1. Boian Andonov Hristov, Prof. (Ph.D) Bulgarian Academy of Sciences
4th International Conference on Photonics July 28029,2016 Berlin, Germany Exact analytical aberration theory of centered optical systems containing conic surfaces
Boian Andonov Hristov, Prof. (Ph.D)
Bulgarian Academy of Sciences

2. Exact Analytical Aberration Theory of Centered Optical Systems Containing Conic Surfaces
To achieve:
Precise analytical correction of individual aberrations;
Exact calculation of the system constructive parameters, such as surface radiuses and conic constants, axial thicknesses, etc.;
Correct pictures of the aberrations space distribution in the whole object space and the whole image space of the optical system.

3. Conic radiuses coefficient- 𝝉 𝒋
1 - angle of incidence (𝜺 𝒋 ); 2 - angle of refraction or reflection (𝜺 𝒋 ′ ); 3 – slope of the surface normal ( 𝝋 𝒋 ) ; 4 – angle between refracted ray and z-axes ( 𝒘 𝒋+𝟏 ) ; 5 – angle between incident ray and z-axes ( 𝒘 𝒋 ). 𝒓 𝒕𝒋 = 𝒓 𝒔𝒋 𝟑 𝒓 𝟎𝒋 𝟐 ; 𝒓 𝟎𝒋 = 𝝉 𝒋 𝒓 𝒔𝒋 ; 𝒓 𝒕 = 𝒓 𝒔𝒋 𝝉 𝒋 𝟐 = 𝒓 𝟎𝒋 𝝉 𝒋 𝟑 ; 𝝉 𝒋 = 1+ K j sin 2 φ j .

4. Exact transformations from the object to the image space in a single conic surface
In the sagittal planes: 𝒛 𝒔𝒋 ′ = 𝒓 𝒔𝒋 𝒑 𝒋 𝒛 𝒔𝒋 + 𝒓 𝒔𝒋 𝟐 𝑸 𝒋 𝒎 𝒋 𝒛 𝒔𝒋 + 𝒓 𝒔𝒋 𝝁 𝒋 = 𝒓 𝒔𝒋 𝒂 𝒋 𝒛 𝒔𝒋 + 𝒃 𝒋 𝒄 𝒋 𝒛 𝒔𝒋 + 𝒅 𝒋 , 𝑸 𝒋 ≡𝟎, 𝐆 𝐣 = 𝐚 𝐣 𝟎 𝐜 𝐣 𝐝 𝐣 = 𝐫 𝐬𝐣 𝐩 𝐣 𝟎 𝐦 𝐣 𝐫 𝐬𝐣 𝛍 𝐣 . Here 𝒑 𝒋 , 𝝁 𝒋 , 𝒎 𝒋 are exact functions depending on the real values of the following parameters: 𝒏 𝒋 , 𝒏 𝒋+𝟏 , 𝜺 𝒋 , 𝝋 𝒋 , 𝒘 𝒋+𝟏 𝒂𝒏𝒅 𝒘 𝒋 . In the tangential planes: 𝒛 𝒕𝒋 ′ = 𝒓 𝒔𝒋 𝒑 𝒋 𝒕 𝒛 𝒔𝒋 + 𝒓 𝒔𝒋 𝟐 𝑸 𝒋 𝒕 𝝉 𝒋 𝟐 𝒎 𝒋 𝒛 𝒕𝒋 + 𝒓 𝒔𝒋 𝝁 𝒋 𝒕 = 𝒓 𝒔𝒋 𝒂 𝒋 𝒕 𝒛 𝒕𝒋 + 𝒃 𝒋 𝒕 𝒄 𝒋 𝒕 𝒛 𝒕𝒋 + 𝒅 𝒋 𝒕 , 𝐆 𝐣 𝐭 = 𝐚 𝐣 𝐭 𝐛 𝐣 𝐭 𝐜 𝐣 𝐭 𝐝 𝐣 𝐭 = 𝐫 𝐬𝐣 𝐩 𝐣 𝐭 𝐐 𝐣 𝐭 𝛕 𝐣 𝟐 𝐦 𝐣 𝐫 𝐬𝐣 𝛍 𝐣 𝐭 . Here 𝒑 𝒋 𝒕 , 𝝁 𝒋 𝒋 𝒕 , 𝑸 𝒋, 𝒕 𝒂𝒏𝒅 𝒎 𝒋 are exact functions depending on the real values of parameters: 𝒏 𝒋 , 𝒏 𝒋+𝟏 , 𝜺 𝒋 , 𝝋 𝒋 , 𝒘 𝒋 , 𝝉 𝒋 𝟐 , 𝒂𝒏𝒅 𝒘 𝒋+𝟏 . In the paraxial region: 𝒛 𝟎𝒋 ′ = 𝒓 𝒔𝒋 𝒑 𝒋 𝟎 𝒛 𝟎𝒋 + 𝒓 𝒔𝒋 𝟐 𝑸 𝒋 𝟎 𝝉 𝒋 𝟐 𝒎 𝒋 𝟎 𝒛 𝟎𝒋 + 𝒓 𝒋 𝝁 𝒋 = 𝐫 𝐬𝐣 𝐚 𝒋 𝟎 𝐳 𝟎𝐣 + 𝐛 𝒋 𝟎 𝐜 𝒋 𝟎 𝐳 𝟎𝐣 + 𝐝 𝒋 𝟎 , 𝑮 𝒋 𝟎 = 𝒂 𝒋 𝟎 𝒃 𝒋 𝟎 𝒄 𝒋 𝟎 𝒅 𝒋 𝟎 = 𝒓 𝒔𝒋 𝒑 𝒋 𝟎 𝑸 𝒋 𝟎 𝒎 𝒋 𝟎 𝒓 𝒔𝒋 𝝁 𝒋 𝟎 . Here 𝒑 𝒋 𝟎 , 𝝁 𝒋 𝒋 𝟎 , 𝑸 𝒋, 𝟎 𝒎 𝒋 𝟎 are exact functions depending on t he real values of parameters: 𝒏 𝒋 , 𝒏 𝒋+𝟏 , 𝜺 𝒋 , 𝝋 𝒋 , 𝒘 𝒋 , 𝝉 𝒋 , 𝒂𝒏𝒅 𝒘 𝒋+𝟏 . 𝑸 𝒋 𝟎 = 𝒒 𝒋𝟑 𝝉 𝒋 𝟑 + 𝒒 𝒋𝟐 𝝉 𝒋 𝟐 + 𝒒 𝒋𝟏 𝝉+ 𝒒 𝒋𝟎 𝝉 𝒋 + 𝐜𝐨𝐬 𝝋 𝒋 𝟐 .
In the saggital planes the image distance is bilinear function of the object distance. It is related to a matrix of this type. It can be proved that coefficient Q is always identical 0. “Rsj” is the well-known abcd matrix formalism in the paraxial theory. I prove that in sagittal planes it is similar.
Here p.mi.m are…….. Patameters shown in the previous slide.
We use identical transforamationa for tangential planes and paraxial regions.

5. Exact transformations from the object to the image space in an optical system of “k” centered conic surfaces
𝑔 𝑘 = 𝐺 𝑘 𝐺 𝑘−1 … 𝐺 𝑗 … 𝐺 2 𝐺 1 = 𝑎 𝑘 0 𝑐 𝑘 𝑑 𝑘 – sagittal matrix;
𝒂 𝒌 , 𝒄 𝒌 , 𝒅 𝒌 - sagittal matrix coefficients of the whole system;
𝑔 𝑘 𝑡 = 𝐺 𝑘 𝑡 𝐺 𝑘−1 𝑡 ,…, 𝐺 𝑗 𝑡 … 𝐺 2 𝑡 𝐺 1 𝑡 = 𝑎 𝑘 𝑡 𝑏 𝑘 𝑡 𝑐 𝑘 𝑡 𝑑 𝑘 𝑡 tangential matrix;
𝒂 𝒕 𝒕 , 𝒃 𝒌 𝒕 , 𝒄 𝒌 𝒕 , 𝒅 𝒌 𝒕 tangential matrix coefficients of the whole system;
𝑔 𝑘 0 = 𝐺 𝑘 0 𝐺 𝑘−1 0 ,…, 𝐺 𝑗 0 ,…, 𝐺 2 0 𝐺 1 0 = 𝑎 𝑘 0 𝑏 𝑘 0 𝑐 𝑘 0 𝑑 𝑘 paraxial matrix;
𝒂 𝒌 𝟎 , 𝒃 𝒌 𝟎 , 𝒄 𝒌 𝟎 , 𝒅 𝒌 𝟎 paraxial matrix coefficients of the whole system;

6. Relative parameters ( 𝒙 𝒋 ) and relative matrix coefficients:
𝒙 𝒋 = 𝒓 𝒔𝒋 𝒓 𝒔(𝒋+𝟏) ;
𝑨 𝒌 , 𝑩 𝒌 , 𝑪 𝒌 , 𝑫 𝒌 : - sagittal relative matrix coefficients;
𝑨 𝒌 𝒕 , 𝑩 𝒌 𝒕 , 𝑪 𝒌 𝒕 , 𝑫 𝒌 𝒕 : - tangential relative matrix coefficients;
𝑨 𝒌 𝟎 , 𝑩 𝒌 𝟎 , 𝑪 𝒌 𝟎 , 𝑫 𝒌 𝟎 : - paraxial relative matrix coefficients;
𝒂 𝒌 = 𝒓 𝒔𝒌 𝑨 𝒌 𝒋=𝟐 𝒌 𝒓 𝒔𝒋 ;
𝒃 𝒌 = 𝒓 𝒔𝟏 𝒓 𝒔𝒌 𝑩 𝒌 𝒋=𝟐 𝒌 𝒓 𝒔𝒋 ;
𝒄 𝒌 = 𝒓 𝒔𝒌 𝑪 𝒌 𝒋=𝟐 𝒌 𝒓 𝒔𝒋 ;
𝒅 𝒌 = 𝒓 𝒔𝟏 𝑫 𝒌 𝒋=𝟐 𝒌 𝒓 𝒔𝒋 .

7. Exact aberration functions and space distribution
Field aberrations: astigmatism, sagittal curvature, tangential curvature, field curvature, distortion, sagittal coma, tangential coma).
On-axes aberrations: paraxial chromatism, spherical aberration, OPD or wave aberration and Abbe’s sine condition)
∆𝒛 𝒌 ′ 𝒓 𝒔𝒌 = 𝑼 𝟏 𝒛 𝟎 𝒓 𝒔𝟏 + 𝑼 𝟎 𝑽 𝟏 𝒛 𝟎 𝒓 𝒔𝟏 + 𝑽 𝟎 - bilinear function to 𝑧 0 for: the distortion, on-axes spherical aberration, OPD and Abbe’s sine condition; Here 𝑈 1 , 𝑈 0 are linear functions to every 𝑥 𝑗 and cubic functions to every 𝜏 𝑗 ;
∆𝒛 𝒌 ′ 𝒓 𝒔𝒌 = 𝑼 𝟐 𝒛 𝟎 𝟐 𝒓 𝒔𝟏 𝟐 + 𝑼 𝟏 𝒛 𝟎 𝒓 𝒔𝟏 + 𝑼 𝟎 𝑽 𝟐 𝒛 𝟎 𝟐 𝒓 𝒔𝟏 𝟐 + 𝑽 𝟏 𝒛 𝟎 𝒓 𝒔𝟏 + 𝑽 𝟎 - quadratic-fractional function to 𝑧 0 for: the image astigmatism, sagittal curvature, tangential curvature and paraxial chromatism; Here 𝑈 2 , 𝑈 1 𝑎𝑛𝑑 𝑈 0 are quadratic functions to every 𝑥 𝑗 and cubic functions to every 𝜏 𝑗 ;
∆𝒛 𝒌 ′ 𝒓 𝒔𝒌 = 𝑼 𝟑 𝒛 𝟎 𝟑 𝒓 𝒔𝟏 𝟑 + 𝑼 𝟐 𝒛 𝟎 𝟐 𝒓 𝒔𝟏 𝟐 + 𝑼 𝟏 𝒛 𝟎 𝒓 𝒔𝟏 + 𝑼 𝟎 𝑽 𝟑 𝒛 𝟎 𝟑 𝒓 𝒔𝟏 𝟑 + 𝑽 𝟐 𝒛 𝟎 𝟐 𝒓 𝒔𝟏 𝟐 + 𝑽 𝟏 𝒛 𝟎 𝒓 𝒔𝟏 + 𝑽 𝟎 - cubic fractional function for: the field curvature, tangential coma and sagittal coma; Here 𝑈 3 , 𝑈 2 , 𝑈 1 , 𝑈 0 are cubic functions to every 𝑥 𝑗 and 𝜏 𝑗 ;
Space distribution: zero points and zero surfaces; vertical and horizontal asymptotes

8. General aberration theorems for a centered optical system containing conic surfaces
Discriminants of aberrations: 𝑫 𝒂𝒔𝒕 = 𝑼 𝟏 𝟐 −𝟒 𝑼 𝟐 𝑼 𝟎 ; Theorem for the paraxial achromatic points:
In every centered optical system containing conic surfaces for every two wavelengths on the optical axis: a) there are two paraxial achromatic points if the paraxial chromatic discriminant is greater than zero; b) there is one achromatic point if the paraxial chromatic discriminant is zero; c) there is no achromatic point if the paraxial chromatic discriminant is less than zero; d) there is an endless set of achromatic points if all the three paraxial chromatic coefficients, 𝑼 𝟐 , 𝑼 𝟏 , 𝑼 𝟎 are zero.
(Ref: HRISTOV, B. Exact Analytical Theory of Aberrations of Centered Optical Systems. OPTICAL REVIEW Vol. 20, No. 5 (2013) 395–419).
 Theorems for the field aberrations: distortion, astigmatism, sagittal curvature, tangential curvature, field curvature, sagittal coma and tangential coma.
General theorem:
When the object space and image space are homogeneous and the object space is aberration-free, no more than one surface in general can be perfectly imaged by a centered optical system containing conic surfaces.
(Ref: HRISTOV, B. Development Of Optical Design Algorithms On The Base Of The Exact (All Orders) Geometrical Aberration Theory, Proc. SPIE 8167 (2011) 12.)

9. Exact analytical correction of individual aberrations and simultaneous correction
By solving one linear equation to the arbitrary chosen relative parameter ( 𝑥 𝑗 ), or one cubic equation to the arbitrary chosen conic radius coefficient ( 𝜏 𝑗 ) on-axes spherical aberration, distortion, OPD, or satisfaction of Abbe’s sine condition become corrected each one separately. By solving one quadratic equation to the arbitrary chosen relative parameter ( 𝑥 𝑗 ), or one cubic equation to the arbitrary chosen conic radius coefficient ( 𝜏 𝑗 ) astigmatism, or sagittal curvature, or tangential curvature, or paraxial chromatism become corrected each one separately. By solving one cubic equation to the arbitrary chosen relative parameter ( 𝑥 𝑗 ), or one cubic equation to the arbitrary chosen conic radius coefficient ( 𝜏 𝑗 ) field curvature, or tangential coma, or sagittal coma become corrected each one separately For simultaneous correction of aberrations we use Sylvester’s matrices by excluding unknown parameters to find exactly all existing solutions.

10. EXAMPLE

11. THANK YOU
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