Model for Calculating Transmission through Various Media

Optical Engineering and Manufacturing



The following results are from a MathCAD program created to simulate the transmission of light through different types of materials. This model is used primarily to evaluate different infrared plastic compounds for white light immunity. This model covers two infrared plastic compounds used in lenses by Infra-ótica, glass plates used for white light immunity tests, and two types of pyro windows . I have assembled raw data from transmittance tests into a transmittance matrix for each material. These matrices can be combined to simulate transmission of light through several different surfaces. One example is an automobile headlight through two panes of glass, a Fresnel lens, and finally through the pyro window.




The main type of light that causes problems in the white light immunity test is in the range of 1 mm to 3 mm, not the visible range. The peak wavelength for light from a tungsten filament source is 1 mm. In this range, our optics allow a tiny portion of this energy to pass through. In this range the pyro window begins to transmit at approximately 0% to 0.3 %. Our current lens HDPE transmits at approximately 50 % in this range while our white pigmented lenses transmit between 3 % to 4 %.


The intensity of the white light source is a major factor. According to the blackbody model, a human radiates 370 mW/sq. cm at 9.5 mm. By the time this energy reaches the pyro elements, 150 to 180 mW/ sq. cm is left, depending on which lens is used. The white light source emits 22,000 W/sq. cm at 1 mm. After passing through our optics 14 W/sq. cm at 1.5 mm is left when a standard lens is used and 500 mW/sq. cm when the zinc sulfide plastic lens is used. Either result is very significant.


Weaknesses of the Model


The model currently does not account for the spectral response of the actual pyro elements. What is shown here is the amount of radiation that actually reaches the elements. The model  loses accuracy as transmittances of the material in question drop below 1 %. This is due to the accuracy of the spectroradiometer at very low levels.


Further Improvements of White Light Immunity


Further efforts to improve white light immunity should be aimed at reducing transmission in the region between 1 to 3 mm. A germanium pyro window may be advantageous. Silicon begins transmitting at 5 mm. Germanium begins transmitting at 6.5 mm. The addition of a second silicon or pyro window, placed directly in front of the pyro would serve to further reduce white light and near infrared by an order of 300 to 500. I have seen this technique in some sensors. The other approach would involve improving the lens plastic to reduce transmission in this range.




This model contains two sources. The sources are blackbodies for 310 K and 2900 K. The 310 K blackbody represents a human. According to the Infrared Handbook, a 2900 K blackbody is an approximation for a tungsten filament lamp. For graphs dealing primarily in the visible and near infrared I have used a logarithmic scale for wavelength and marked the boundaries of the visible region.


Most of our targets are human, so we model them as blackbodies. Humans usually are clothed so we adjust for the surface area and emissivity of clothing and exposed skin. M is the standard calculation for radiant exitance of a blackbody.  This calculation is set up to accept a variety of wavelengths and temperatures. We will use 310 K as the temperature. A blackbody will produce variable radiant exitance according to wavelength, so λ, wavelength is left as a variable. L is the calculation for irradiance of a surface by the blackbody. The variables ε and A are the emissivity of the surface and its surface area. They adjust the irradiance according to the area of the target and adjust for whether that area is human skin (h) or clothing (c). The denominator has a distance, r, which adjusts for the distance of the source to the detector.


The final calculation, H, produces a vector that represents the irradiance by a human source. It is set for a temperature of 310 K and is scaled to produce units in terms of Watts per square centimeter. The vector is n rows long. Each row represents the irradiance at a given wavelength.

The material transmittance data for the polymers, glass, and detector windows is stored in a long array n rows long. By placing the source model, H, in a vector form we can model different combinations of glass, plastics, and detector windows by using simple vector multiplication. For example, transmittance of the human source through glass is H X G, where H is the irradiance vector above, and G is the glass transmittance data placed into a matrix form. Similarly, H X G X S would be transmittance of  the human source through glass, then through a silicon window.

To model white light immunity we need a white light source. The source we used was a Tungsten Halogen automotive headlamp. A standard UL test is to position the headlamp some distance away from the sensor. A glass window is placed in front of the sensor. Then the headlamp is pulsed at different frequencies, 0.1 Hz, 1 Hz, and 10 Hz. The glass represents that the sensor will usually be installed inside, and most intense white light sources, such as headlamps or reflected sunlight will have to pass through a window first. The source model is a bit more complex. It accounts for radiated light, blackbody radiation from the headlamp lens, and blackbody radiation from the headlamp lens, the glass window, and the optics in the sensor. The blackbody radiation is created when the lens and glass are heated. This heat generates infrared radiation that is then detected by the sensor.

Lg is the irradiance from the heated glass and headlamp lens. Ab and Ag are the surface areas of the headlamp lens and illuminated glass plane, respectively. The distance from the headlamp to the glass is r. The distance from the glass to the sensor is d. ε is the emissivity of the glass.

Ll is the irradiance from the heated sensor lens. Al is the area of the lens, dlens is the distance between the lens and the pyroelectric detector, εl is the emissivity of the lens.

Lb is the irradiance from the tungsten elements in the headlamp. They are treated as a blackbody at 2900 K. The w and ha variables account for the width and height of the beam. When the beam reaches the glass, only the center portion of the beam will actually strike the glass and factor into the problem.

Ln is the total irradiance from the headlamp. It is in vector form so it can be combined with the material data matrices. The total irradiance includes: blackbody radiation from the actual lighting elements, radiated IR light from the heated lens and glass, and radiated IR light from the sensor´s own lens. It is scaled to produce units of Watts per square cm.

Transmittance Curves for Several Basic Elements of the Model

Text Box: Human source
Text Box: Headlamp source
Text Box: Natural material
Text Box: Pigmented material

1/4 inch thick Glass Plate

Silicon Pyro Window

Germanium Pyro Window


These graphs represent scenarios that are applicable to passive infrared sensors. The first scenario is the transmission of infrared energy from a human target to the pyro. The elements considered are the source, lens, and pyro window. The other scenario is the unwanted transmission of energy from a white light source onto the pyro. This model uses the headlight source, plate glass window, lens, and pyro window.

Infrared Transmission, Source = 310 K Blackbody

Standard Lens

White Light Pigmented Lens and Pyro Window

White Light Immunity, Source = 2900 K Blackbody, Lens = Standard Lens

Transmittance through glass plate.

Transmittance through glass plate, standard lens, and silicon pyro window.

White Light Immunity, Source = 2900 K Blackbody, Lens = White Light Pigmented Lens

Transmittance through glass plate, white light pigmented lens, and silicon pyro window. Transmittance is reduced from a peak of 15 W/sq. cm to 0.5 W/sq. cm.

15 East 26th Street, Suite 18C ˇ New York, NY 10010, USA

Telephone: +1.916.834.9283 ˇ Brasil: +55.11.3304.9840 ˇ Hong Kong +852.2127.0512ˇ Fax +1.917.472.7359

The final calculation H1 produces a vector that represents the irradiance by a human source. It is set for a temperature of 310 K and is scaled to produce units in terms of Watts per square centimeter. The vector is n rows long. Each row represents the irradiance at a given wavelength.

c= speed of light                     h= Planck´s constant            k= Boltzmann´s constan