Optical probe for spatially resolved plasma emission and absorption spectroscopy Gerald L. Rogoff

GTE Laboratories, 40 Sylvan Road, Waltham, Massachu­setts 02254. Received 10 January 1985. 0003-6935/85/121733-03$02.00/0. © 1985 Optical Society of America. Spatially resolved plasma emission and absorption mea­

surements can provide valuable data for characterizing local conditions and processes. Two-dimensional resolution can be obtained with external collimators,1 while 3-D resolution can be obtained in some cases by using laser-induced fluo­rescence2 or, if the plasma is optically thin and rotationally symmetric, by mathematically inverting external emission scans by Abel inversion.3 Optical fiber and fine rod probes have also been used to obtain 2-D spatial resolution in plasma emission studies with the probe external to the plasma4,5 and with the probe inserted into the plasma.6 This Letter intro­duces an optical-probe method for obtaining 3-D resolution rather than 2-D resolution. Furthermore, it can be used for measuring both emission and absorption coefficients, and it can be applied to plasmas of arbitrary shape and optical thickness. The method utilizes an optical probe extended into the plasma and moved along its direction of view; the differential equation of radiative transfer is used to relate measured intensities to local emission and absorption coef­ficients. For discussion purposes the probe is considered to be an optical fiber or fine rod. As discussed later, however, it can also take other forms.

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Fig. 1. Schematic illustration of the experimental arrangement. For the calculations the origin is located on the optical axis at or beyond the plasma boundary. The end of the movable probe is located at x = p. A spectral filter or monochromator may be located between the probe and the detector, which are shown connected by the fiber for

illustrative purposes.

The experimental arrangement is shown schematically in Fig. 1. An optical fiber or rod probe is inserted into a plasma which may be nonuniform and of arbitrary shape. The plasma is characterized by an emission coefficient ε(x,y,z) (power radiated per unit volume per unit solid angle per unit wavelength interval) and an absorption coefficient α(x,y,z) (fractional decrease in power per unit distance traveled). For simplicity, the plasma is taken to be isotropic and of uniform refractive index. Also, the plasma boundaries are assumed to be sufficiently diffuse and the refractive index sufficiently close to unity for reflections at the boundaries to be neglected. For convenience in this discussion rectangular coordinates are used, with the origin located at or beyond the plasma edge farthest along the line of sight of the probe. The location of the probe's end is indicated by p.

The probe transmits incident radiation falling on its flat endface within an effective acceptance solid angle Ω, which must be sufficiently small that ε and α do not vary signifi­cantly perpendicular to the optical axis within the field of view of the probe7; that is, ε and α are functions only of x in the field of view. If ap is the cross-sectional area of the end of the probe, for small Ω the total power accepted is apΩI(p), where I(p) is the intensity associated with a given ray (power per unit area perpendicular to the x direction per unit solid angle per unit wavelength interval). With I0 representing the intensity of an external radiation source,8 I(p) can be written

where Iext(p) is the contribution due to the presence of the external source given by

and Iint) is the contribution from the plasma emission given by

The integrand in Eq. (3) represents the contribution from a volume element at location x, the exponential factor ac­counting for absorption of the radiation between the emitters at x and the probe at p.

If the probe is moved along the x direction, the derivative of I(p) with respect to p can be measured. To obtain an ex­pression for that derivative we can differentiate Eq. (1) by using the Leibniz formula for the derivative of an integral. With the derivative dI(p)/dp at location p denoted by I'(p), the result can be written

This is the equation of radiative transfer9*10 in the plasma, an exact relationship. Equation (4) is the basic equation for the probe operation, relating the quantities I′, I, α, and ε locally in the plasma.

The measured signal is proportional to the power I(p) in­tegrated over the spectral bandwidth of the detection system Δλ. If ε, α, and I0 do not vary with wavelength within that bandwidth, the measured power is apΩI(p)Δλ, where I(p)Δλ represents the integral over wavelength. In this case Eq. (4) can be used directly. If ε, α, or I0 does vary with wavelength, Eq. (4) is replaced by a more general form obtained by inte­grating Eq. (4) over wavelength. That more general expres­sion can be written

where Ia(p) is the average of I(p) over wavelength, εα(p) is the average of ε(p) over wavelength, I'a(p) is the derivative of Ia (p) with respect to p, and α(p) is an average absorption coefficient given by

As indicated above, if e, α, and I0 do not vary with wave­length within the system bandwidth, Eq. (4) may be used to determine ε and α. For a steady discharge I'(p) can be de­termined from a plot of I(p) vs p or from measurements of ƒ(p) at two slightly different values of p. Alternatively, for a time-varying plasma, I′(p) may be determined by rapid modulation of the probe position or by moving the probe be­tween cycles of a repetitive plasma.11 Equation (4) can then be used to determine the plasma radiative characteristics in a variety of ways:

To determine both α and ε at a particular location p, mea­surements of I(p) and I′(p) can be made twice, each time with a different external intensity level I0. That is, I(p) is varied by varying I0. A plot of I′(p) vs I(p) yields a linear relation­ship with -′α(p) as the slope and e(p) as the intercept on the V axis. Algebraically, with the two measurements denoted by subscripts 1 and 2, we have for a given location p, α= (I′1 - I′2)/(I2 - I1) and ε = (I′1I2 - I'1I′2)/(I2 - I1).

If the plasma is optically thin, α is negligible, I0 can be omitted, and ε(p) is given directly by I'{p).

If the plasma is optically thick, I p) will not change with I0. In this case ƒ'(p) = 0, and ƒ(p) corresponds to a local black-body intensity given by I(p) = ε(p)/α(p).

If ε is negligible, an external source I0 can be used to mea­sure α(p) directly.12

If either e(p) or α(p) is known, Eq. (4) can be used to de­termine the other.

If ε, α, or I0 does vary with wavelength within the spectral bandwidth, Eq. (5) must be used, and the extraction of ε and α becomes more difficult. The interpretation of εα is clear, however, and, for example, if the plasma is optically thin, εa(p) is easily obtained from I′a(p). The difficulty of determining a meaningful absorption coefficient from Eq. (6) can be avoided if I0is provided by a source with a linewidth that is small compared with the detector bandwidth—by a laser, for example. If the externally generated beam is chopped to permit selective detection of that radiation, Eq. (4) can be used to measure α(p) with ε(p) set to zero.

Since the optical probe represents an intrusive diagnostic technique, an important consideration is its perturbation of the plasma and the effect of that perturbation on the mea­surement. Since the probe is not electrically conductive, the electrical perturbation13-14 of the plasma should be similar to that of a floating electrostatic probe of the same dimensions.

1734 APPLIED OPTICS / Vol. 24, No. 12 / 15 June 1985

This perturbation can be minimized by reducing the diameter of the probe as much as possible. The optical perturbation depends on the extent to which obstruction of radiation and particles by the probe interferes with the radiative excitation, deexcitation, and transport processes that are important in a particular plasma of interest. That issue remains to be examined, either theoretically or by experimental comparison of probe results with independent radiative measurements for selected test conditions.

Some practical issues for implementing this technique ex­perimentally include the probe construction and probe ma­terials, especially for high-temperature gases. As indicated previously, the probe need not be a single optical fiber or rod. It may consist of a bundle of fibers, each with the same ac­ceptance solid angle, or it may be a small-bore tube that acts as either a light pipe or collimator. (A light pipe preserving ray angles may require collimation by apertures after the light leaves the tube.) Concerning high-temperature applications, if suitable solid fiber or rod materials are unavailable, the use of a metal light pipe or collimator might be considered. Also, the probe might be moved through the discharge rapidly.

As an intrusive diagnostic instrument this optical probe has some disadvantages and limitations relative to laser diagnostic techniques. However, for those situations for which this optical probe approach is appropriate, it should be consider­ably simpler, less expensive, and easier to interpret than most laser approaches, and it should be a useful addition to the repertoire of available diagnostic techniques.

The optical probe analysis can be extended for application to plasmas that are anisotropic or refractive, bending the rays reaching the probe. Furthermore, the diagnostic method as described here can be used for obtaining 3-D spatial resolution in nonionized media as well as in plasmas. It may be useful, for example, for hot gases or liquids.

References 1. G. L. Rogoff, "Optical System for Spatial Discrimination of Ra­

diation from Extended Bodies," Appl. Opt. 8, 723 (1969); "Optical System for Direct Spatially Resolved Measurements of Radiative Emission and Self-Absorption of Extended Bodies," Rev. Sci. Instrum. 42, 99 (1971).

2. R. A. Gottscho and T. A. Miller, "Optical Techniques in Plasma Diagnostics," Pure Appl. Chem. 56, 189 (1984).

3. See, e.g., W. L. Barr, "Method for Computing the Radial Distri­bution of Emitters in a Cylindrical Source," J. Opt. Soc. Am. 52, 885 (1962).

4. G. S. Selwyn and E. Kay, "Spatially-Resolved Optical Emission Studies of Fluorocarbon RF Plasmas through the Use of UV Transmitting Optical Fibers," in Proceedings, Sixth Interna­tional Symposium on Plasma Chemistry, M. I. Boulos and R. J. Munz, Eds. (Montreal, 1983), pp. 716-721.

5. T. D. Simons, "Optical Fiber Luminosity Probe for Plasma Temperature Fluctuation Measurements," Appl. Opt. 23, 1807 (1984).

6. D. Field, A. J. Hydes, and D. F. Klemperer, "Spatially Resolved Optical Spectroscopy of Plasma Etching Systems," Vacuum 34, 347 (1984); "Spectroscopic Studies of Fluorescent Emission in Plasma Etching of Si and SiO2 and the Mechanism of Gas-Surface Interactions," Vacuum 34, 563 (1984).

7. The field of view at any x can be described by the radius r given by r − θ(p - x), where θ is half of the effective acceptance angle of the probe. [For small Ω, θ is the numerical aperture of the fiber probe and is given by θ − (Ω/π)1/2.] Thus ε and αε can be con­sidered functions only of x if their fractional variations over the radial distance r perpendicular to the x axis are much less than unity. For ε (or α) this criterion can be written

Note that the acceptance angle of the detection system need not be determined at the face of the probe. If the probe preserves ray angles, the effective acceptance angle may be determined by optics located anywhere between the face of the probe and de­tector.

8. The external source is assumed to be sufficiently extended to fill the field of view of the probe and to be uniform over that field.

9. H. R. Griem, Plasma Spectroscopy (McGraw-Hill, New York, 1964), pp. 180-181.

10. G. Bekefi, Radiation Processes in Plasmas (Wiley, New York, 1966), p. 38.

11. The accuracy of the measurement increases as the probe move­ment distance decreases. The extent to which this distance can be reduced depends on the sensitivity of the particular experi­mental system used. The probe can affect this sensitivity through reflection losses from its face as well as through internal transmission losses.

12. If the external source does not fill the field of the probe, that source can be chopped and selectively detected. In this case Ω is given by the solid angle subtended at the probe by the source.

13. T. Okuda and K. Yamamoto, "Disturbance Phenomena in Probe Measurement of Ionized Gases," J. Phys. Soc. Jpn. 13, 1212 (1958).

14. J. F. Waymouth, "Perturbation of a Plasma by a Probe," Phys. Fluids 7, 1843 (1964); "Perturbation of Electron Energy Distri­bution by a Probe," J. Appl. Phys. 37, 4492 (1966).

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