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Part II

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Figure 5. Explanation to optical ray definition.

Figure 6. Explanation to paraxial ray' parameters.

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Figure 7. Ray propagation in free space on distance d.

Figure 8. Ray propagation through a thin lens of focal length f .

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Figure 9. Ray reflection from a spherical mirror of radius R.

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Figure 10. Ray reflection from a plane mirror.

Figure 11. Ray inversion (or coordinate inversion) on reflection (after [2], p.591).

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Figure 12. Dielectric interface.

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Figure 13. Sketch of quadratic ducts: (a) – optically stable duct (a ray is permanently confined (guided) in the duct), (b) – optically unstable duct (a ray escapes (leaks) from the duct).

Figure 14. Examples of mechanical systems in stable & unstable equilibrium.

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Figure 15. Sketches of ray traces in stable and unstable ducts.

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Summary Table of ABCD matrices [2] (for reduced slope).

Optical element

description Explanation to the element’ parameters

ABCD matrix

1

Straight section: Length d in media n

10

1 nd

2a

Thin convex (converging) lens: Focal length f>0

− 11

01

f

2b

Thin concave (diverging) lens: Focal length f<0

+ 11

01

f

3

Spherical interface between 2 dielectric media: Refractive indices n1, n2, radius of curvature R

− 1

0112

Rnn

4a

Spherical concave mirror (normal incidence): Radius of curvature R>0

− 12

01

R

d x

n

x f

x f

R

R

z

z

z

z

x

x

z

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4b

Spherical convex mirror (normal incidence): Radius of curvature R>0

+ 12

01

R

5a

Stable duct (or gain):

( )

121

10

220

>=

−=

Constn

xnnxn

0

2

22 0

nn

Constn

=

>=

γ

( ) ( )

( ) ( ) ( )

∆∆−

∆∆

zzn

nzz

γγγ

γγγ

cossin

sincos

0

0

5b

Unstable duct (or loss):

( )

121

10

220

>=

−=

Constn

xnnxn

0

2

22 0

nni

Constn

=

<=

γ

( ) ( )

( ) ( ) ( )

∆∆−

∆∆

zzn

nz

z

γγγ

γγ

γ

coshsinh

sinhcosh

0

0

6

Curved mirror (arbitrary angle of incidence, EM field is in the plane of incidence (“tangential”)): Radius of curvature R angle of incidence θ

−1

cos2

01

θR

R

z

Δz

Δz

R

z

θ

θ

incident axis

exit axis

x z

x

x z

x

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Figure 16. Optical system containing N optical elements.

7

Curved mirror (arbitrary angle of incidence, EM field is ┴ to the plane of incidence (“sagittal”)): Radius of curvature R angle of incidence θ

−1

cos2

01

θR

R

z

θ

θ

incident axis

exit axis

x

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Figure 17. Optical system - Problem II-1.

Figure 18. Optical system - Problem II - 2.

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Figure 19. Lens guide equivalent of the optical system - Problem II - 2

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Figure 20. Sketch of a beam expander – Problem II - 3

Figure 21. Explanation to the case A = 0.

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Figure 22. Explanation to the case B = 0.

Figure 23. Explanation to the case C = 0.

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Figure 24. Explanation to the case D = 0.

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Figure 25. Examples of periodic focusing systems.

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Figure 26. Sketch of a stable resonator.

Figure 27. Ray trajectory in a stable periodic system (after [2], p.602).

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Figure 28. Left: optical waveguide (credits : http://www.sprengel-elektronik.net/lightCABLE-

KunstoffPMMA-Lichtleiter-Staerke-3mm-klar-farblos-flexibel-ohne-Ummantelung-pro-lfm ). Right : Vertical Cavity Surface Emitting Laser (VCSEL) (credits: https://www.google.co.il/url?sa=i&source=images&cd=&cad=rja&uact=8&ved=0ahUKEwjeqeql1pLeAhVQp4sKHe0zDHAQMwg8KAAwAA&url=http%3A%2F%2Fiopscience.iop.org%2F14644266%2F2%2F4%2F310%2Fmedia%2Fvcsel.html&psig=AOvVaw2ec51EHtTH6CR1CBPAF8Dx&ust=15400443

19614962&ictx=3&uact=3 )

Figure 29. Sketch of an unstable resonator.

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Figure 30. Unstable periodic focusing systems of the "positive-branch" and "negative-branch" types.

Figure 31. Left: Explosive Photodissociation Iodine Laser (EPIL), used in cosmic interferomentry experiments (credits : http://militaryrussia.ru/blog/topic-

620.html). Right : The Laser Weapon System (LaWS) installed aboard the guided-missile destroyer USS Dewey (DDG-105) (credits:

https://news.usni.org/2014/02/28/document-report-navy-shipboard-lasers )

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Figure 32. Sketches of lensguides (Problem II - 4).

Figure 33. Lensguides’ unit cell and its two-mirror cavity equivalent (Problem II - 4).

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Figure 34. Optical system - Problem II - 5.

Figure 35 . The stability diagram for a two-mirror optical resonator (after [2]).

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Figure 36. Lens guide equivalent of the resonator – Problem II - 5.

Figure 37. Z-cavity (Problem II - 6).

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Figure 38. Cavity sketch – Problem II – 7.

Figure 39. Cavity sketch – Problem II – 8.

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Figure 40. Lensguide equivalent of the cavity – Problem II – 8.

Figure 41. Single GRIN lens based probe sketch – Problem II – 9.

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Figure 42. Examples of single GRIN lens based Optical Coherence Tomography probes (after [10]).

Figure 43. Example of laser beam, injected into an optical system (after [2]).

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Figure 44. Explanation to the choice of θ angle.

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References : [1]. http://en.wikipedia.org/wiki/Ray_(optics) [2]. A. Siegman. Lasers.(University Science books 1986) [3]. A. Yariv. Quantum electronics (3rd edition, Wiley, 1989) [4]. T. Gavlin, G.Eden (ECE Illinois) Optical Resonator Modes ECE 455 Optical Electronics.

https://courses.engr.illinois.edu/ece455/Files/Galvinlectures/02_CavityModes.pdf [5]. Lecture 25. Lens imaging II. [6]. Geoffrey Brooker. Modern Classical Optics. (Oxford Master Series in Atomic, Optical and Laser

Physics) (2003) (google book link) [7]. S. Rushin. Introduction to Lasers (course # 05124601- EE – Physical Electronics, Tel Aviv

University) – course materials [8]. Z. Yun, M. Iskander, Ray Tracing for Radio Propagation Modeling: Principles and Applications,

IEEE Access, pp. 1089 – 1100, July 2015, DOI 10.1109/ACCESS.2015.2453991 [9]. A. Yariv, P. Yech, Photonics. Oxford University Press, Chapter 2 (Lecture on the basis of this

chapter) [10]. W. Jung, W. Benalcazar et. al, Numerical analysis of gradient index lens–based optical

coherence tomography imaging probes, Journal of Biomedical Optics 15(6), 066027-1 - 066027-10 (2010).