Gaussian beams. w e. z z. tan. Solution of scalar paraxial wave equation (Helmholtz equation) is a Gaussian beam, given by: where

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1 ECE 566 OE System Design obert Mceo 93 Gaussian beams Saleh & Teich chapter 3, M&M Appenix k k e A r E tan Solution of scalar paraxial ave euation (Helmholt euation) is a Gaussian beam, given by: here Paraxial ray tracing of optical systems Note that () oes not obey ray tracing sign convention. Unfortunately there s no particularly goo ay to fix this. Sie aius of curvature Gouy phase

2 ECE 566 OE System Design Gaussian beams Detaile vie eal part of E vs. raius an / At =, I(, ) = I(,)/ (min value) Any constant times () is a ray path. Note that the rays are converging an iverging spherical ave except near the focus here they ben. Ergo rays o not alays travel in straight lines the region near the focus violates the slolyvarying envelope approximation. Conversion formulas obert Mceo 94

3 ECE 566 OE System Design obert Mceo 95 Gaussian beam parameter () k k D e A r E / k k D e A r E / The complete Gaussian beam expression normalie to intensity Define the complex raius of curvature What is ()? Can no rite Gaussian above as Note that phase of /() is Paraxial ray tracing of optical systems tan arg D is number of transverse imensions =,

4 ECE 566 OE System Design Ho oes change ith transfer an refraction? Free space: Start ith expression for () so = + Thin lens Start ith expression for /() f Thin lens euation expresse as change in curvature of ave NOTE HOW GAUSSIAN BEAM SIGN CONVENTION HAS CHANGED THE SIGN f Apply to / f Solve for obert Mceo 96

5 ECE 566 OE System Design obert Mceo 97 ABCD approach to k k t T k k D C B A k k t t Check for free space: f k Check for thin lens: emember the ABCD matrices for thin lens refraction an free-space transfer an efine the evolution euation for Which says, rather remarkably, that e can moel the propagation of a Gaussian beam through a paraxial optical system using ray matrices. But there s a better ay to o so (at least I like it s better). Paraxial ray tracing of optical systems

6 ECE 566 OE System Design epresentation of Gaussian beams by complex rays () Define the folloing three rays. Note their suggestive names an relationship to the Gaussian beam. Paraxial ray traectory form Waist ray Waist location Define the complex ray traectory You can then sho that this ray contains () y u Chief ray y u f J. Arnau, Applie Optics, V4, N4, p. 538, 5 Feb 985 A. W. Greynols, SPIE V 56, p. 33, 985 M&M A.5 obert Mceo 98 Waist location Paraxial image plane ABCD vector form This is Greynols efinition an yiels the proper form of. Arnau s efinition yiels *. ay heights over ray slopes E.g. at =

7 ECE 566 OE System Design epresentation of Gaussian beams by complex rays () First e note: y u y u n n agrange invariant (N spots =) By brute for tracing of the rays, e can fin the folloing Gaussian parameters base on the to rays at that point: u u y y Which gives all other beam parameters /e fiel raius at this We coul use these to an the expressions for the Gaussian beam parameters to generate the complete Gaussian, but this oul be a bit teious. A more elegant ay is to use the complex ray formalism: y u u y u u Which, apart from the on-axis phase k S gives the full Gaussian beam at this plane.. Herloski, S. Marshall,. Antos, Applie Optics, V, N8, p 68, 5 Apr 983 obert Mceo 99 At plane

8 ECE 566 OE System Design epresentation of Gaussian beams by complex rays (3) On-axis examples: ) = m, =, f = 5 m, f f system. ) = m, =, f = 5 m, f f system. 3) = m, =, f = 5 m, 3 f 3/ f system. Notes In (), secon aist is at Fourier plane, as expecte. In (), secon aist occurs before image plane, as expecte. In (3), as istance to lens increases, aist moves to paraxial image plane obert Mceo

9 ECE 566 OE System Design epresentation of Gaussian beams by complex rays (5) Off-axis examples: ) = m, =, f = 5 m, f f system. ) = m, =, f = 5 m, f f system. 3) = m, =, f = 5 m, 3 f 3/ f system. Notes In (), aist is centere at ero (as expecte of Fourier trransform) In (), image is at - m, expecte from magnification M=-. This type of problem is not possible ith the ABCD formalism. obert Mceo

10 ECE 566 OE System Design Do Gaussian beams obey paraxial imaging? (/3) Obect at aist f= M=- imaging Image not at aist eal obect f= Demagnifying imaging eal image Anser: Yes. The image is also a Gaussian E fiel istribution in amplitue an any point on the obect on from the peak by some value, say /e for the point (), ill image to the point on the image on from the peak by the same value. Shon above only for real obects conugate to real images ( -t f ). obert Mceo

11 ECE 566 OE System Design Do Gaussian beams obey paraxial imaging? (/3) Ho about for virtual obects? f= Virtual obect 4 eal image Anser: Still orks. obert Mceo 3

12 ECE 566 OE System Design Do Gaussian beams obey paraxial imaging? (3/3) Ho about for virtual images? 7.5 f= 5 Virtual image.5 eal obect Conclusion: All parts of the obect Gaussian image correctly to the appropriate parts of the image Gaussian incluing both real an virtual obects an images. Corollary: If you apply paraxial imaging to the obect Gaussian over all, you generate the image Gaussian over all. Gaussian beams obey paraxial imaging exactly. obert Mceo 4-7.5

13 ECE 566 OE System Design obert Mceo 5 Design example Collimator lens T T M Design of ieal imaging systems ith geometrical optics Gaussian beam propagation ABCD from start to center at center starting ith = NEW e Where is aist? What is ne ayleigh range?