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
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