand H22 ……
' 1'
f fs s
''
f fn n
1 1 1 if '
'n n
s s f
Cardinal points and planes: 1st Focal points (F1) Cardinal points and planes: 1st Focal points (F1)
제 1 초점 (first focal point, object side focal point): F1
무한대에 상이 생기는 축 상 물체 점
상 측에서 광 축과 평행하게 입사한 광선이 모이는 점, 또는, 모이는 것처럼 보이는 점.
u'k=0
1 k
...
F1
u'k=0
1 k
...
F1
Cardinal points and planes: 2nd Focal points (F2) Cardinal points and planes: 2nd Focal points (F2)
. . .
F'
1 k (마지막 면)
u =01
F'
1 k
u =01
제 2 초점 (second focal point, image side focal point) : F2
무한대에 있는 축 상 물체 점의 상점
광축과 평행하게 입사한 광선이 모이는 점(실상), 또는, 모이는 것처럼 보이는 점(허상)
F2
F2
1st Principal planes (PP1) and points(H1)1st Principal planes (PP1) and points(H1)
nnLLnn nn’’
HH11
ƒƒ
FF11
PPPP11제 1 주요면 (물체측 주요면) : PP1
-상측에서 광축과 평행하게 입사한 광선을 물체측에서 보아 굴절되는 것처럼 보이는 가상면.
제1 주요점 (물체측 주요점 ): H1 – 제1 주요면과 광축의 교점.
2nd principal planes (PP2) and points (H2)2nd principal planes (PP2) and points (H2)
nnLLnn nn’’
HH22
ƒ’ƒ’
FF22
PPPP22
제 2 주요면 (상측 주요면) : PP2- 물체측에서 광축과 평행하게 입사한 광선을 상측에서 보아 굴절되는 것처럼 보이는 가상면.
제2 주요점 (상측 주요점 ): H2- 제2 주요면과 광축의 교점.
Objective distance, image distanceObjective distance, image distance
물체거리 (object distance)
: 제1 주요면에서 물체까지의 거리
s = HO
상거리 (image distance)
: 제2 주요면에서 상면까지의 거리
s’ = H’O’
l l'
1 k
H H'
h h
P P'
u1
u'k
o o'
s s’
n1
n'1
h1
A1= H
1= H'
1
u'1
u1
l1
l'1
면의 물체거리
면의 상거리
o
1 2
* 면에서 물체까지의 거리 l1, l2 등과는 다름.
l1
l2
Effective focal length & back focal lengthEffective focal length & back focal length
1 k......
u1=0
h1
hk
F'u'k
Ak
H'HA 1
bfl
f' b
efl, f'
f b
f
F u1
h 1
hk
u'k=0
유효 초점거리(effective focal length, efl): f'- 제2 주요점에서 제2초점까지의 거리 : efl = f’ = H’F’
후 초점거리(back focal length, bfl): f‘b-광학계의 마지막 면의 정점에서 제2 초점까지의 거리 : bfl = f’b = AkF’
제2 주요면의 위치 = bfl - efl
efl, f’
bfl, f’b
Front focal length = Working distance of a lensFront focal length = Working distance of a lens
물체측 초점거리(Object side focal length): f- 제1 주요점에서 제1 초점까지의 거리 : f = HF
앞 초점거리(front focal length): fb = 작동거리 (working distance)- 제1면의 정점에서 제1초점까지의 거리 : fb = A1F
제1 주요면의 위치 : fb - f1 k......
u1=0
h1
hk
F'u'k
Ak
H'HA 1
bfl
f' b
efl, f'
f b
f
F u1
h 1
hk
u'k=0
efl, f’
bfl, f’b
f
fb
Utility of principal planesUtility of principal planes
HH22
ƒ’ƒ’
FF22
HH11
ƒƒ
FF11
s s’
nnLLnn nn’’
hh
hh’’
Suppose Suppose s, ss, s’’, f, f, f, f’’ all measured from Hall measured from H11 and Hand H22 ……
' 1'
f fs s
''
f fn n
1 1 1 if '
'n n
s s f
Nodal points (N1, N2) and Nodal planes (NP1, NP2)Nodal points (N1, N2) and Nodal planes (NP1, NP2)
nn nn’’
NN22
NPNP22
NN11
NPNP11
nnLL
절점(nodal point : N1, N2)광학계는 입사각과 출사각이 같은 광선이 1개 존재. 제1절점:이 광선을 물체측에서 보아 입사하는 것처럼 보이는 점. 제2절점:이 광선을 상측에서 보아 출사하는 것처럼 보이는 점.
Nodal point (N) and optical center (c) Nodal point (N) and optical center (c)
N' Nu
u' c
1 2
광심(optical center : C) :
절점(nodal point)을 정의하는 하나의 광선 (입사각과 출사각이 같은 광선)이실제로 광 축과 교차하는 점.
Nodal point 의 성질
i) 제1절점으로 입사한 광선은 제2절점에서 출사 (제1절점 - 광심 (C) - 제2절점) ii) 제1절점으로 입사한 광선은 입사각과 출사각이 같다. iii) 상측 매질의 굴절률과 물체측 매질의 굴절률이 같으면 절점과 주요점은 같다.
N1 = H1 , N2 = H2
iv) 제2절점을 기준으로 광학계를 회전시키면 상의 위치는 변화하지 않는다.
F'
F'N2 N1
C
Cardinal planes of simple systemsThin lens
Cardinal planes of simple systemsThin lens
Psn
sn
''
Principal planes, nodal planes, Principal planes, nodal planes,
coincide at centercoincide at center
VV
H, HH, H’’
VV’’
VV’’ and V coincide andand V coincide and
is obeyed.is obeyed.
Cardinal planes of simple systemsSpherical refracting surface
Cardinal planes of simple systemsSpherical refracting surface
nn nn’’
Gaussian imaging formula Gaussian imaging formula obeyed, with all distances obeyed, with all distances measured from Vmeasured from V
VV
Psn
sn
''
Combination of two systems: e.g. two spherical interfaces, two thin lenses …
Combination of two systems: e.g. two spherical interfaces, two thin lenses …
nn22nn nn’’HH11’’HH11
HH22 HH22’’
HHtt’’
yyYY
dd
ƒƒtt’’
ƒƒ11’’
FF’’ FF11’’
1. Consider F1. Consider F’’ and Fand F11’’
hh’’
Find Find hh’’
Combination of two systems:Combination of two systems:
nn22nn nn’’
HH11’’HH11
HH22 HH22’’HH
yyYY
ddƒƒ
1. Consider F and F1. Consider F and F22
FF22
ƒƒ22
hh
FF
Find Find hh
SummarySummary
HH11’’HH11 HH22 HH22’’HH HH’’
ƒƒ ƒ’ƒ’hh hh’’
FF FF’’
dd
nn22nn nn’’
SummarySummary
2
2121
211
2
2
2
12
1
2
21
2
,''
''
'''
'''''
nPPdPPP
orfn
ffdn
fn
fn
fn
nn
PPdHH
ffdh
nn
PPdHH
ffdh
Hecht, 6.1, p.214
Thick LensThick Lens
nn22
RR11 RR22
HH11,H,H11’’ HH22,H,H22’’
In air n = nIn air n = n’’ =1=1
Lens, nLens, n22 = 1.5= 1.5
RR11 = = -- RR22 = 10 cm= 10 cm
d = 3 cmd = 3 cm
Find Find ƒƒ11,,ƒƒ22,,ƒƒ, h and h, h and h’’
Construct the Construct the principal planes, H, principal planes, H, HH’’ of the entire of the entire systemsystem
nn nn’’
Principal planes for thick lens (n2=1.5) in airPrincipal planes for thick lens (n2=1.5) in air
EquiEqui--convex or convex or equiequi--concave and moderately thick concave and moderately thick PP11 = P= P22 ≈≈ P/2P/2
3' dhh
12
22
'ff
ndh
ff
ndh
HH HH’’ HH HH’’
Principal planes for thick lens (n2=1.5) in airPrincipal planes for thick lens (n2=1.5) in air
PlanoPlano--convex or convex or planoplano--concave lens with Rconcave lens with R22 = =
PP22 = 0= 0
dh
h
32'
0
12
22
'ff
ndh
ff
ndh
HH HH’’ HH HH’’
Principal planes for thick lens (n=1.5) in airPrincipal planes for thick lens (n=1.5) in air
For meniscus lenses, the principal planes move For meniscus lenses, the principal planes move outside the lensoutside the lens
RR22 = 3R= 3R11 (H(H’’ reaches the first surface)reaches the first surface)
P Same for all lensesSame for all lenses
12
22
'ff
ndh
ff
ndh
HH HH’’ HH HH’’ HH HH’’HH HH’’
Examples: Two thin lenses in airExamples: Two thin lenses in air
2
2
ffd
PPdh
ƒƒ11 ƒƒ22
dd
HH11’’HH11 HH22 HH22’’
n = nn = n2 2 = n= n’’ = 1= 1
Want to replace HWant to replace Hii, H, Hii’’ with H, Hwith H, H’’
1
1'ffd
PPdh
hh hh’’
HH HH’’
Examples: Two thin lenses in airExamples: Two thin lenses in air
ƒƒ11 ƒƒ22
dd
n = nn = n2 2 = n= n’’ = 1= 1
2121
2
2121
111,
ffd
fff
ornPPdPPP
HH HH’’
FF FF’’
ƒƒ ƒ’ƒ’ fss1
'11
ss’’ss
Two separated lenses in airTwo separated lenses in air
If fIf f11’’=2f=2f22’’
d = 0.5 d = 0.5 ff22’’
HHHH’’
FF’’FF
ff’’
d = d = ff22’’
HHHH’’
FF’’FF
ff’’
Two separated lenses in airTwo separated lenses in air
ff11’’=2f=2f22’’
d = 2d = 2ff22’’
HHHH’’
FF’’FF
ff’’
d = 3d = 3ff22’’
Principal points at Principal points at
e.g. Astronomical telescopee.g. Astronomical telescope
Two separated lenses in airTwo separated lenses in airff11’’=2f=2f22’’
d = 5d = 5ff22’’
ff’’
e.g. Compound microscopee.g. Compound microscope
HH
FF’’FF
HH’’
Two separated lenses in airTwo separated lenses in air
ff11’’==--2f2f22’’
d = d = --ff22’’
e.g. Galilean telescopee.g. Galilean telescope
Principal points at Principal points at
Two separated lenses in airTwo separated lenses in air
ff11’’==--2f2f22’’
d = d = --1.51.5ff22’’e.g. Telephoto lense.g. Telephoto lens
HH HH’’
FF’’
ff’’
FF
우와아~ !!! 복잡하다.우와아우와아~ !!! ~ !!! 복잡하다복잡하다..
HH11’’HH11 HH22 HH22’’HH HH’’
ƒƒ ƒ’ƒ’hh hh’’
FF FF’’
dd
nn22nn nn’’
렌즈렌즈 22개가개가 있는있는 경우도경우도 힘들다힘들다..
좋은좋은 방법이방법이 없을까없을까??HH11’’HH11 HH22 HH22’’
HH HH’’
ƒƒ ƒ’ƒ’hh hh’’
FF FF’’
dd
nn22nn nn’’HH11’’HH11 HH22 HH22’’
HH HH’’
ƒƒ ƒ’ƒ’hh hh’’
FF FF’’
dd
nn22nn nn’’
33개개 이상이상 있으면있으면? ? 못하겠다못하겠다..
포기포기??
Matrix Methods in paraxial opticsMatrix Methods in paraxial optics
• Development of systematic methods of analyzing optical systems with numerous elements
• Matrices for analyzing the translation, refraction, and reflection of optical rays
• Matrices for thick and thin lenses, optical systems with numerous elements
Matrix MethodMatrix Method
1
1
2
2
y
DCBAy
112
112
DCyBAyy
ABCD Matrix
What is the ray-transfer matrix What is the ray-transfer matrix
tansin
How to use the ray-transfer matrices How to use the ray-transfer matrices
How to use the ray-transfer matrices How to use the ray-transfer matrices
translation refraction translation translation
Translation Matrix Translation Matrix
0 01 1 0
0 01
1 10 1 0 1
y yy L x x
( yo, o )
( y1, 1 )
L
1 0 1 0 0 0 0tany y L y L
1 0 0
1 0 0
1
0 1
y y L
y
Refraction Matrix Refraction Matrix
' :
1 1
Paraxial Snell s Law n n
y n y n y y n nyR n R n R R R n n
1 0
1 0: 0
1 1 : 0
y y
y y Concave surface Rn n Convex surface R
R n n
y=y’
yR
y yR R
n n'
Reflection Matrix Reflection Matrix
:
2
1 0
2 1
1 02 1
Law of Reflection
y y yR R R
y y
yR
y y
R
y=y’
y y yR R R
Thick Lens Matrix IThick Lens Matrix I
0 011
0 011
1 0: L
L L
y yyRefraction at first surface Mn n n
n R n
2 1 12
2 1 1
11 2 :
0 1y y yt
Translation from st surface to nd surface M
3 2 23
3 2 22
1 0: L L
y y yRefraction at second surface Mn n n
n R n
Thick Lens Matrix IIThick Lens Matrix II
1
21
1
2 1 1 2
:
11 0
1
1 1
L
L LL L
L
L L
L
L L
LL L L
L L
Assuming n n
t n n t nn R n
M n n nn n nn R nn R n
t n n t nn R n
t n nn n n n n n tn R n R n R n R
2 1
1 0 1 010 1L L L
L L
tM n n n n n n
n R n n R n
3 2 1:Thick lens matrix M M M M
Thin Lens MatrixThin Lens Matrix
2 1
1 2
:1 0
1 1 1
1 1 1
1 01 1
L
L
Thin lens matrix
M n nn R R
n nbutf n R R
Mf
The thin lens matrix is found by setting t = 0:
nL
Summary of Matrix MethodsSummary of Matrix Methods
0
System Ray-Transfer Matrix System Ray-Transfer Matrix
Introduction to Matrix Methods in Optics, A. Gerrard and J. M. Burch
1
1
y
2 2
2 2
n
n
y
System Ray-Transfer Matrix System Ray-Transfer Matrix Any paraxial optical system, no matter how complicated, can be represented by a 2x2 optical matrix. This matrix M is usually denoted
: system matrixA B
MC D
A useful property of this matrix is that
0Detf
nM AD BCn
where n0 and nf are the refractive indices of the initial and final media of the optical system
0Det 1f
nM AD BCn
Usually, the medium will be air on both sides of the optical system and
Significance of system matrix elements
Significance of system matrix elements
The matrix elements of the system matrix can be analyzed to determine the cardinal points and planes of an optical system.
0
0
f
f
y yA BC D
Let’s examine the implications when any of the four elements of the system matrix is equal to zero.
0 0
0 0
f
f
y Ay B
Cy D
D=0 : input plane = first focal planeA=0 : output plane = second focal planeB=0 : input and output planes correspond to conjugate planesC=0 : telescopic system
D=0 A=0
B=0 C=0
System Matrix with D=0System Matrix with D=0Let’s see what happens when D = 0.
0
0
0 0
0
0f
f
f
f
y yA BC
y Ay B
Cy
When D = 0, the input plane for the optical system is the first (object) focal plane.
Ex) Two-Lens System : find the 1st focal plane Ex) Two-Lens System : find the 1st focal plane
f1 = +50 mm f2 = +30 mm
q = 100 mmr s
InputPlane
OutputPlane
F1 F2F1 F2
T1 R1 R2 T3T2
1 1
2 1 1 21 1
11 0 1 1 01 1 1
1 1 11 1 10 1 0 1 0 1 1 1
q q rr qrf fs q s
M rr
f f f ff f
03 2 2 1 1
02 1
1 0 1 01 1 1
1 11 10 1 0 1 0 1f
f
y y s q rM M T R T R T
f f
3 2 2 1 1
1 1
2 1 1 2 1 1
1 2 1 1 2 2 1 1
2 1 1 2 1 1
110 1 1 1 11 1
1 1 1
1 1 11 1
M T R T R Tq q rr qf fsq q r rr q
f f f f f f
q s s q q r r q q r rr q sf f f f f f f f
q q r rr qf f f f f f
2 1 1
2 1 1
1 2
1 1 0
30 50 100 50175
100 50 30
q r rD r qf f f
f f q frq f f
r mm
ƒƒ11 ƒƒ22
dd
HH HH’’
FF FF’’
ƒƒ ƒ’ƒ’
ss’’ss
hh
rr
1 2
1 2 1 2 1 2
1 1 1 f fd ff f f f f f f d
2
2
ffd
PPdh
2 1 2 1
2 1 2
f d f f f dr f h ff f f d
< check! >
System Matrix with A=0, C=0System Matrix with A=0, C=0
0
0
0
0 0
0f
f
f
f
y yBC D
y B
Cy D
When A = 0, the output plane for the optical system is the output focal plane.
When C = 0, collimated light at the input plane is collimated light at theexit plane but the angle with the optical axis is different. This is a telescopic arrangement, with a magnification of D = f/0.
0
0
0 0
0
0f
f
f
f
y yA BD
y Ay B
D
0
0
0
0 0
0
0f
f
f
f
f
y yAC D
y Ay
Cy D
ym A
y
When B = 0, the input and output planes are object and image planes, respectively, and the transverse magnification of the system m = A.
System Matrix with B=0System Matrix with B=0
Conjugate planes
Ex) Two-Lens System with B=0Ex) Two-Lens System with B=0
f1 = +50 mm f2 = +30 mm
q = 100 mmr s
ObjectPlane
ImagePlane
F1 F2F1 F2
T1 R1 R2 T3T2
1
1 2 2 1 1
2 2 1 1
1 2 2 1 2 2 1 2
1 1 2 2 1 2 1 1 2
1 2 1
1 01
1 1
q rr qq r r q q r r fB r q s s r q q r rf f f f f
f f f ff f r q f qr r f f f q f f q
f r q q r f f f r r f q f f q f f
q s s qm Af f f
Location of Cardinal Points (Planes)for an Optical System
Location of Cardinal Points (Planes)for an Optical System
Distances measured to the right of the respective reference plane are positive, distances measured to the left are negative. As shown:
p < 0 q > 0f1 < 0 f2 > 0r > 0 s < 0v > 0 w < 0
Principal planesNodal planesFocal planes
유도해보자!
Ex) Thick Lens AnalysisEx) Thick Lens Analysis
R1 = +30 mm R2 = +45 mm
Input plane(RP1 )
V1 V2
t = 50 mm
nL = 1.8n0 = 1.0 n0 = 1.0
Find for the lens:
(a) Principal Points(b) Focal Points(c) Focal Length(d) Nodal Points
output plane(RP2 )
1
2 1 1 2
, :
1
1 1
50* 0.8 50*1.011.8*30 1.8
50* 0.80.8 0.8 0.8*501 145 1.8*30 30 1.8*45
L
L L
LL L L
L L
Thick lens matrix assuming n n
t n n t nn R nA B
MC D t n nn n n n n n t
n R n R n R n R
0.2593 27.770.02206 1.494
: det 1Check M AD BC
Thick Lens AnalysisThick Lens Analysis
10.2593 27.77 0.02206 1.494A B mm C mm D
Thick Lens AnalysisThick Lens Analysis
1
2
1.494 67.720.02206
0.2593 11.750.02206
1.494 1 22.390.02206
1 0.2593 33.580.02206
1.494 1v 22.390.02206
1 0.2593 33.580.02206
45.33
45.33
p mm
q mm
r mm
s mm
mm
w mm
f p r mm
f q s mm
Thick Lens AnalysisThick Lens Analysis
RP1 RP2
t = 50 mm
PP1
F1
F2
PP2
H1H2
si = +86.7 mmso = -95 mm
In general, for any optical system:
1 2
0 0
1 20 0
1
1 1 1:
i
i
i
i
n sf f ms s n s
sfor n n f f f ms s f s
f1
f2
y 축
x 축
z 축A
2)
1 2
A 2A 1 c1c2
(+)r1(-)r2
3)
k+1
k-1 k1 2 3 4
o o'
d1 d2
y1
z1z k-1
yk-1
y2
z 2
물체면 상면
Ray tracingRay tracing부호에 관한 규약
1) 광선은 최초에 좌 → 우로 진행좌 → 우 : 순방향 우 → 좌 : 역방향
2) 좌표계의 원점은 면의 정점에 둔다. 광축은 z-축
3) 곡률 반경의 부호는 곡률 중심의 위치에 따른다.
4) 여러 개의 면이 있으면 각 면의 정점을 원점으로 하는 좌표계를 사용.
※ 면 번호(Surface Number) : 광선이 만나는 순서대로 붙인 면의 번호
Ray tracingRay tracing부호에 관한 규약
5) 굴절률의 부호 ---- 순방향 : + 역방향 : -
n=1
1
2
n =11
n '=12
순방향일때 굴절률도 (+)역방향일때 굴절률도 (-)
n'1=n
2=-1
Paraxial ray tracingParaxial ray tracing
C o'o
u u'
n n'i
i'
h r
A
sin 'sin ' ' ' ( , ' 1)' - ' , -' '- ' -' ' ( ' )
sin ~ /'' ' , : power of refraction
n i n i ni n i i ii u i un u n nu nn u nu n nh r r h r
n nn u nu hk kr
근축 광선 추적(Paraxial Ray Tracing) = Gaussian ray tracing (GRT) 근축 광학적 근사를 통하여 계산된 광선의 진행경로.
굴절 방정식(refraction equation)
l l’
결상 방정식(imaging equation)
' ' '' , , or, ' ' '
h h n n n n n nu u K Kl l l l l l r
면 불변량 (surface invariant) = (Abbe’s Zero invariant)
From Snell's law, ' ' '
1 1 1 1' , , = '' '
n u n nu n
h h hu u Q n nl l r l r l r
Paraxial invarianceParaxial invariance
sin 'sin 'n i n i
면 불변량 (surface invariant) = (Abbe’s Zero invariant)
불변량 (invariant) : 굴절 전후에 변화하지 않는 물리량
굴절 불변량 (snell’s law) :
From Snell's law, ' ' '
1 1 1 1' , , = '' '
n u n nu n
h h hu u Q n nl l r l r l r
' 'ni n i근축 굴절불변량 (snell’s law) :
Lagrange 불변량
'' ' '
, ' , ''
H= ' ' '
n n n nl l
h h h hu u l ll l u u
nu n u
n n'
u'
h
uo o'
l l'
o'1
o1
A
'
'
