Group velocity dispersion memo
Mode-lock pulsed laser has broad wavelength spectrum due to the Heisenberg principle:
Δν τ = τ cΔλ/(λ2)~ 0.44 (FWHM). For 100fs, the wavelength width (Δλ) is ~8nm at 800nm wavelength.
Because light with different wavelength propagates with different velocity, the dispersion of pulsewidth occurs.
Phase shift of light in some material is:
Φ (ω) = n(ω)Lω/c (1)
where L is the length of material, n is the refractive index and c is the speed of light. Φ(ω) can be described as a power seriese as:
Φ(ω) = Φ(ω0) + (dΦ(ω)/dω) *(ω-ω0) + (d2Φ(ω)/dω2) * (ω-ω0)2
The first and seond term does not change the pulse shape. The third term causes pulse broadening. The second order differentiation of Φ can be obtained from equation (1) as:
(d2Φ(ω)/dω2)L-1 = λ3(2πc2) -1 (d2n/dλ2)
The left term is refered to as group velocity dispersion constant, D. Typical value is 100-400 fs^2/cm for silica or BK7 and 1000-2000 fs^2/cm for SF10 or SF11. Pulse broadening is expressed as:
τout = τin[1 + 7.68*(DL)2 τin-4]1/2
=τin[1+205*(DL)2 (cΔλ / λ2)4]1/2
Here 7.68 = (2*sqrt(ln2))^4 is to covert gaussian width parameter to FWHM. DL of typical microscope setting is 3,000-10,000 fs2. Compensation can be done using a pair of prism with distance of Lprism obtained by:
Dprism = -2Lprismλ3(πc2)-1(dn/dλ)2
For example, high dispersive materials such as SF-10 or SF-11 have dn/dλ = 40,000-80,000 m-1 at 800-900nm. Thus, Lprism~30-100cm is required to compensate the dispersion caused by a microscope.
---
Sellmeier equation
λ dependency of dispersive materials can be described as:
n2=1+Σ(Aiλ2)/(λ2-Bi)
A1,A2,A3,B1,B2,B3
SF10 1.61625977 0.25922933 1.07762317 0.0127535 0.05819840 116.607680
SF11 1.73848403 0.31116897 1.17490871 0.0136069 0.06159605 121.922711
References:
V.Iyer et al. J.Biomed.Optic 2006
(http://sensor.bcm.tmc.edu/saglab/pdf/JBO_Iyer.pdf)
Newport web page
(http://www.newport.com/store/genproduct.aspx?id=141161〈=1033§ion=Detail
)
Mellesgriot web page
http://www.mellesgriot.com/products/optics/mp_3_1.htm
Wikipedia Sellmeier equation
http://en.wikipedia.org/wiki/Sellmeier_equation
BCP webpage
http://bcp.phys.strath.ac.uk/ultrafast/dictionary/dispersion%20and%20pulse%20broadening/dispersion%20and%20pulse%20broadening.html
Δν τ = τ cΔλ/(λ2)~ 0.44 (FWHM). For 100fs, the wavelength width (Δλ) is ~8nm at 800nm wavelength.
Because light with different wavelength propagates with different velocity, the dispersion of pulsewidth occurs.
Phase shift of light in some material is:
Φ (ω) = n(ω)Lω/c (1)
where L is the length of material, n is the refractive index and c is the speed of light. Φ(ω) can be described as a power seriese as:
Φ(ω) = Φ(ω0) + (dΦ(ω)/dω) *(ω-ω0) + (d2Φ(ω)/dω2) * (ω-ω0)2
The first and seond term does not change the pulse shape. The third term causes pulse broadening. The second order differentiation of Φ can be obtained from equation (1) as:
(d2Φ(ω)/dω2)L-1 = λ3(2πc2) -1 (d2n/dλ2)
The left term is refered to as group velocity dispersion constant, D. Typical value is 100-400 fs^2/cm for silica or BK7 and 1000-2000 fs^2/cm for SF10 or SF11. Pulse broadening is expressed as:
τout = τin[1 + 7.68*(DL)2 τin-4]1/2
=τin[1+205*(DL)2 (cΔλ / λ2)4]1/2
Here 7.68 = (2*sqrt(ln2))^4 is to covert gaussian width parameter to FWHM. DL of typical microscope setting is 3,000-10,000 fs2. Compensation can be done using a pair of prism with distance of Lprism obtained by:
Dprism = -2Lprismλ3(πc2)-1(dn/dλ)2
For example, high dispersive materials such as SF-10 or SF-11 have dn/dλ = 40,000-80,000 m-1 at 800-900nm. Thus, Lprism~30-100cm is required to compensate the dispersion caused by a microscope.
---
Sellmeier equation
λ dependency of dispersive materials can be described as:
n2=1+Σ(Aiλ2)/(λ2-Bi)
A1,A2,A3,B1,B2,B3
SF10 1.61625977 0.25922933 1.07762317 0.0127535 0.05819840 116.607680
SF11 1.73848403 0.31116897 1.17490871 0.0136069 0.06159605 121.922711
References:
V.Iyer et al. J.Biomed.Optic 2006
(http://sensor.bcm.tmc.edu/saglab/pdf/JBO_Iyer.pdf)
Newport web page
(http://www.newport.com/store/genproduct.aspx?id=141161〈=1033§ion=Detail
)
Mellesgriot web page
http://www.mellesgriot.com/products/optics/mp_3_1.htm
Wikipedia Sellmeier equation
http://en.wikipedia.org/wiki/Sellmeier_equation
BCP webpage
http://bcp.phys.strath.ac.uk/ultrafast/dictionary/dispersion%20and%20pulse%20broadening/dispersion%20and%20pulse%20broadening.html
posted by Ryohei at 8:03 PM
<< Home