How do I solve the following problem step by step? and what are the solution? I

Question

How do I solve the following problem step by step? and what are the solution? I am really lost, and do not understand the whole exercise..

A femtosecond laser generated light pulses with a duration of only 100 fs (ie 1.00 · 10-13 s). With an ordinary lens, such a laser beam focused down to a spot with a diameter of 10 m. A typical energy of a laser pulse from such lasers is 1 mJ. Suppose that energy is uniformly distributed over the laser spot, and the effect is constant during the pulse (a square pulse).

1.Find the intensity of the laser focus.

2. Find the amplitude, Emax, of the electric field in focus.

3. Compare this amplitude with the size of the electric field acting on an electron around the nucleus of a hydrogen atom in ground state (Hint: the distance between nucleus and electron is given by the size a0 = 5.29 · 10-11m) What does the comparison of the laser's effect on the atom

4. The laser light at a wavelength of 800 nm. Calculate how many oscillations the electromagnetic field undergoes during the pulse duration. Do you think that the frequency of such a laser will be more or less clearly defined than it is for a more traditional laser, such as a helium-neon laser? Argument and explain your answer, please.

Explanation / Answer

1) The intensity (irradiance) is the energy per unit area per unit time usually in Watts/cm^2 10 microns is .01mm. .01 mm is .001cm Here we have .001J/[1X10^-13s*Pi*(.001cm)^2] = 3 x 10 ^15 W/cm^2 =3 x 10^11W/m^2 2) The amplitude of the electric field. Irradiance is proportional to the square of the amplitude of the electric field I = e0c E = SQRT(I/e0c) = SQRT(3x10^11W/m^2/[8.85 x 10-12 C^2/Nm^2*3x10^8m/s]) lets look at units W = (kgm/s^2)*(m)/s so inside the SQRT we have {kgm^2/(m^2s^3)}/{(C^2m)/(Nm^2s)} = (Nskgm/s^3)/C^2 = (N kgm/s^2)/C^2 = N^2/C^2 E = 11,000,000 N/C 3) This is a big e-field, but it is hard to beat an atom. force on an electron in the hydrogen atom = (1/4Pi e0)*q1q2/r^2 = 9.0 x 10^9N*m^2/C^2(1.6 x 10^-19 C)^2/(5.29 x 10^-11m)^2 = 8.1 x 10-8 N Field is F/q = 8.1 x 10-8N/1.6 x 10^-19 C = 5.1 x 10^11 N/C = 510,000,000 N/C Other atoms in air are larger. Focused laser beams can break down air and form plasmas. 4) How far does light travel in 100 fs? Distance = rate times time D = 3 x 10^8m/s*1 x 10^-13s = 0.00003 m = 30 microns. The wavelength is 0.8 microns There are 37.5 oscillations. This is much less well defined than a CW laser such as a He:Ne laser. To form such a short pulse, more wavelengths are needed (if you know Fourier series, this is where it comes in, to make a pulse, one needs many different frequency components.) When working with ultra short pulse lasers, special broadband optical coatings are needed to have the same effect on all of the wavelength components in the pulse. This is a real-world application of the less clearly defined frequency of femto second lasers. Stated another way: Many nearby frequencies are more or less in step for 40 cycles before a beat is detected.

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