### Laser lecture 04

• 1. 14/06/2015 1 6/14/20151 412 PHYS Lasers and their Applications Department of Physics Faculty of Science Jazan University KSA Propagation of Gaussian Beams & Optical Resonators Lecure-4 Propagation of laser beams cab be described by the Helmholtz equation one of the possible solutions to this equation is given by 2 2 0 ( ) ( , , ,) exp { ( ) } 2 ( ) k x y x y z i p z q z       0 Constant depends on the beam amplitude and can be determined by boundary conditions zki ezyxzyxU   ),,(),,( 0),,()( 22  zyxUk
• 2. 14/06/2015 2 ( ) ln(1 / )op z i z q  )(zp ( )q z complex beam parameter, given as 2 0 0 0( ) i w q z z iz z q z         Complex function given as Rayleigh range at which he beam waist is given by0z 0( ) 2w z w 0w Is the minimum beam waist For a spherical wave 0 2 2 2 2 2 0 0 0 1 1 1 ( ) ( ) zz i i q z iz z z z z R z n w z           The real part represents the wavefront curvature, with a radius given by 2 0( ) [1 ( / ) ]R z z z z  )(z Beam waist radius, given by 2 1/2 0 0( ) [1 ( / ) ]w z w z z  0w Is the minimum beam waist radius, given in terms of the beam divergence as 1/2 0 0 log(2) 2 tan( / 2) z w                 
• 3. 14/06/2015 3 Laser resonators  The feedback in lasers is achieved by placing the amplifier (active medium) between mirrors, a construction we call an optical cavity or resonator. The resonator is the space of optical amplifier that contains the feedback elements The resonator When the population inversion occurs in the active medium, the spontaneous emission produces a photon that propagates along the optical axis of the active medium and the resonator The photons interacts with the excited atoms and the stimulated emission will occur and hence a wave with amplified amplitude will propagate through the medium towards one of the mirrors Upon reflection from the mirror, the wave will be further amplified by passing through the medium due to the resonance with the excited atoms (because they both have same energy ) Eventually the wave will be oscillated between mirrors and get amplified in every pass and loss some photons in the output mirror as the output beam
• 4. 14/06/2015 4 Laser resonator stability 1 2 0 1 1 1L L r r            12 r،r Radii of mirror’s curvatures L Length of the resonator The Condition for resonator stability • The cavity is an essential part of a laser. It provides the positive feedback that turns an amplifier into an oscillator. • The design of the cavity is therefore very important for the optimal operation of the laser. Types of resonators Plane parallel resonator Confocal resonator Hemispherical resonator Large radius resonator Concentric resonator
• 5. 14/06/2015 5 Example: Sol. Determine whether or not the following mirror arrangements lead to stability: a. Two mirrors with radii of curvature of 1.8 m, separated by a distance of 2 m b. One mirror with radius of curvature of 2m and the other with radius of 3m, separated by a distance of 2.3m c. One mirror with radius of curvature of 5m and the other with radius of 3m, separated by a distance of 4m d. Two mirrors with radius of curvature of 0.5 m, separated by a distance of 0.5m 1 20 (1 / )(1 / ) 1L r L r    a. r1=r2=1.8m, L=2m 2 2 1 2(1 / )(1 / ) (1 2 /1.8) (1 1.11) 0.121L r L r        Cavity is stable b. r1=2 m, r2=3m, L=2.3m 1 20 (1 / )(1 / ) 1L r L r    (1 2.3/ 2)(1 2.3/3) ( 0.15)(0.25) 0.0345      Cavity is unstable c. r1=5 m, r2=3m, L=4m (1 4/5)(1 4/3) (0.2)( 0.33) 0.067      Cavity is unstable d. r1=0.5 m, r2=0.5m, L=0.5m (1 0.5/ 0.5)(1 0.5/ 0.5) 0   Cavity is on edge of stability ---confocal cavity
• 6. 14/06/2015 6 Laser modes •The cavity determines the properties of the beam of light that is emitted by the laser. • This beam is characterized by its transverse and longitudinal mode structure. Transverse modes are created in cross section of the beam, perpendicular to the optical axis of the laser. Longitudinal modes only specific frequencies are possible inside the optical cavity of a laser, according to standing wave condition. Transverse mode structure A transverse mode is a field configuration on the surface of one reflector that propagates to the other reflector and back, returning in the same pattern, apart from a complex amplitude factor (that gives the total phase shift and loss of the round trip.
• 7. 14/06/2015 7 Separating the fundamental Gaussian mode Lasers operating with the fundamental Gaussian mode TEM00 are preferred due to the following reasons: 1. TEM00 has a symmetrical, uniform circular configuration with the greatest intensity at its center: this suits many applications that requires high accuracy 2. Contains about 85% of the total output intensity 3. Can be easily separated from higher order modes by using a pin hole aperture with a diameter that allow only photons propagated along the optical axis to be incident on mirrors It is preferred to get the laser operated in the fundamental mode TEM00 The intensity of the Gaussian mode is given by   2 2 0 0exp( 2 / )I r I w w  The total power 2 / 2P w I 02 /div w  Divergence 0w Radius of Gaussian beam (the radius at which the intensity reduced by 2 /1 e
• 8. 14/06/2015 8 Example 1: A He-Ne Laser with a Gaussian fundamental mode TEM00 operates with a wavelength of 632.8 nm . A lens is used to collimate the beam to pass through a glass plate of a thickness of 12 mm and a refractive index of 1.46. suppose the plate is place at the focal point of the lens where the beam diameter is 2m Calculate the beam diameter at the other end of the plate? 0 0 2 12 0.012 , 632.8 , 2 1 2 2 d m z mm m nm d w w m             1 2 2 0 0 1 z w z w z                 2 62 60 0 9 1 2 2 6 3 6 3 1.46 10 7.25 10 632.8 10 0.012 10 1 1.66 10 7.25 10 beam dimeter 2 ( ) 3.32 10 n w z m w z m w z m                                  Sol. We have A fundamental Gaussian beam from a Ti:Sapphire laser with a wavelength of 759nm and a power of 1mW incident on a target far from the minimum waist point by 100m. If the radius of the minimum waist is 2mm, find the beam waist at the target? Calculate the radius of wavefront curvature and peak intensity of the beam Sol. Waist radius at the target 1/22 2 3 2 0 0 0 9 0 (2 10 ) ( ) 1 , ( ) 3.14 16.55 759 10 wz w z w z Rayleigh range z                     1/22 3 100 ( ) 2 10 1 12 16.55 w z mm               Example 2:
• 9. 14/06/2015 9 Usually the spot size is represented by the beam waist and the area of the laser spot (if it is circular) is give by 2 3 2 ( ) 3.14 12 10 0.0314A w z m       The radius of wavefront curvature is 2 2 0 16.55 ( ) [1 ( / ) ] 100 1 102.74 100 R z z z z m               peak intensity 3 3 2 2 3 2 1 10 4.4 10 / ( ) / 2 3.14 (12 10 ) / 2 p P P I W m A w z            Note: is the effective area 2 { ( ) / 2}A w z  The longitudinal modes determine the emission spectrum of the laser. The light bouncing repeatedly off the end mirrors sets up standing waves inside the cavity. LONGITUDINAL MODES More general where n is the average refractive index of the cavity. The last Equation implies that only certain frequencies which satisfy will oscillate
• 10. 14/06/2015 10 Plane-mirror resonators • This type of cavity consists of two flat, parallel mirrors separated by a distance L . It is also called a Fabry-Perot resonator (F-P). • In this type of cavity, the beam fills the space between the mirrors nearly uniformly, unlike in the spherical mirror cavities where the beam is focused somewhere inside (or outside of) the cavity. The finesse coefficient of the Fabry-Perot cavity Is given as 1 2 1/4 1/4 1/2 1/2 1 21 c R R f R R    If the two mirrors have similar reflectivities R1=R2 then 1 c R f R    The finesse coefficient has great importance for determining the fluorescence line shape  /cf     Frequency spacing between modes  Laser fluorescence line width 2 (2 )kL m  2 4 2 (2 ) n L L kL m C        Note, that the condition of standing wave is 2 m L   2 m c m n L   Frequency spacing between modes or Free Spectral Range 1 2 m m c n L     Phase displacement for one round trip oscillation Frequency of the mode of order m Derivation of the number of Longitudinal modes
• 11. 14/06/2015 11 2 22 c or n Ln L    The number of operating longitudinal modes is given by ( / 2 ) m c n L   The quality factor of Fabry-Perot resonator is 0 Q     Calculate the width of the frequency mode for a Fabry-Perot resonator consists of two similar plane mirrors separated by a distance of 1 cm Assume the reflectivities 70% 99.9% 95% Calculate the finesse factor for the wavelength of 800 nm from a GaAs laser And find also the value of the resonator quality factor for each case? 8 103 10 1.5 10 2 2 1 0.01 c Hz n L         8 14 9 3 10 3.75 10 800 10 c Hz          1 0.9999R For 10 4 5 4 0.9999 1.5 10 3.141 10 4.78 10 1 0.9999 3.14 10 c c f Hz f                Example 3: Sol.
• 12. 14/06/2015 12 Resonator quality 14 80 5 3.75 10 7.85 10 3.78 10 Q          2 0.95R For 10 80.95 1.5 10 61.2 2.45 10 1 0.95 61.2 c c f Hz f              14 60 8 3.75 10 1.53 10 2.45 10 Q          For 3 0.75R  10 9 14 5 9 0.75 1.5 10 10.87 1.38 10 1 0.75 10.87 3.75 10 2.72 10 1.38 10 c c f Hz f Q                        Example 4: For a He-Ne laser with a wavelength of 632.8 nm, if the length of the resonator is 30 cm, find: a. Frequency difference between longitudinal modes (mode spacing b. Number of modes c. Frequency of the laser light Sol. 8 3 10 0.5 2 2 0.3 c GHz L       a. Mode spacing b. Number of modes 6 6 2 2 2 0.3 0.948 10 0.6328 10 m L L m m            c. Frequency of the laser 6 9 14 0.948 10 0.5 10 4.74 10m Hz         or 8 14 6 3 10 4.74 10 0.6328 10 c Hz        
• 13. 14/06/2015 13 Note: 1) and 2) allow only low powers to be obtained (of no practical use) 1. Reducing laser cavity length to make the mode spacing large and hence allow only one mode to operate 2 c n L   2. Reduce pumping power and hence allow the amplification only for the central mode Laser operation in single longitudinal mode There are many methods that can be used to force the laser oscillation in single longitudinal mode 4. Using Prism to select one longitudinal mode 5. Using grating to select one longitudinal mode 3. Using Etalon Generate additional losses for the extra modes by placing frequency selective optical elements in the laser resonator The lasing mode gets some of the gain of the killed modes higher power/mode
• 14. 14/06/2015 14 Example 5: An ion Argon laser with a wavelength of 514.5 nm with a spectral bandwidth of 2GHz and a cavity length of 50 cm. find the number of possible longitudinal modes? Sol. 9 8 2 2 10 6mod / 2 3 10 / (2 50 10 ) m es c L           Refractive index of argon gas =1 Example 6: If a spectral filter with a bandwidth of 0.1 nm is used to obtain a single longitudinal mode from a He-Ne laser, what should be the length of the laser cavity? 8 9 10 2 9 2 3 10 (0.1 10 ) 7.5 10 (632.8 10 ) c Hz              Sol. The fluorescence frequency bandwidth is given by For a single mode oscillation, we should have 8 10 3 10 0.002 2 2 7.5 10 c L L m L          