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Introduction to Geometrical Optics:
Math resources
Apratim MajumderOptics for Energy 2020
2020/09/03Thursday
Part II
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Example Problem:Thick Lens
(Slides 38-40 in Lecture notes PDF)
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𝑛𝑛𝑜𝑜𝑜𝑜𝑜𝑜𝛼𝛼𝑜𝑜𝑜𝑜𝑜𝑜𝑥𝑥𝑜𝑜𝑜𝑜𝑜𝑜 =
Refractionat
spherical surface (R2)
Free spacepropagation inside lens
(𝐷𝐷𝐷)
Refractionat
spherical surface (R1)
𝑛𝑛𝑖𝑖𝑖𝑖𝛼𝛼𝑖𝑖𝑖𝑖𝑥𝑥𝑖𝑖𝑖𝑖
From Output side to Input side
Solution:Only consider ray entering and exiting lens(ignore Free Space Prop. In air outside lens)
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𝑛𝑛𝑜𝑜𝑜𝑜𝑜𝑜𝛼𝛼𝑜𝑜𝑜𝑜𝑜𝑜𝑥𝑥𝑜𝑜𝑜𝑜𝑜𝑜 =
Refractionat
spherical surface (R2)
Free spacepropagation inside lens
(𝐷𝐷𝐷)
Refractionat
spherical surface (R1)
𝑛𝑛𝑖𝑖𝑖𝑖𝛼𝛼𝑖𝑖𝑖𝑖𝑥𝑥𝑖𝑖𝑖𝑖
Here, 𝑛𝑛𝑜𝑜𝑜𝑜𝑜𝑜 = 𝑛𝑛𝑖𝑖𝑖𝑖 = 𝐷
From Output side to Input side
Solution:Only consider ray entering and exiting lens(ignore Free Space Prop. In air outside lens)
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𝑛𝑛𝑜𝑜𝑜𝑜𝑜𝑜𝛼𝛼𝑜𝑜𝑜𝑜𝑜𝑜𝑥𝑥𝑜𝑜𝑜𝑜𝑜𝑜 =
Refractionat
spherical surface (R2)
Free spacepropagation inside lens
(𝐷𝐷𝐷)
Refractionat
spherical surface (R1)
𝑛𝑛𝑖𝑖𝑖𝑖𝛼𝛼𝑖𝑖𝑖𝑖𝑥𝑥𝑖𝑖𝑖𝑖
Here, 𝑛𝑛𝑜𝑜𝑜𝑜𝑜𝑜 = 𝑛𝑛𝑖𝑖𝑖𝑖 = 𝐷Hence:
𝛼𝛼𝑜𝑜𝑜𝑜𝑜𝑜𝑥𝑥𝑜𝑜𝑜𝑜𝑜𝑜 =
Refractionat
spherical surface (R2)
Free spacepropagation inside lens
(𝐷𝐷𝐷)
Refractionat
spherical surface (R1)
𝛼𝛼𝑖𝑖𝑖𝑖𝑥𝑥𝑖𝑖𝑖𝑖
From Output side to Input side
Solution:Only consider ray entering and exiting lens(ignore Free Space Prop. In air outside lens)
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𝛼𝛼𝑜𝑜𝑜𝑜𝑜𝑜𝑥𝑥𝑜𝑜𝑜𝑜𝑜𝑜 =
Refractionat
spherical surface (R2)
Free spacepropagation inside lens
(𝐷𝐷𝐷)
Refractionat
spherical surface (R1)
𝛼𝛼𝑖𝑖𝑖𝑖𝑥𝑥𝑖𝑖𝑖𝑖
From Output side to Input side
Solution:Only consider ray entering and exiting lens(ignore Free Space Prop. In air outside lens)
𝐷 0𝐷𝐷𝑛𝑛
𝐷
Free space propagation
matrix
𝐷 −𝑛𝑛𝑜𝑜𝑜𝑜𝑜𝑜 − 𝑛𝑛𝑖𝑖𝑖𝑖
𝑅𝑅0 𝐷
Refraction at spherical surface
matrix𝛼𝛼𝑜𝑜𝑜𝑜𝑜𝑜𝑥𝑥𝑜𝑜𝑜𝑜𝑜𝑜 =
𝐷 −𝐷 − 𝑛𝑛′
𝑅𝑅20 𝐷
𝐷 0𝐷𝐷1𝑛𝑛′
𝐷𝐷 −
𝑛𝑛′ − 𝐷𝑅𝑅1
0 𝐷
𝛼𝛼𝑖𝑖𝑖𝑖𝑥𝑥𝑖𝑖𝑖𝑖
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Solve the equation:
𝛼𝛼𝑜𝑜𝑜𝑜𝑜𝑜𝑥𝑥𝑜𝑜𝑜𝑜𝑜𝑜 =
𝐷 −𝐷 − 𝑛𝑛′
𝑅𝑅20 𝐷
𝐷 0𝐷𝐷1𝑛𝑛′
𝐷𝐷 −
𝑛𝑛′ − 𝐷𝑅𝑅1
0 𝐷
𝛼𝛼𝑖𝑖𝑖𝑖𝑥𝑥𝑖𝑖𝑖𝑖
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Solve the equation:
𝛼𝛼𝑜𝑜𝑜𝑜𝑜𝑜𝑥𝑥𝑜𝑜𝑜𝑜𝑜𝑜 =
𝐷 −𝐷 − 𝑛𝑛′
𝑅𝑅20 𝐷
𝐷 0𝐷𝐷1𝑛𝑛′
𝐷𝐷 −
𝑛𝑛′ − 𝐷𝑅𝑅1
0 𝐷
𝛼𝛼𝑖𝑖𝑖𝑖𝑥𝑥𝑖𝑖𝑖𝑖
𝛼𝛼𝑜𝑜𝑜𝑜𝑜𝑜𝑥𝑥𝑜𝑜𝑜𝑜𝑜𝑜 =
𝐷𝑛𝑛′ − 𝐷𝑅𝑅2
0 𝐷
𝐷𝐷 − 𝑛𝑛′
𝑅𝑅1𝐷𝐷1𝑛𝑛′
𝐷𝐷1𝑛𝑛′
𝐷 − 𝑛𝑛′
𝑅𝑅1+ 𝐷
𝛼𝛼𝑖𝑖𝑖𝑖𝑥𝑥𝑖𝑖𝑖𝑖
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Solve the equation:
𝛼𝛼𝑜𝑜𝑜𝑜𝑜𝑜𝑥𝑥𝑜𝑜𝑜𝑜𝑜𝑜 =
𝐷 −𝐷 − 𝑛𝑛′
𝑅𝑅20 𝐷
𝐷 0𝐷𝐷1𝑛𝑛′
𝐷𝐷 −
𝑛𝑛′ − 𝐷𝑅𝑅1
0 𝐷
𝛼𝛼𝑖𝑖𝑖𝑖𝑥𝑥𝑖𝑖𝑖𝑖
𝛼𝛼𝑜𝑜𝑜𝑜𝑜𝑜𝑥𝑥𝑜𝑜𝑜𝑜𝑜𝑜 =
𝐷𝑛𝑛′ − 𝐷𝑅𝑅2
0 𝐷
𝐷𝐷 − 𝑛𝑛′
𝑅𝑅1𝐷𝐷1𝑛𝑛′
𝐷𝐷1𝑛𝑛′
𝐷 − 𝑛𝑛′
𝑅𝑅1+ 𝐷
𝛼𝛼𝑖𝑖𝑖𝑖𝑥𝑥𝑖𝑖𝑖𝑖
𝛼𝛼𝑜𝑜𝑜𝑜𝑜𝑜𝑥𝑥𝑜𝑜𝑜𝑜𝑜𝑜 =
𝐷 +𝐷𝐷1𝑛𝑛′
𝑛𝑛′ − 𝐷𝑅𝑅2
𝐷 − 𝑛𝑛′
𝑅𝑅1+𝑛𝑛′ − 𝐷𝑅𝑅2
(𝐷𝐷1𝑛𝑛′
𝐷 − 𝑛𝑛′
𝑅𝑅1+ 𝐷)
𝐷𝐷1𝑛𝑛′
𝐷𝐷1𝑛𝑛′
𝐷 − 𝑛𝑛′
𝑅𝑅1+ 𝐷
𝛼𝛼𝑖𝑖𝑖𝑖𝑥𝑥𝑖𝑖𝑖𝑖
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Solve the equation:
𝛼𝛼𝑜𝑜𝑜𝑜𝑜𝑜𝑥𝑥𝑜𝑜𝑜𝑜𝑜𝑜 =
𝐷 −𝐷 − 𝑛𝑛′
𝑅𝑅20 𝐷
𝐷 0𝐷𝐷1𝑛𝑛′
𝐷𝐷 −
𝑛𝑛′ − 𝐷𝑅𝑅1
0 𝐷
𝛼𝛼𝑖𝑖𝑖𝑖𝑥𝑥𝑖𝑖𝑖𝑖
𝛼𝛼𝑜𝑜𝑜𝑜𝑜𝑜𝑥𝑥𝑜𝑜𝑜𝑜𝑜𝑜 =
𝐷𝑛𝑛′ − 𝐷𝑅𝑅2
0 𝐷
𝐷𝐷 − 𝑛𝑛′
𝑅𝑅1𝐷𝐷1𝑛𝑛′
𝐷𝐷1𝑛𝑛′
𝐷 − 𝑛𝑛′
𝑅𝑅1+ 𝐷
𝛼𝛼𝑖𝑖𝑖𝑖𝑥𝑥𝑖𝑖𝑖𝑖
𝛼𝛼𝑜𝑜𝑜𝑜𝑜𝑜𝑥𝑥𝑜𝑜𝑜𝑜𝑜𝑜 =
𝐷 +𝐷𝐷1𝑛𝑛′
𝑛𝑛′ − 𝐷𝑅𝑅2
𝐷 − 𝑛𝑛′
𝑅𝑅1+𝑛𝑛′ − 𝐷𝑅𝑅2
(𝐷𝐷1𝑛𝑛′
𝐷 − 𝑛𝑛′
𝑅𝑅1+ 𝐷)
𝐷𝐷1𝑛𝑛′
𝐷𝐷1𝑛𝑛′
𝐷 − 𝑛𝑛′
𝑅𝑅1+ 𝐷
𝛼𝛼𝑖𝑖𝑖𝑖𝑥𝑥𝑖𝑖𝑖𝑖
𝛼𝛼𝑜𝑜𝑜𝑜𝑜𝑜𝑥𝑥𝑜𝑜𝑜𝑜𝑜𝑜 =
𝐷 +𝐷𝐷1𝑛𝑛′
𝑛𝑛′ − 𝐷𝑅𝑅2
𝐷 − 𝑛𝑛′
𝑅𝑅1+𝐷𝐷1(𝐷 − 𝑛𝑛′)(𝑛𝑛′−𝐷)
𝑛𝑛′𝑅𝑅1𝑅𝑅2+𝑛𝑛′ − 𝐷𝑅𝑅2
𝐷𝐷1𝑛𝑛′
𝐷𝐷1𝑛𝑛′
𝐷 − 𝑛𝑛′
𝑅𝑅1+ 𝐷
𝛼𝛼𝑖𝑖𝑖𝑖𝑥𝑥𝑖𝑖𝑖𝑖
-
Solve the equation:
𝛼𝛼𝑜𝑜𝑜𝑜𝑜𝑜𝑥𝑥𝑜𝑜𝑜𝑜𝑜𝑜 =
𝐷 −𝐷 − 𝑛𝑛′
𝑅𝑅20 𝐷
𝐷 0𝐷𝐷1𝑛𝑛′
𝐷𝐷 −
𝑛𝑛′ − 𝐷𝑅𝑅1
0 𝐷
𝛼𝛼𝑖𝑖𝑖𝑖𝑥𝑥𝑖𝑖𝑖𝑖
𝛼𝛼𝑜𝑜𝑜𝑜𝑜𝑜𝑥𝑥𝑜𝑜𝑜𝑜𝑜𝑜 =
𝐷𝑛𝑛′ − 𝐷𝑅𝑅2
0 𝐷
𝐷𝐷 − 𝑛𝑛′
𝑅𝑅1𝐷𝐷1𝑛𝑛′
𝐷𝐷1𝑛𝑛′
𝐷 − 𝑛𝑛′
𝑅𝑅1+ 𝐷
𝛼𝛼𝑖𝑖𝑖𝑖𝑥𝑥𝑖𝑖𝑖𝑖
𝛼𝛼𝑜𝑜𝑜𝑜𝑜𝑜𝑥𝑥𝑜𝑜𝑜𝑜𝑜𝑜 =
𝐷 +𝐷𝐷1𝑛𝑛′
𝑛𝑛′ − 𝐷𝑅𝑅2
𝐷 − 𝑛𝑛′
𝑅𝑅1+𝑛𝑛′ − 𝐷𝑅𝑅2
(𝐷𝐷1𝑛𝑛′
𝐷 − 𝑛𝑛′
𝑅𝑅1+ 𝐷)
𝐷𝐷1𝑛𝑛′
𝐷𝐷1𝑛𝑛′
𝐷 − 𝑛𝑛′
𝑅𝑅1+ 𝐷
𝛼𝛼𝑖𝑖𝑖𝑖𝑥𝑥𝑖𝑖𝑖𝑖
𝛼𝛼𝑜𝑜𝑜𝑜𝑜𝑜𝑥𝑥𝑜𝑜𝑜𝑜𝑜𝑜 =
𝐷 +𝐷𝐷1𝑛𝑛′
𝑛𝑛′ − 𝐷𝑅𝑅2
𝐷 − 𝑛𝑛′
𝑅𝑅1+𝐷𝐷1(𝐷 − 𝑛𝑛′)(𝑛𝑛′−𝐷)
𝑛𝑛′𝑅𝑅1𝑅𝑅2+𝑛𝑛′ − 𝐷𝑅𝑅2
𝐷𝐷1𝑛𝑛′
𝐷𝐷1𝑛𝑛′
𝐷 − 𝑛𝑛′
𝑅𝑅1+ 𝐷
𝛼𝛼𝑖𝑖𝑖𝑖𝑥𝑥𝑖𝑖𝑖𝑖
𝛼𝛼𝑜𝑜𝑜𝑜𝑜𝑜𝑥𝑥𝑜𝑜𝑜𝑜𝑜𝑜 =
𝐷 +𝐷𝐷1𝑛𝑛′
𝑛𝑛′ − 𝐷𝑅𝑅2
−(𝑛𝑛′ − 𝐷){𝐷𝑅𝑅1
−𝐷𝑅𝑅2
+𝐷𝐷1(𝑛𝑛′−𝐷)𝑛𝑛′𝑅𝑅1𝑅𝑅2
}
𝐷𝐷1𝑛𝑛′
𝐷𝐷1𝑛𝑛′
𝐷 − 𝑛𝑛′
𝑅𝑅1+ 𝐷
𝛼𝛼𝑖𝑖𝑖𝑖𝑥𝑥𝑖𝑖𝑖𝑖
-
𝛼𝛼𝑜𝑜𝑜𝑜𝑜𝑜𝑥𝑥𝑜𝑜𝑜𝑜𝑜𝑜 =
𝐷 +𝐷𝐷1𝑛𝑛′
𝑛𝑛′ − 𝐷𝑅𝑅2
−(𝑛𝑛′ − 𝐷){𝐷𝑅𝑅1
−𝐷𝑅𝑅2
+𝐷𝐷1(𝑛𝑛′−𝐷)𝑛𝑛′𝑅𝑅1𝑅𝑅2
}
𝐷𝐷1𝑛𝑛′
𝐷𝐷1𝑛𝑛′
𝐷 − 𝑛𝑛′
𝑅𝑅1+ 𝐷
𝛼𝛼𝑖𝑖𝑖𝑖𝑥𝑥𝑖𝑖𝑖𝑖
In System Matrix, term −𝑀𝑀12 is the power term
Hence, Power of the lens = (𝑛𝑛′ − 𝐷){ 1𝑅𝑅1
− 1𝑅𝑅2
+ 𝐷𝐷1(𝑖𝑖′−1)
𝑖𝑖′𝑅𝑅1𝑅𝑅2
Also, Power of lens (𝑃𝑃) = 1/Focal length of lens (𝑓𝑓)and in the case of a thick lens, focal length is the Effective Focal Length (EFL)
Thus, EFL = 𝑓𝑓 and 1𝑓𝑓
= (𝑛𝑛′ − 𝐷){ 1𝑅𝑅1
− 1𝑅𝑅2
+ 𝐷𝐷1(𝑖𝑖′−1)
𝑖𝑖′𝑅𝑅1𝑅𝑅2
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Example Problem:Imaging Condition(Slides 41-44 in Lecture notes PDF)
-
S S’
n n’
-
𝑛𝑛𝑜𝑜𝑜𝑜𝑜𝑜𝛼𝛼𝑜𝑜𝑜𝑜𝑜𝑜𝑥𝑥𝑜𝑜𝑜𝑜𝑜𝑜 =
Free spacepropagation
(𝑆𝑆𝑆)
Refraction insidethe optical system
Free spacepropagation
(𝑆𝑆)
𝑛𝑛𝑖𝑖𝑖𝑖𝛼𝛼𝑖𝑖𝑖𝑖𝑥𝑥𝑖𝑖𝑖𝑖
From Output side to Input side
Solution:The optical system may be complicated and have many lenses inside.Hence, simple consider a “black box” with 1st and 2nd Principal planes,
where the rays suffer refraction for the first and last times. Also, let this system have a power P that defines how the rays behave.
Complicated Optical System with power P
S S’
n n’
-
𝑛𝑛𝑜𝑜𝑜𝑜𝑜𝑜𝛼𝛼𝑜𝑜𝑜𝑜𝑜𝑜𝑥𝑥𝑜𝑜𝑜𝑜𝑜𝑜 =
Free spacepropagation
(𝑆𝑆𝑆)
Refraction insidethe optical system
Free spacepropagation
(𝑆𝑆)
𝑛𝑛𝑖𝑖𝑖𝑖𝛼𝛼𝑖𝑖𝑖𝑖𝑥𝑥𝑖𝑖𝑖𝑖
Here, 𝑛𝑛𝑜𝑜𝑜𝑜𝑜𝑜 = 𝑛𝑛’ and 𝑛𝑛𝑖𝑖𝑖𝑖 = 𝑛𝑛Hence:
𝑛𝑛’𝛼𝛼𝑜𝑜𝑜𝑜𝑜𝑜𝑥𝑥𝑜𝑜𝑜𝑜𝑜𝑜
=Free space
propagation(𝑆𝑆𝑆)
Refraction insidethe optical system with power P
Free spacepropagation
(𝑆𝑆)
𝑛𝑛𝛼𝛼𝑖𝑖𝑖𝑖𝑥𝑥𝑖𝑖𝑖𝑖
From Output side to Input side
Solution:The optical system may be complicated and have many lenses inside.Hence, simple consider a “black box” with 1st and 2nd Principal planes,
where the rays suffer refraction for the first and last times. Also, let this system have a power P that defines how the rays behave.
Complicated Optical System with power P
S S’
n n’
-
𝑛𝑛’𝛼𝛼𝑜𝑜𝑜𝑜𝑜𝑜𝑥𝑥𝑜𝑜𝑜𝑜𝑜𝑜
=Free space
propagation(𝑆𝑆𝑆)
Refraction insidethe optical system
Free spacepropagation
(𝑆𝑆)
𝑛𝑛𝛼𝛼𝑖𝑖𝑖𝑖𝑥𝑥𝑖𝑖𝑖𝑖
From Output side to Input sideSolution:
Complicated Optical System with power P
S S’
n n’
𝑛𝑛’𝛼𝛼𝑜𝑜𝑜𝑜𝑜𝑜𝑥𝑥𝑜𝑜𝑜𝑜𝑜𝑜
=𝐷 0𝑆𝑆𝑆𝑛𝑛′
𝐷𝐷 −𝑃𝑃0 𝐷
𝐷 0𝑆𝑆𝑛𝑛
𝐷𝑛𝑛𝛼𝛼𝑖𝑖𝑖𝑖𝑥𝑥𝑖𝑖𝑖𝑖
-
Solve the equation:
𝑛𝑛’𝛼𝛼𝑜𝑜𝑜𝑜𝑜𝑜𝑥𝑥𝑜𝑜𝑜𝑜𝑜𝑜
=𝐷 0𝑆𝑆𝑆𝑛𝑛′
𝐷𝐷 −𝑃𝑃0 𝐷
𝐷 0𝑆𝑆𝑛𝑛
𝐷𝑛𝑛𝛼𝛼𝑖𝑖𝑖𝑖𝑥𝑥𝑖𝑖𝑖𝑖
-
Solve the equation:
𝑛𝑛’𝛼𝛼𝑜𝑜𝑜𝑜𝑜𝑜𝑥𝑥𝑜𝑜𝑜𝑜𝑜𝑜
=𝐷 0𝑆𝑆𝑆𝑛𝑛′
𝐷𝐷 −𝑃𝑃0 𝐷
𝐷 0𝑆𝑆𝑛𝑛
𝐷𝑛𝑛𝛼𝛼𝑖𝑖𝑖𝑖𝑥𝑥𝑖𝑖𝑖𝑖
𝑛𝑛’𝛼𝛼𝑜𝑜𝑜𝑜𝑜𝑜𝑥𝑥𝑜𝑜𝑜𝑜𝑜𝑜
=𝐷 0𝑆𝑆𝑆𝑛𝑛′
𝐷
𝐷 −𝑃𝑃𝑆𝑆𝑛𝑛
−𝑃𝑃
𝑆𝑆𝑛𝑛
𝐷
𝑛𝑛𝛼𝛼𝑖𝑖𝑖𝑖𝑥𝑥𝑖𝑖𝑖𝑖
-
Solve the equation:
𝑛𝑛’𝛼𝛼𝑜𝑜𝑜𝑜𝑜𝑜𝑥𝑥𝑜𝑜𝑜𝑜𝑜𝑜
=𝐷 0𝑆𝑆𝑆𝑛𝑛′
𝐷𝐷 −𝑃𝑃0 𝐷
𝐷 0𝑆𝑆𝑛𝑛
𝐷𝑛𝑛𝛼𝛼𝑖𝑖𝑖𝑖𝑥𝑥𝑖𝑖𝑖𝑖
𝑛𝑛’𝛼𝛼𝑜𝑜𝑜𝑜𝑜𝑜𝑥𝑥𝑜𝑜𝑜𝑜𝑜𝑜
=𝐷 0𝑆𝑆𝑆𝑛𝑛′
𝐷
𝐷 −𝑃𝑃𝑆𝑆𝑛𝑛
−𝑃𝑃
𝑆𝑆𝑛𝑛
𝐷
𝑛𝑛𝛼𝛼𝑖𝑖𝑖𝑖𝑥𝑥𝑖𝑖𝑖𝑖
𝑛𝑛’𝛼𝛼𝑜𝑜𝑜𝑜𝑜𝑜𝑥𝑥𝑜𝑜𝑜𝑜𝑜𝑜
=𝐷 −
𝑃𝑃𝑆𝑆𝑛𝑛
−𝑃𝑃
𝑆𝑆′
𝑛𝑛′𝐷 −
𝑃𝑃𝑆𝑆𝑛𝑛
+𝑆𝑆𝑛𝑛
−𝑃𝑃𝑆𝑆′
𝑛𝑛′+ 𝐷
𝑛𝑛𝛼𝛼𝑖𝑖𝑖𝑖𝑥𝑥𝑖𝑖𝑖𝑖
-
Solve the equation:
𝑛𝑛’𝛼𝛼𝑜𝑜𝑜𝑜𝑜𝑜𝑥𝑥𝑜𝑜𝑜𝑜𝑜𝑜
=𝐷 0𝑆𝑆𝑆𝑛𝑛′
𝐷𝐷 −𝑃𝑃0 𝐷
𝐷 0𝑆𝑆𝑛𝑛
𝐷𝑛𝑛𝛼𝛼𝑖𝑖𝑖𝑖𝑥𝑥𝑖𝑖𝑖𝑖
𝑛𝑛’𝛼𝛼𝑜𝑜𝑜𝑜𝑜𝑜𝑥𝑥𝑜𝑜𝑜𝑜𝑜𝑜
=𝐷 0𝑆𝑆𝑆𝑛𝑛′
𝐷
𝐷 −𝑃𝑃𝑆𝑆𝑛𝑛
−𝑃𝑃
𝑆𝑆𝑛𝑛
𝐷
𝑛𝑛𝛼𝛼𝑖𝑖𝑖𝑖𝑥𝑥𝑖𝑖𝑖𝑖
𝑛𝑛’𝛼𝛼𝑜𝑜𝑜𝑜𝑜𝑜𝑥𝑥𝑜𝑜𝑜𝑜𝑜𝑜
=𝐷 −
𝑃𝑃𝑆𝑆𝑛𝑛
−𝑃𝑃
𝑆𝑆′
𝑛𝑛′𝐷 −
𝑃𝑃𝑆𝑆𝑛𝑛
+𝑆𝑆𝑛𝑛
−𝑃𝑃𝑆𝑆′
𝑛𝑛′+ 𝐷
𝑛𝑛𝛼𝛼𝑖𝑖𝑖𝑖𝑥𝑥𝑖𝑖𝑖𝑖
𝑛𝑛’𝛼𝛼𝑜𝑜𝑜𝑜𝑜𝑜𝑥𝑥𝑜𝑜𝑜𝑜𝑜𝑜
=𝐷 −
𝑃𝑃𝑆𝑆𝑛𝑛
−𝑃𝑃
𝑆𝑆′
𝑛𝑛′+𝑆𝑆𝑛𝑛−𝑃𝑃𝑆𝑆𝑆𝑆′
𝑛𝑛𝑛𝑛′𝐷 −
𝑃𝑃𝑆𝑆′
𝑛𝑛′
𝑛𝑛𝛼𝛼𝑖𝑖𝑖𝑖𝑥𝑥𝑖𝑖𝑖𝑖
System Matrix
-
Solution:
Complicated Optical System with power P
S S’
n n’
𝑛𝑛’𝛼𝛼𝑜𝑜𝑜𝑜𝑜𝑜𝑥𝑥𝑜𝑜𝑜𝑜𝑜𝑜
=𝐷 −
𝑃𝑃𝑆𝑆𝑛𝑛
−𝑃𝑃
𝑆𝑆′
𝑛𝑛′+𝑆𝑆𝑛𝑛−𝑃𝑃𝑆𝑆𝑆𝑆′
𝑛𝑛𝑛𝑛′𝐷 −
𝑃𝑃𝑆𝑆′
𝑛𝑛′
𝑛𝑛𝛼𝛼𝑖𝑖𝑖𝑖𝑥𝑥𝑖𝑖𝑖𝑖
𝑛𝑛’𝛼𝛼𝑜𝑜𝑜𝑜𝑜𝑜 = 𝐷 −𝑃𝑃𝑆𝑆𝑛𝑛
𝑛𝑛𝛼𝛼𝑖𝑖𝑖𝑖 − 𝑃𝑃𝑥𝑥𝑖𝑖𝑖𝑖
𝑥𝑥𝑜𝑜𝑜𝑜𝑜𝑜 =𝑆𝑆′
𝑛𝑛′+𝑆𝑆𝑛𝑛−𝑃𝑃𝑆𝑆𝑆𝑆′
𝑛𝑛𝑛𝑛′𝑛𝑛𝛼𝛼𝑖𝑖𝑖𝑖 + (𝐷 −
𝑃𝑃𝑆𝑆′
𝑛𝑛′)𝑥𝑥𝑖𝑖𝑖𝑖
Equating the terms:
-
Complicated Optical System with power P
S S’
n n’
𝑛𝑛’𝛼𝛼𝑜𝑜𝑜𝑜𝑜𝑜 = 𝐷 −𝑃𝑃𝑆𝑆𝑛𝑛
𝑛𝑛𝛼𝛼𝑖𝑖𝑖𝑖 − 𝑃𝑃𝑥𝑥𝑖𝑖𝑖𝑖
𝑥𝑥𝑜𝑜𝑜𝑜𝑜𝑜 =𝑆𝑆′
𝑛𝑛′+𝑆𝑆𝑛𝑛−𝑃𝑃𝑆𝑆𝑆𝑆′
𝑛𝑛𝑛𝑛′𝑛𝑛𝛼𝛼𝑖𝑖𝑖𝑖 + (𝐷 −
𝑃𝑃𝑆𝑆′
𝑛𝑛′)𝑥𝑥𝑖𝑖𝑖𝑖
For proper imaging to take place, all rays from object point must meet at image point. Hence, 𝑥𝑥𝑜𝑜𝑜𝑜𝑜𝑜 cannot have any angle (𝛼𝛼𝑖𝑖𝑖𝑖) dependence. Hence:
𝑆𝑆′
𝑖𝑖′+ 𝑆𝑆
𝑖𝑖− 𝑃𝑃𝑆𝑆𝑆𝑆
′
𝑖𝑖𝑖𝑖′= 0
or, 𝑛𝑛𝑆𝑆
+𝑛𝑛′
𝑆𝑆′= 𝑃𝑃 =
𝐷𝑓𝑓
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