Abstract
The image orientation change (IOC) of an object following its reflection by a system comprising an arbitrary number of flat boundary surfaces can be described using a merit function (Γ) expressed in the form of a 3×3 matrix. The present study proposes a design methodology for stable-IOC reflector and prism systems in which the merit function is solved using an eigenvalue-based approach. It is shown that a reflector system remains IOC-stable following its rotation about the eigenvector of the IOC merit function, provided that the image can still physically enter the system’s aperture. Furthermore, it is shown that an IOC-stable prism can be obtained by adding two refracting flat boundary surfaces at the entrance and exit positions of the light ray in an optical system comprising multiple reflectors provided that the condition n n =Γn 1 is maintained. Illustrative examples are provided to demonstrate the validity of the proposed design approach.
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Abbreviations
- (xyz)0 :
-
coordinate frame of object.
- (xyz)′0 :
-
coordinate frame of imaged object.
- (xyz) i :
-
coordinate frame imbedded in ith boundary surface.
- r i :
-
ith boundary surface with unit normal vector n i .
- ℓ i :
-
unit directional vector of light ray reflected/refracted at r i .
- N i :
-
N i =ξ i−1/ξ i , where ξ i is the refractive index of medium i.
- n i :
-
unit normal vector of boundary r i .
- n si :
-
unit normal vector of boundary r i after the system rotates.
- Γ:
-
merit function to be used to represent a specific IOC.
- Γs :
-
merit function after the system rotates.
- ∂ ℓ i (n i )/∂ ℓ i−1 :
-
reflector matrix.
- ∂ ℓ i (n i ,N i )/∂ ℓ i−1 :
-
refraction matrix.
- λ :
-
eigenvalue.
- X :
-
normalized eigenvector of the merit function Γ.
- X s :
-
normalized eigenvector of the rotated merit function Γs.
- I :
-
identity matrix.
References
W.J. Smith, Modern Optical Engineering, 3rd edn. (Edmund Industrial Optics, Barrington, 2001), pp. 100–121
R.H. Ginsberg, Appl. Opt. 33, 8105 (1994)
N. Lin, Opt. Eng. 33, 2400 (1994)
W. Mao, Opt. Eng. 34, 79 (1995)
E.J. Galvez, C.D. Holmes, J. Opt. Soc. Am. A 16, 1981 (1999)
C.Y. Tsai, P.D. Lin, Appl. Opt. 45, 3951 (2006)
C.Y. Tsai, P.D. Lin, Appl. Opt. 46, 3087 (2007)
R.P. Paul, Robot Manipulators—Mathematics, Programming and Control (MIT Press, Cambridge, 1982)
E. Kreyszig, Advanced Engineering Mathematics, 9th edn. (Wiley, New York, 2006), pp. 350–351
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Tsai, CY. An eigenvalue-based approach for the design of reflector systems and prisms with a specified image orientation change. Appl. Phys. B 102, 235–242 (2011). https://doi.org/10.1007/s00340-010-4139-y
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DOI: https://doi.org/10.1007/s00340-010-4139-y