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Polar nephelometer | Table of contents |
▶ Polar nephelometer, alignment
▶ Polar nephelometer, calibration
▶ Polar nephelometer, descriptions
▶ Polar nephelometer, measurand
▶ Polar nephelometer, radiometry
▶ Polar nephelometer, small angle
▶ Polar nephelometer, small angle, calibration
▶ Polar nephelometer, small angle, optical Fourier transform
Polar nephelometer, alignment
Wilson JD 2007 (pp. 204-206), Jonasz M and Fournier 2007 (pp. 162-163)
Polar nephelometer, calibration
review: Jonasz M and Fournier 2007 (pp. 163-172), Holland AC 1980, Holland AC and Gagne 1970
methods: Jonasz M and Fournier 2007 (pp. 163-172; translucent screen), Leong KH et al 1994 (molecular scattering), Kullenberg G 1984 (fluorescence), Bell BW and Bickel 1981 (scattering by a fiber), Holland AC 1980 (translucent screen), Aas E 1979 (molecular scattering), Fry ES 1974 (translucent screen)
Polar nephelometer, descriptions
Tan Hiroyuki et al 2013 (hydrosol, angle 8° to 172°, wavelength 400 to 700 nm, imaging),
Dolgos G et al 2010 (aerosol, polarized, linear polarization, angle 1.5°-178.5°, wavelength 532 nm, imaging),
Bunkin NF et al 2009b (hydrosol, polarized light, scattering matrix, angle 0°-90°, wavelength 532 nm),
Gogoi A et al 2009b, Gogoi A et al 2009a (hydrosol, polarized light, two orthogonal polarizations, angle 10° to 170°, rotating polar array of detectors),
Ma Lin et al 2009 (aerosol, polarized light, discrete angle),
Curtis DB et al 2008 and Curtis DB et al 2007 (aerosol, polarized, linear polarization, angle 15°-175°, wavelength 550 nm, elliptical mirror),
Zugger ME et al 2008 (hydrosol, two orthogonal polarizations1°-170°, wavelength 532 nm),
Michels R et al 2008 and Michels R et al 2007 (correction for refraction at plane cuvette),
Barkey B et al 2007 and Barkey B and Liou 2001 (aerosol, polarized light),
Castagner JL and Bigio 2007 and
Castagner JL and Bigio 2006 (confocal imaging)
Daugeron D et al 2007(aerosol, polarized light, angle 20°-160°, wavelength 802 nm, two perpendicular polarizations),
Lotsberg JK et al 2007 (cylindrical cuvette),
Pinto PN et al 2007 (hydrosol, unpolarized light, angle 15°-45°, wavelength 633 nm, rectangular, i.e. plane cuvette, correction for refraction at plane cuvette)
Shao Bing et al 2006 (hydrosol, unpolarized light, angle 0.5°-179.5°, wavelength 658 nm),
Passos D et al 2005 (hydrosol, unpolarized light, angle 10°-160°, wavelength 635, 670 nm, cylindrical sample cuvette serves as lens focusing scattered light at detector),
Kaller W 2004 (aerosol, unpolarized light, angle 3°-177°),
Lee ME and Lewis 2003 (hydrosol, unpolarized light, angle 0.6°-173°, wavelength 532 nm),
Bolt RA and de Mul 2002a (tissue, angle -90 to +90°),
Witkowski K et al 1998 (hydrosol, polarized light, scattering matrix, angle 8°-155°, wavelength 488 nm, 514 nm, 633 nm),
Dueweke PW et al 1997 (aerosol, angle ~0.2° to ~5° in 0.1° steps off axis, 45°, 90°, and 135°, wavelength 532 nm, imaging),
Chang Hyuksang et al 1995 and Chang Hyuksang and Biswas 1992 (aerosol, polarized light, angle 25°-125°, wavelength 833 nm)
Leong KH et al 1994 (aerosol, unpolarized light, angle fixed in range of 23°-128°, wavelength 855 nm),
Kuik F et al 1991 (aerosol, polarized light, scattering matrix, angle 5°-175°, wavelength 633 nm),
Quinby-Hunt MS et al 1989 (hydrosol, polarized light, angle ~10° to ~160°, wavelength 514 nm, scattering matrix, cylindrical cuvette),
Nakajima T et al 1989 (aerosol, polarized light, top-left quadrant of the scattering matrix, angle 7°-170°, wavelength 633 nm),
Hansen MZ and Evans 1980 (aerosol, polarized light, scattering matrixangle 2°-178°, wavelength 514 nm),
Thompson RC et al 1980 (hydrosol, polarized light, scattering matrix, angle 5°-170°, wavelength 442 nm, 633 nm),
Gucker FT et al 1973 (aerosol, single-particle, unpolarized light, angle 7°-173°, 183°-353°, elliptical mirror),
Holland AC and Gagne 1970 (aerosol, polarized light, scattering matrix, angle 18°-166°, wavelength 546 nm)
Polar nephelometer, errors
effect of backscattering by forward scattered light reflected on cuvette wall: Gogoi A et al 2009b, Gogoi A et al 2009a, Volten H et al 1998, Schnablegger H and Glatter 1995, Schnablegger H and Glatter 1993, Sugihara S et al 1982
effect of finite angular resolution: Jonasz M and Fournier 2007 (pp. 172-177), Grasso V et al 1997, Grasso V et al 1995, Jonasz M 1990
Polar nephelometer, history
Kaye PH et al 2007, Kerker M 1997
Polar nephelometer, measurand
A polar nephelometer is intended to measure the volume scattering function of a turbid medium posessing axial symmetry of light scattering, or light scattering by a particle posessing radial symmetry (such as homogeneous of coated sphere). If the intent is to measure differential light scattering by a single particle which does not have radial symmetry, a two-dimensional nephelometer should be an instrument of choice because light scattering by such a particle does not have axial symmetry.
Polar nephelometer, radiometry
The polar nephelometer radiometry is discussed here for a nephelometer intended for measuring the volume scattering function of turbid medium (Fig. 1). This function is defined as follows:
β(θ) = dI(θ) / ( E dV ) | (1) |
where β [m^{-1} sr^{-1}] is the volume scattering function (for example, Jonasz M and Fournier 2007), θ is the scattering angle, i.e. the angle between the observation direction and the incident light beam direction, dI [W sr^{-1}] is the intensity of light scattered at the scattering angle, E [W m^{-2}] is the irradiance of the turbid medium at the elementary volume dV [m^{3}]. The volume scattering function is assumed to be axially symmetrical, hence it is defined as being dependent on the scattering angle only (see, for example, Yanovitskij EG 1998 for a condition of axial symmetry of the scattering function). The scattering function may in general also depend on the azimuth angle (i.e. the angle between the plane containing the observation and incidence directions and an arbitrary reference plane).
Fig. 1. Geometry of the polar nephelometer.
As it follows from Fig.1, the polar nephelometer measures an average value of the scattering function within its field of view (FOV), defined by the field stop and the aperture stop. The aperture stop is usually defined by the active area of the nephelometer light detector. Given that scattering functions of many natural and industrial turbid media typically vary rapidly with the angle θ in the forward angular range, the intrinsic characteristics of the nephelometer geometry causes measurement errors in this range that are dependent on the magnitude of the FOV (Jonasz M 1990).
The smallest measurable signal is determined by the noise of the light detection system. Depending on design, this noise may consist of photodetector noise alone or of the noise of photodetector, amplifier, and other signal processing systems, which constitute the light detection system.
The noise of a photodetector is specified by its Noise Equivalent Power (NEP). NEP is equal to the signal power, Φ, numerically equal to the power of noise contained in a 1 Hz bandwidth about a frequency at which the received light power is modulated. We refer here to signal modulation caused by intentional modulation of the incident light, a typical means of rejecting ambient light and electrical noise in the synchronous detection technique. Thus, with the signal power equal to the NEP, the signal-to-noise ratio (SNR) is unity. The detector SNR at an incident light power of Φ can thus be defined as follows:
SNR_{max} = Φ / ( NEP B^{ 1/2} ) | (2) |
where B [Hz] is the electrical bandwidth. It is assumed here that the NEP density is uniform within a bandwidth B. This equation is a special case of the SNR definition:
SNR = Φ / Φ_{n} | (3) |
where Φ_{n} is the noise power>/p>
High-quality solid-state photodiodes have NEPs on the order of 10^{-15} W Hz^{ -1/2} at 633 nm and 1 Hz bandwidth. The wavelength is important as the sensitivity of photodetectors, such as photodiode and photomultiplier, varies with wavelength. Photomultipliers have their maximum NEPs on the order of 10^{-16} W Hz^{ -1/2}. As follows from Eq. 2, the SNR can be increased by reducing the electrical frequency bandwith. For example, a reduction of the bandwidth by a factor of 10 increases the SNR by a factor of 10^{1/2} = 3.16.
A photodiode-based detection system is simple, inexpensive, and robust. However, a photodiode lacks the inherent amplification of a photomultiplier and its minute output current must be amplified externally. That current can be calculated by using the following equation:
i = Φ r | (4) |
where r [A / W] is the photofiode reponsivity.
It turns out, that an external amplifier is a major source of electrical noise in a photodiode-based detection system. Thus, the amplifier noise must be evaluated. The frequency density of this noise can be expressed as follows:
v_{tn} = { (i_{dn} R_{f} )^{ 2} + | |
( i_{n} R_{s })^{ 2} + | |
[v_{n} ( 1 + C_{j} / C_{f} ) ] ^{2} + | |
4 kT R_{s} }^{1/2} | (5) |
where
i_{dn} = NEP r | (6) |
is the noise current density of the detector, and
R_{s} = R_{j} R_{f} / ( R_{j} + R_{f} ) | (7) |
R_{s} is the source resistance of the transimpedance amplifier, with R_{j} being the photodiode junction (also referred to as "shunt") resistance, and R_{f} being the feedback resistance. Parameter i_{n} is the input noise density of the amplifier. We use the maximum noise gain for the input voltage noise of a transimpedance amplifier ( a factor of 1 + C_{j} / C_{f} ). The last term in Eq. 5 is the thermal noise density [ V Hz^{ -1/2} ], with the Boltzmann constant k = 1.38×10^{-23} J / K, and T [K] being the absolute temperature of the amplifier.
Assuming the following (representative) values of key parameters (the detector is a GaAsP photodiode: Hamamatsu G1115):
Hence, i_{dn} = 10^{-15} W Hz^{ -1/2} × 0.3 A / W = 3 × 10^{-16} A Hz^{ -1/2} and R_{s} = 45 GΩ × 1 GΩ / ( 45 GΩ + 1 GΩ ) ≅ 1 GΩ and the voltage density of the total output noise the detection system is evaluated as follows:
v_{ tn} = { [ ( 3 × 10^{-16} A Hz^{ -1/2} × 10^{9} Ω ] ^{2} + | |
[ 10^{-15} A Hz^{ -1/2} × 10^{9} Ω ] ^{2} + | |
[ 10^{-8} V Hz^{ -1/2} × ( 1 + 300 pF / 10 pF )]^{ 2} + | |
( 4 × 1.38 × 10^{-23 } J / K × 293 K × 10^{9} Ω ) }^{1/2} | |
~ { 9 × 10^{-14} + 1 × 10^{-12} + 9 × 10^{-14} + 1.6 × 10^{-11} }^{1/2} V Hz^{ -1/2} | |
~ 4 × 10^{-6} V Hz^{ -1/2} | (8) |
With a lock-in amplifier as a bandwidth-limiting voltmeter set for measurement time, t = 25 s, one can easily achieve a bandwidth B = 0.01 Hz, according to the following formula (for example, Gualtieri DM 1987):
t = 1 / ( 4 B ) | (9) |
With that bandwidth, the total output voltage noise, V, equals 10^{-7} V. This output voltage is equivalent to an input current of
i = V / R_{f} | |
= 1×10^{-16} A | (10) |
A photocurrent of 1×10^{-16} A is equivalent to the scattered light power of 1×10^{-16} A / 0.3 ≅ 3×10^{-16} W at which a SNR of unity is achieved. These calculations neglect any power losses which may be introduced by optical elements between the scattering volume and the photodetector (such as vignetting by apertures, attenuation by lenses, polarizers, etc.).
Minimum scattered light power available for this sensor is that scattered at 90° by pure sea water when the polarization of the incident light is parallel to the scattering plane. This plane contains the direction of the incident beam and that of observation. Assume that illumination is provided by a low-power HeNe laser (wavelength = 633 nm). The minimum power, F_{min} [W] can be calculated using the following equation:
Φ_{min} = β_{w, 633} (90°) E V Ω_{w} | (11) |
where β_{w, 633} (90°) is the volume scattering function of pure sea water at 90°, Ω_{w} [sr] is the acceptance solid angle of the detector (in water) (Fig. 1), and V [m^{3}] is the scattering volume of sea water illuminated by the laser beam of irradiance, E [W m^{-2}]. According to Morel A 1974 β_{w, 633} (90°) = 7×10^{-5} m^{-1} sr^{-1}. Assume the scattering volume on the order of 10^{-9} m^{3} (1 mm^{3}), the detector's acceptance angle of 2.4×10^{-4} sr (corresponding to an angular resolution of 1 °), and irradiance of about 1300 W m^{-2} (1 mW for a 1 mm diameter beam, typical of low-power HeNe lasers). By using these values one obtains Φ_{min} = 2×10^{-14} W (17 fW ) from Eq. 11.
The minimum power yields a SNR of 2×10^{-14} / 3×10^{-16} i.e. about 60 at a bandwidth of 0.01 Hz. However, inapropriate photodiode amplifier circuit design, layout, construction, as well as inappropriate coupling between the photodiode, amplifer, and the lock-in amplifier may yield SNR which is orders of magnitude lower than that value and may even make it impossible to measure light power of the Φ_{min} magnitude.
The Φ_{min} corresponds to a photon flux of N = 2×10^{-16} W / 3.14×10^{-19} J ~ 7×10^{4} photons / s at 633 nm. The coefficient of variation (relative noise) of this photon flux equals N^{1/2} / N, that is about 0.4% in this example. This inherent photon-flux noise is insignificant in comparison to the noise due to the fluctuations in the number of particles in the scattering volume during the measurement time and the signal-independent noise of the light detection system: a photodetector and an amplifier.
The largest signal to measure is that resulting from the detector intercepting the laser beam. The magnitude of this signal is required for the calibration of the sensor. According to Eq. 4, a laser beam with a power of 1 mW incident on a photodiode with a responsivity of 0.3 A / W would generate a photocurrent of 0.3 mA.
Two problems arise here. First, the photodiode may become non-linear at this photocurrent, with nonlinearity on the order of couple percent (for example, Fischer J and Fu 1993). It may even become saturated. Second, when the photodiode is connected to a transimpedance amplifier with a feedback resistor of 10^{9} Ω, the amplifier would generate a voltage of only about 10 V (supply voltage) rather than linearly amplified output voltage of 3×10^{6} V that would reflect the input light power magnitude. A linear output voltage on the order of 10 V corresponds to an incident light power on the order of 30 nW.
These problems require that the laser beam power be attenuated for the beam power measurement by a factor on the order of 3×10^{-8} / 10^{-3} = 3×10^{-5}. This can be done, for example, by using neutral density filters. Switching in a smaller feedback resistor to reduce the amplifier gain should be avoided because the switching circuit may introduce additional noise or prevent one from optimizing the circuit layout in order to minimize the electronics noise at the low end of the scattered power range.
Polar nephelometer, reviews
Kaye PH et al 2007 (single particle), Jonasz M and Fournier 2007 (pp. 154-180), Prentice J et al 2003, Bickel WS and Bailey 1985
Polar nephelometer, small angle
Brogioli D et al 2003 and Brogioli D et al 2002 (speckle-field analysis), Laux A et al 2002, Wax A et al 2001 (low-coherence interferometry), Padmabandu GG and Fry 1990 (photorefraction (at scattering angle of 0), Spinrad RW et al 1978, Spinrad RW 1978
▶ Polar nephelometer, small angle, calibration
▶ Polar nephelometer, small angle, optical Fourier transform
Polar nephelometer, small angle, calibration
◀ Polar nephelometer, small angle
■ Polar nephelometer, calibration
Polar nephelometer, small angle, optical Fourier transform
Eliçabe GE et al 2007, Roßkamp D et al 2007, Agrawal YC 2005, Constantinides GN et al 1998, Laux A et al 2002, Estes LE et al 1997 (aquatic, in situ), Dueweke PW et al 1997 (~0.2° to ~5° in 0.1° steps, off axis forward and backward directions, 45°, 90°, and 135°, wavelength 532 nm), Forand JL et al 1993, Petzold TJ 1972, Bauer D and Morel 1967, Bauer D and Ivanoff 1965
◀ Polar nephelometer, small angle
CITATION: Jonasz M. 2016. Polar nepheometer (www.mjcopticaltech.com/Publications/PolarNeph.php). |
Published: 20-Jun-2016 |
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