Photothermal Lens Spectrometry
Within a Cylindrical Cell

A synopsis of the paper presented at the 9th International Conference on Photoacoustic and Photothermal Phenomena, Nanjing, P. R. China, 1996

Agnès Chartier and Stephen Bialkowski

email: agnes@cc.usu.edu , Stephen.Bialkowski@usu.edu

This work was susbsequently published in Optical Engineering, 36 303-311 1997, under the title "Photothermal lens spectrometry of homogeneous fluids with incoherent white-light excitation using a cylindrical sample cell".

Difference Between Laser- and Lamp-Excited Apparatus

With the laser-excited photothermal lens apparatus, the laser beam is focused to a small volume. This spatial anisotropy results in an anisotropic temperature profile. This, in turn, results in thermal lens formation.

In the cylindrical sample cell apparatus, a small volume cell is illuminated by the image of an excitation source. The sample cell is illuminated with spatially constant irradiance. Thermal diffusion results in the spatial anisotropic temperature change. This, in turn, produces the thermal lens element.

Features of Photothermal Lens Spectrometry Using the Cylindrical Sample Cell

• Excitation source may be either conventional (incoherent) or laser
  
• Incoherent "white light" excitation provides non-selective analyte detection
  
• High sensitivity follows from the use of laser probes of the resulting photothermal lens
  
• Photothermal lens signals are independent of excitation source beam parameters
  
• Photothermal lens signals are produced only by thermal diffusion
  
• Volume changes from excited states or photolysis products do not affect signal

Cylindrical Cell Thermal Lens Theory

1 - The thermal diffusion equation

is solved for zero temperature change at the cylinder boundary.

2 - The temperature change, dT (K), for continuous excitation is

where a is the cell radius, K_f is thermal conductivity of the sample medium, D_T is thermal diffusivity, and the X_n are roots of the Bessel's function equation, J₀(X_n)=0.

3 - The thermal lens strength obtained from

is

where t_c=a²/4D_T is the characteristic thermal diffusion time constant and

is the heat generated along the optical path. Y_H is the quantum yield for heat production.

4 - The maximum thermal lens strength for times much longer than the characteristic thermal diffusion time is

5 - The decay of the lens can be similarly modeled for chopped sources.

6 - Accounting for the finite thermal conductivity of the cell (radiative heat transfer solution) yields the more complex solution

where h~K_s/K_fd (m^-1), is a relative thermal diffusion parameter, K_s is sample cell thermal conductivity, and d (m) is the thermal diffusion layer thickness.

7 - The long-time limited signal is the same as that for the "infinite" thermal conductivity approximation given in Point 4. However, the time constant for lens formation and decay depends on the thermal transfer characteristics of the sample cell.

Important Points

• Lens formation is due to thermal diffusion out of a cylindrically illuminated region.
  
• Refractive index changes due to physical properties other than temperature change will not result in a lens, i.e., they don't diffuse through the sample cell wall.
  
• The maximum (long-time) thermal lens strength is:
  1. independent of sample radius
  2. scales linearly with excitation irradiance through the q_H term
  3. independent of sample cell thermal conductivity.
     
• The maximum thermal lens strength result is equivalent to that obtained for Gaussian laser-beam excitation with same irradiance.
  
• The theoretical enhancement factor is equivalent to that of the two-laser photothermal lens apparatus with equivalent irradiance.

Experimental Apparatus

Schematic of the apparatus used in these experiments. The optically filtered Xe arc lamp excitation source passes through a shutter and is imaged onto the entrance of the cylindrical sample cell. The near infrared diode probe laser is paraxially combined with the excitation source at the hot mirror, and is subsequently focused through the sample cell. The probe laser passes through the laser line filter and the pinhole aperture to develop the photothermal lens signal at the photodiode detector. The electronic signal is captured with an analog-to-digital converter and subsequently processed using the personal computer.

White-Light Lens Experiment

• Uses a 1 kW broad-band, "Cermax," light source.
  
• A 6 µL (a=0.5 mm, l=1 cm) HPLC optical flow cell is used with continuous sample flow.
  
• Time-dependent "chopped" data collected with analog-to-digital converter.
  
• Time-dependent signals processed with matched filter or ensemble averaging.

White Light Photothermal Lens Results^*

• Measured photothermal lens signal magnitudes correspond well to those calculated based on sample optical absorbance and the thermal conductivity of the solvent
  
• Signal to noise ratios were better than conventional spectrophotometry.
  
• This was attributed to:
  a - relatively high average transmission across the excitation wavelength range
  b - relatively low probe laser shot noise
  
• The integrated absorbance detection limit was 2x10^­5 AU for 100 transient averages
  
• Transient signal analysis was used to determine both the sample and the sample cell thermal conductivities (K_f and K_s)

* Measurements made using 7.5x10³ W m^­2 (~6 mW through the 1 mm internal diameter stainless steel sample cell) distributed from 700 nm to 375 nm on pseudoisocyanin at average transmission of T~0.9 across the wavelength range.