Ray transfer martix of interface

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SUMMARY

The discussion centers on the differing descriptions of the ray transfer matrix of an interface as presented in two textbooks: "Lasers" by A.E. Siegman and "Laser Resonators and Beam Propagation" by N. Hodgson. The primary distinction lies in the use of the paraxial approximation and ray notation, specifically the (y, nu) form for rays in Siegman's work. The confusion arises from the interpretation of the optical path, which may lead to variations in the ray transfer matrix representations. Understanding these differences is crucial for accurate application in optical systems.

PREREQUISITES
  • Familiarity with ray transfer matrix theory
  • Understanding of paraxial approximation in optics
  • Knowledge of ray notation, specifically (y, nu) form
  • Basic concepts of optical path length
NEXT STEPS
  • Study the ray transfer matrix in detail using "Lasers" by A.E. Siegman
  • Examine the differences in ray notation in "Laser Resonators and Beam Propagation" by N. Hodgson
  • Research the implications of the paraxial approximation in optical systems
  • Explore practical examples of ray transfer matrices in plane dielectric slabs
USEFUL FOR

Optical engineers, physicists, and students studying laser optics or beam propagation who seek to deepen their understanding of ray transfer matrices and their applications.

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I found two different descriptions about ray transfer martix of interface in two textbook. I am confused about this. which one is correct? I hope someone can help me.
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A:Lasers, A.E.Siegman,pp586
B:Laser Resonators and Beam Propagation, N.Hodgson, pp16
 

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Siegman's is the one I am used to seeing- that is the refraction matrix written out using the paraxial approximation and the (y, nu) form for rays. I don't immediately see how to transform Siegman's into Hodgson's, but maybe Hodgson is using a different ray notation?
 
I check these two descriptions again, and I found that the difference may be caused by the understanding about optical path.

example:Plane Dielectric Slab
 

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