激光原理(英文)
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Laser gain and losses
Ruby laser example
Laser turn-on and gain saturation
Gain decreases as output power increases • Saturation
Fabry-Perot cavity for feedback
– field is zero on mirrors
• Multiple wavelengths possible
– agrees with resonance conditioຫໍສະໝຸດ Baidus
Classical mechanics analog
Fabry-Perot boundary conditions
Multi-mode laser
Laser basics
Optics, Eugene Hecht, Chpt. 13; Optical resonator tutorial:
http://www.dewtronics.com/tutorials/lasers/leot/
Laser oscillation
Laser is oscillator • Like servo with positive feedback • Greater than unity gain
• Zero order mode is Gaussian
• •
Intensity profile: beam waist: w0
I
I e2r2 / w2 0
2
w w0
1
lz w02
• confocal parameter: z
zR
w02 l
• far from waist
• Wave equation looks like harmonic oscillator
• Ex: E = E e -iwt 2E nw 2 E 0
c
d2x k
dt 2
x0 m
• Separate out z dependence
2E z 2
2ik
E z
Multiple resonant frequencies
Single longitudinal mode lasers
• Insert etalon into cavity • Use low reflectivity etalon
– low loss
Laser transverse modes
• High reflectivity mirrors • Low loss per round trip • Must remember resonance conditions
– round trip path is multiple of l
Laser longitudinal modes
• High reflectivity Fabry-Perot cavity • Boundary conditions
• Relation is dFWHM = w 2 ln2 ~ 1.4 w • Define average intensity
• Iavg = 4 P / ( d2FWHM) • Overestimates peak: I0 = Iavg/1.4
Resonator options
• Best known -- planar, concentric, confocal • Confocal unique
2E x 2
2E y 2
wn c
2
k2
E
0
• Solutions for x and y are Hermite polynomials
Transverse laser modes
Frequencies of transverse modes
Single transverse mode lasers
• Put aperture in laser • Create loss for higher order modes
Multi-longitudinal Multi-transverse&long. Single mode
Gaussian beams
w lz w0
• divergence angle
2l 0.637 l
w0
w0
Gaussian propagation
Power distribution in Gaussian
•
Intensity distribution:
I
I e2r2 / w2 0
• Experimentally to measure full width at half maximum (FWHM) diameter
– mirror alignment not critical – position is critical – transverse mode frequencies identical
Special cases
Types of resonators
Ruby laser example
Laser turn-on and gain saturation
Gain decreases as output power increases • Saturation
Fabry-Perot cavity for feedback
– field is zero on mirrors
• Multiple wavelengths possible
– agrees with resonance conditioຫໍສະໝຸດ Baidus
Classical mechanics analog
Fabry-Perot boundary conditions
Multi-mode laser
Laser basics
Optics, Eugene Hecht, Chpt. 13; Optical resonator tutorial:
http://www.dewtronics.com/tutorials/lasers/leot/
Laser oscillation
Laser is oscillator • Like servo with positive feedback • Greater than unity gain
• Zero order mode is Gaussian
• •
Intensity profile: beam waist: w0
I
I e2r2 / w2 0
2
w w0
1
lz w02
• confocal parameter: z
zR
w02 l
• far from waist
• Wave equation looks like harmonic oscillator
• Ex: E = E e -iwt 2E nw 2 E 0
c
d2x k
dt 2
x0 m
• Separate out z dependence
2E z 2
2ik
E z
Multiple resonant frequencies
Single longitudinal mode lasers
• Insert etalon into cavity • Use low reflectivity etalon
– low loss
Laser transverse modes
• High reflectivity mirrors • Low loss per round trip • Must remember resonance conditions
– round trip path is multiple of l
Laser longitudinal modes
• High reflectivity Fabry-Perot cavity • Boundary conditions
• Relation is dFWHM = w 2 ln2 ~ 1.4 w • Define average intensity
• Iavg = 4 P / ( d2FWHM) • Overestimates peak: I0 = Iavg/1.4
Resonator options
• Best known -- planar, concentric, confocal • Confocal unique
2E x 2
2E y 2
wn c
2
k2
E
0
• Solutions for x and y are Hermite polynomials
Transverse laser modes
Frequencies of transverse modes
Single transverse mode lasers
• Put aperture in laser • Create loss for higher order modes
Multi-longitudinal Multi-transverse&long. Single mode
Gaussian beams
w lz w0
• divergence angle
2l 0.637 l
w0
w0
Gaussian propagation
Power distribution in Gaussian
•
Intensity distribution:
I
I e2r2 / w2 0
• Experimentally to measure full width at half maximum (FWHM) diameter
– mirror alignment not critical – position is critical – transverse mode frequencies identical
Special cases
Types of resonators