all dimensions are mm the minus means meniscus lens f 2

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1 TEM Gauss-beam described with ray-optics. F.A. van Goor, University of Twente, Enschede The Netherlands. December 8, 994 as significantly modified by C. Nelson - 26 Example of an optical system consisting of two lenses with focal lengths f and f 2 at s and 2 respectively. These lenses are the uncorrected focal beam of a laser-optic system. all dimensions are mm f f 2 the minus means meniscus lens The calculation starts with a Gaussian beam with wavelength λ and a waist w at : λ. w 5 wavelength is micron The system can be described using ABCD matrices for the propagation (M, M, and M 2 ) and for the two lenses (M L and M L2 ): ( ) ( ) B ( ) if,, M ( ) B ( ) ( ) B ( ) if,, if 2,, 2 M ( ) B ( ) ( ) B 2 ( ) if 2, 2, M 2 ( ) B 2 ( ) M L M L2 f f 2 The beam is calculated from to max, using N rays: max N 2.. max i.. N At the waist at the Gaussian beam can be described with rays of which the divergence and s lying on an upright ellipse (χ=): λ θ max χ x( φ) w sin( φ + χ) θ φ π w φ,. π.. 2 π ( ) θ max cos( φ)

2 θ( φ) x( φ) Fig.. The s and divergences lie on an ellipse that is upright in the beam waist. Definition of the s and divergences of the N rays at : 2 i Θ i N π θ θ Θ i i ( ) x i x Θ i ( ) Or in vector notation: i R x i θ i After propagation of the beam from to the and divergency follow from: R( ) M 2 ( ) M L2 M ( ) M L M ( ) R The s and divergences of the N rays at are: ( ) ( ) x( ) R( ) T θ ( ) R( ) T i.. max Z i + i X i, i x Z i i

3 Fig. 2a. x() as a function of for the N rays demonstrating the propagation of a Gaussian beam. Expanded vertical axis. This is the ray trace for the unmodified fiber optic laser head Fig. 2b. x() as a function of for the N rays demonstrating the propagation of a Gaussian beam. True Scale This is the ray trace for the unmodified fiber optic laser head Fig. 2c. x() as a function of for the N rays demonstrating the propagation of a Gaussian beam. Highly expanded focal one. This is the ray trace for the unmodified fiber optic laser head.

4 Now we do the whole thing all over again. This time we insert a negative (meniscus) lens into the optic path in order to extend the focal distance out to a more useful length. We see that a focal length of -2 mm provides a useful solution. Example of an optical system consisting of two lenses with focal lengths f and f 2 at s and 2 respectively: all dimensions are mm 5 f 2 2 f 2 2 the minus means meniscus lens The calculation starts with a Gaussian beam with wavelength λ and a waist w at : λ. w 5 wavelength is micron The system can be described using ABCD matrices for the propagation (M, M, and M 2 ) and for the two lenses (M L and M L2 ): ( ) ( ) B ( ) if,, M ( ) B ( ) ( ) B ( ) if,, if 2,, 2 M ( ) B ( ) ( ) B 2 ( ) if 2, 2, M 2 ( ) B 2 ( ) M L M L2 f f 2 The beam is calculated from to max, using N rays: max N 2.. max i.. N At the waist at the Gaussian beam can be described with rays of which the divergency and s lying on an upright ellipse (χ=): λ θ max χ x( φ) w sin( φ + χ) θ φ π w φ,. π.. 2 π ( ) θ max cos ( φ)

5 θ( φ) x( φ) Fig.. The s and divergences lie on an ellipse that is upright in the beam waist. Definition of the s and divergences of the N rays at : 2 i Θ i N π θ θ Θ i i ( ) x i x Θ i ( ) Or in vector notation: i R x i θ i After propagation of the beam from to the and divergency follow from: R( ) M 2 ( ) M L2 M ( ) M L M ( ) R The s and divergences of the N rays at are: ( ) ( ) x( ) R( ) T θ ( ) R( ) T i.. max Z i + i X i, i x Z i i

6 Fig. 2a. x() as a function of for the N rays demonstrating the propagation of a Gaussian beam. Expanded vertical axis. This is the ray trace for the modified fiber optic laser head Fig. 2b. x() as a function of for the N rays demonstrating the propagation of a Gaussian beam. Scale. This is the ray trace for the modified fiber optic laser head. True Fig. 2c. x() as a function of for the N rays demonstrating the propagation of a Gaussian beam. Highly expanded focal one. This is the ray trace for the modified fiber optic laser head. Notice that the "one of confusion" seems to show a useful "circular ring" pattern.

7 θ( ) i θ( )i x( ) i x( )i..5 Fig. 3. The s and divergences at several values of. θ( 2 )i x( 2 )i It is illustrative to calculate the beam quality which remains constant according to Liouvilles theorem and should be unity for a Gaussian beam (diffraction limited): For this we need the first moment of xθ and the second moments of x and θ: xθ mean ( ) N N x( ) θ N ( ) ( i ) i x var ( ) N ( x( ) i ) 2 θ var ( ) i = i = The beam quality (or M 2 factor, or 'Times Diffraction Limited' factor) follows from: N N i = ( θ( ) i ) 2 BeamQuality( ).2 2 π λ x var ( ) θ var ( ) xθ mean ( ) 2 5 θ var ( ). x var ( ) 5 xθ mean ( ) Fig. 4. The first and second moments as a function of

8 2 BeamQuality( ) Fig. 5. The beam quality as a function of (this should be unity for a diffraction limited Gaussian beam) The beam radius (), and width (envelope), w(), can be calculated from the moments: x var ( ) R( ) w( ) 2 x var ( ) xθ mean ( ) 2 5 R( ) w( ) w( ) Fig. 6a. The beam radius and width calculated from the moments.

9 w( ).2 w( ) Fig. 6b. The beam radius and width calculated from the moments w( ).2 w( ) Fig. 6c. The beam radius and width calculated from the moments. Last we calculate the diffraction limited beam waist for an ideal lens with parameters as follows (from O'Shea page 232): convergence double angle: theta_degrees 5 π theta_rad theta_degrees theta_rad =.87 rad λ = 3 mm 8 4λ Dwaist Dwaist =.5 mm π theta_rad the Rayleigh range is: Dwaist Range Range =.67 mm theta_rad The calculated beam waist and Rayleigh range are in good agreement with the previous calculations

21. Propagation of Gaussian beams

21. Propagation of Gaussian beams 1. Propagation of Gaussian beams How to propagate a Gaussian beam Rayleigh range and confocal parameter Transmission through a circular aperture Focusing a Gaussian beam Depth of field Gaussian beams and