Question: For this problem, although it is incorrect, treat the plasma diameter as equal to the spot size generated by the laser.

The actual diameter of the plasma is related to the incident field power and approaches the spot diameter as the power of the incident field is increase several times over threshold.

For a calculation at the level that we are analyzing this problem at, we can assume that the source size is equal to the laser spot size.

Laser-induced breakdown spectroscopy (LIDS) is a type of spectroscopy that uses plasma generated by a laser pulse. A schematic of a LIBS system is shown in the figure 1. The laser is focused on the sample and forms plasma, which atomizes and excites samples. LIBS has the capabilities of analyzing any material regardless of its composition (solid, liquid or gas). When a material breaks down and forms plasma of sufficient temperature, it emits light at characteristic frequencies (spectrum) that corresponds to the material.

LABS can (in principle) detect all elements, limited only by the power of the laser and the spectral sensitivity of the spectrometer & detector.

The detection limit of the LIBS spectrometer is a function of

a) the plasma temperature

b) spectral pass band of the optical system, and

c) the strength of the spectral emission being viewed.

The LIBS system presented in Figure loperates by focusing the laser onto a small part of the sample with sufficient power to ablate a very small amount of material. Typically, only a few nanograms of material ablated from the sample. This ablated material generates a plasma plume with temperatures in excess of 100,000 K.

This plasma quickly cools to a temperature in the range of 5,000-20,000 K. At these temperatures the plasma plume is imaged. When the plasma is first farmed it emits a continuum of light (i.e. white light). When the plasma cools and expands the spectral lines of the elements present in the pl asma start to be observable.

The delay between viewing the emission of continuum radiation and the spectral signature of the materials present is approximately 10 μs from the laser pulse.

Design

System Understanding

Referring to Figure describe the purpose of the following subsystems, identifying each component in the path.

Transnnission Path

The laser source is a 1064nm Nd:YAG laser with a pulse width of 10ns. The average output power of the laser is 750W. The waist location is at the output window of the laser and has a diameter of 1mm.

Assuming that it takes an intensity of 3.4 GW/cm² FWHM to ablate the surface of the material of interest, what is the diameter of the beam on the sample?

What F# is necessary to achieve this spot size? If the focal length of the lens is 100mm what is the Diameter of the lens (f3)? If the output of the laser has a diameter of 1mm and is a waist of the laser, determine the distances Z₁ and Z₂. Determine the distance Z₃.

Calculate the radius of the beam assuming a Gaussian profile. Assuming it takes more than 75% of the peak intensity to form plasma, determine the tolerance on the sample location. (How accurate do you need to place the sample to get a signal?)

Receiver Path

One of the advantages of LIB S is that the spectrometer that is used to analyze the plasma spectrum can be separated from the event by a distance with the use of a fiber optic cable. Assume for this design that the fiber optic cable is an 81125 (8 !um core/ 125 inn cladding) cable made of fused silica.

For the purpose of this problem take the index of refraction of the core to be 1.46 and the cladding to be 1.42. What is the NA, the acceptance angle, θ_c, and θ'_c, of the fiber?

Using this information, what is the focal length of the lens used to focus the light from the plasma into the fiber? At the other end of the fiber is a grating, lens, and linear CCD array.

Sketch the configuration of this system used to collect the light diffracted by the diffraction grating onto the CCD Array. The grating has a A = 2000nm. What is the range of angles diffracted for a wavelength range from 400nm to 1000nm?

Assuming that each pixel of the CCD array is 5 microns and that there are 4096 pixels in the linear array.

What is the focal length of the lens? What is the wavelength resolution for the spectrometer?