### Group velocity dispersion memo

Mode-lock pulsed laser has broad wavelength spectrum due to the Heisenberg principle:

Δν τ = τ cΔλ/(λ

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(ω

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:

(d

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:

τ

=τ

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 fs

D

For example, high dispersive materials such as SF-10 or SF-11 have dn/dλ = 40,000-80,000 m

---

λ dependency of dispersive materials can be described as:

n

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

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/(λ

^{2})~ 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(ω

_{0}) + (dΦ(ω)/dω) *(ω-ω_{0}) + (d^{2}Φ(ω)/dω^{2}) * (ω-ω_{0})^{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:

(d

^{2}Φ(ω)/dω^{2})L^{-1}= λ^{3}(2πc^{2})^{-1}(d^{2}n/dλ^{2})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:

τ

_{out}= τ_{in}[1 + 7.68*(DL)^{2}τ_{in}^{-4}]^{1/2}=τ

_{in}[1+205*(DL)^{2}(cΔλ / λ^{2})^{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 fs

^{2}. Compensation can be done using a pair of prism with distance of L_{prism}obtained by:D

_{prism}= -2L_{prism}λ^{3}(πc^{2})^{-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, L_{prism}~30-100cm is required to compensate the dispersion caused by a microscope.---

**Sellmeier equation**λ dependency of dispersive materials can be described as:

n

^{2}=1+Σ(A_{i}λ^{2})/(λ^{2}-B_{i})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