{MJC_Logo}

Home Publications Contact Feedback Disclaimer

 

An angular light scattering sensor: Radiometric considerations

Miroslaw Jonasz

Introduction

This document addresses radiometric considerations relevant in designing an angular light scattering sensor. Such a sensor may be referred to as a polar nephelometer if the sensor's detector unit revolves about an axis intersecting the incident light beam. Thus, position of that unit can be best expressed in polar coordinates with a variable angle and a fixed radius of rotation. A sensor intended for measurement of the volume scattering function of sea water is used as an example.

In the following text I refer to absolute values of light power in order to establish a reference scale and carry sample calculations for a specific light detection system. However, only relative power values are needed to determine the scattering function.
[Menu]

Sensor geometry

This sensor is intended to measure axially symmetrical volume scattering function beta(theta) [m-1 sr-1]. Such scattering function depends on only the scattering angle (Fig.1). It is defined as follows (for example, Jerlov, 1976):

 beta(theta) = dI(theta) / E dV  (1)

where theta [rad] is the scattering angle, measured from the direction of propagation of the incident light, dI [W sr-1] is the intensity of light scattered by a volume element dV [m3], and E [W m-2] is the irradiance of the scattering volume due to the incident beam.

{Polar nephelometer geometry}

Fig. 1. Geometry of an angular light scattering sensor (polar nephelometer).

The scattering function of a medium containing scattering centers (particles) that are not randomly oriented and positioned may not necessarily be axially symmetrical and may depend on an azimuthal angle in addition to the scattering angle.

As seen in Fig.1, the sensor measures an average value of the scattering function within its field of view (FOV) as defined by the field stop and the aperture stop. The aperture stop (Fig.1) is usually defined by the active area of the sensor's light detector. With a scattering function rapidly varying with the angle, especially in the forward and backward angle regions the intrinsic characteristics of the sensor geometry may cause significant measurement errors for sensors with a wide FOV (Jonasz, 1990).
[Menu]

The smallest measurable signal

The smallest measurable signal is determined by the sensor noise. Here we analyze the noise related to the light detection system of the sensor. 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, F, 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 F can thus be defined as follows:

 SNRmax = F / ( NEP B1/2 )  (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 = F / Fnoise  (3)

The best 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 detectors, 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 at which measurements are performed. For example, a reduction of the bandwidth by a factor of 10 increases the SNR by a factor of 101/2 = 3.16

A photodiode-based detection system is simple, inexpensive, and robust. However, a photodiode lacks an inherent amplification of a photomultiplier and its minute current must be amplified externally. That current can be calculated by using the following equation:

 i = F r  (4)

where r [A / W] is the detector 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:

 vtotal_output_noise ={ ( idet_noiseRf) 2 + ( iinput_noiseRsource ) 2 +  
  [vinput_noise( 1 + Cj / Cf ) ] 2 +  (5)
  4 kT Rsource}1/2  

where

 inoise_det = NEP r  (6)

is the noise current density of the detector

 Rsource = R jR f / (R j+ R f)  (7)

Rsource 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. We use the maximum noise gain for the input voltage noise of a transimpedance amplifier ( a factor of 1 + Cj / Cf ). The last term in Eq. 5 is the thermal noise with the Boltzmann constant k = 1.38×10-23J/K, and T is temperature of the amplifier in K.

By assuming the following values of key parameters (the detector is a GaAsP photodiode: Hamamatsu G1115)

  • photodiode responsivity, R = 0.3 A / W
  • photodiode NEP, 10-15 W Hz-1/2
  • photodiode (junction) capacitance, Cj = 300 pF
  • photodiode shunt resistance, Rj = 45 GOhm
  • transimpedance amplifier parameters
    • feedback capacitance, Cf = 10 pF
    • feedback resistance, Rf = 1 GOhm
  • operational amplifier input noise densities
    • current noise density, iinput_noise = 1 fA Hz-1/2
    • voltage noise density, vinput_noise = 10 nV Hz-1/2

we obtain an estimate of the voltage density of the total output noise the detection system:

 vtotal_output_noise = { [ ( 10-15 0.3 ) 109 ] 2 +  
  [10-15 45 109 109 / (45 109 + 109 )] 2 +  
  [10-8 ( 1 + 300 / 10 )] 2 +  
  4 (1.38 10-23 ) 293 (45 109 )}1/2  
  ~ { 10-13 + 10-12 + 2.5 10-22 }1/2  
  ~ 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 of 0.01 Hz, according to the following formula (Gualtieri, 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 / Rf  
  = 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, Fmin [W] can be calculated using the following equation:

 Fmin = betaw,633 (90°) E V omegaw  (11)

where betaw,633 (90°) is the volume scattering function of pure sea water at 90°, omegaw [sr] is the acceptance solid angle of the detector (in water) (Fig. 1), and V [m3] is the scattering volume of sea water illuminated by the laser beam of irradiance, E [W m-2]. According to Morel (1974) betaw,633 (90°) = 7×10-5 m-1 sr-1. One can assume as representative a scattering volume of 10-9 m3 (1 mm3), a detector's acceptance angle of 2.4×10-4 sr (corresponding to an angular resolution of 1o), 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 Fmin = 2×10-14 W (17 fW ) from Eq. 11.

The minimum power thus corresponds to a SNR of 2×10-14 / 3×10-16 i.e. about 60 at a bandwidth of 0.01 Hz. However, inapropriate photodiode amplifier circult 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 make it impossible to measure light power of the Fmin magnitude.

The Fmin corresponds to a photon flux of N = 2×10-16 W / 3.14×10-19 J ~ 7×104 photons/s at 633 nm. The coefficient of variation (relative noise) of this photon flux equals N1/2 /N, that is about 0.4% in this example. However, 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.
[Menu]

The largest measurable signal

The maximum light scattered power which needs to be measured is that of the laser beam. The magnitude of this power 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 a non-linear detector at this photocurrent, with nonlinearity on the order of couple percent (for example, Fischer and Fu, 1993). It may even become saturated. Second, when the photodiode is connected to a transimpedance amplifier with a feedback resistor of 109 Ohms, the amplifier would generate a voltage of only about 10 V (supply voltage) rather than linearly amplified output voltage of 3×106 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.
[Menu]

References

Fisher J, Fu L. 1993. Photodiode nonlinearity measurements with an intensity stablized laser as a radiation source. Applied Optics 32: 4187-4190.

Gualtieri D. M. 1987. Precision of lock-in amplifiers as a function of signal-to-noise ratio. Review of Scientific Instruments 58: 299-300.

Jerlov N. G. 1976. Marine optics. Elsevier, Amsterdam

Jonasz M. 1990. Volume scattering function measurement error: effect of angular resolution of the nephelometer. Applied Optics 29: 64-70.

Morel A. 1974. Optical properties of pure water and pure sea water. In: Optical aspects of oceanography. Jerlov, N. G. and Steeman-Nielsen, E. (eds.). Academic Press, London.

[Menu]

Last modified: . Copyright 2000 MJC Optical Technology. All rights reserved.