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This is the explanation for designing the reflector that is essential for spot heating in halogen lamp heaters and other radiative heat applications

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Optics of Lamp Heating

Table of Contents 1. Overview of the reflective optical system for lamp heating 2. Basic concept on how to form a light source image 3. Calculation of the light energy handled by a micro mirror S 4. Method to calculate the luminous intensity distribution on the screen 5. Example and concept of luminous intensity distribution based on a ellipsoidal mirror (for point condensing) 6. Example and concept of luminous intensity distribution based on a parabolic mirror (for parallel light) 7. Example and concept of mirror structure with uniform luminous intensity distribution 8. How to determine the mirror coordinates 9. Evaluation based on prototype


Example: Design example using computer software

1. 1. Overview of the reflective optical system for lamp heating

In lamp heating using a halogen lamp, a light source image of the halogen lamp filament (approximately 2102ºF (2800ºC)) is created and used for heating.

Moreover, the light source image creation method is used when a high temperature is required, and light source image (single) is not created when a relatively wide surface has to be heated.

The focal distance is comparatively long with respect to the lens aperture when a single lens is used to create the light source image and is linked to a clean light source image. This makes the calculation of luminous intensity distribution and power density comparatively simple.

However, the optical system using only a lens can utilize only 20 to 30% of the light emitted from the light source, and this is not economical for both lighting and heating.

A reflective optical system can be used such as a deep mirror to cover the light source that enables utilizing 70% of the light, and many reflective optical systems are available.

However, the deep reflectors have some problems. The explanation is simple if the light source is a point, but the light source always has an area. In case of a deep mirror completely covering the light source, a clean light source image like a lens cannot be created. Aberration increases significantly and is not practical at all with the optical design method using lens.

The method for mirror design, accurate calculation of luminous intensity distribution and target luminous intensity distribution is introduced here. This method is perhaps not the standard for optical design, but since the basic theory is extremely simple and clear, there is little doubt on the theoretical validity. Design without any problems is also possible with this method based on actual results.

The easiest way to simplify the calculations is to treat the light source as spherical. In many cases where the light source is combined with a mirror for heating, the horizontal vertical ratio is maintained close to 1:1 so that it is as close as possible to a sphere. Hence a practical luminous intensity distribution that is adequately accurate can be calculated even when designed as a spherical light source.

Luminous intensity distribution can be calculated accurately even for light sources that cannot be considered as spherical by approximating as a collection of multiple spherical sources. This approach is needed in some cases of mirrors used for lighting. When multiple spherical sources are considered, the process is a little complicated as the overlapping portions have to be eliminated.

Related information → Physics of light heating Sharp light distribution mirror focusing system

2. Basic concept on how to form a light source image

* Here, the design is based on dividing the mirror surface into multiple flat microscopic mirrors. If the number of divisions is around 500 then the optical performance is similar to a continuous surface, and almost exactly the same when the number of divisions is above a few thousands. With the present day computers, calculation is almost instantaneous even when the division of tens of thousands or more.

If the number of divisions is around 200, because there is a moderately blurred effect, mirror with a multi surface structure (multi-mirror) having comparatively lesser number of divisions is used especially in mirrors that are used for lighting. Also, when designed with this design method, rotation polyhedral mirrors that can be manufactured easily with lathes during actual fabrication is widely used for heating (Even small quantities are easy to manufacture at low cost)


Fig 1: Basic concept on how to form a light source image



* Basic idea of design

As shown in the above Figure 1, if one point is taken on the mirror, and considered as a micro mirror, then the image of the light source is magnified in the ratio L2/L1 with the same principle as a pin hole camera and projected on the screen. The area of the light source projected on the screen can be determined from this.

The diameter of the light source image of the micro-mirror that is formed on the screen [dia] D: [dia]D = [dia]d x (L2/L1)

Strictly speaking, the light source image projected on the screen is not a perfect circle as the light enters at an angle, also, illuminance is the highest at the center and the periphery is slightly darker, but if a margin of error is permitted, then the calculation is simple when calculated as uniform illuminance of diameter [dia]d x (L2/L1) on the screen.

If a really strict light distribution is required to be determined, then the exact luminous intensity distribution can also be determined by strictly determining the shape of the light source image and illumination intensity. However, in the range of applications handled in Fintech, we have not felt the need to this extent. → This is determined as not required considering the effect that is obtained when compared to the time and cost.

3. Calculation of the light energy handled by a micro mirror S

* The light energy incident on the micro mirror S is calculated in the method shown in Figure 2 below, and by dividing this with the light source image area on the screen calculated using the above method, the power density Btu (th)/second/in2 (W/cm2) irradiated on the screen with this micro mirror can be calculated. (Since the exact incident angle differs, the power density cannot be said to be the same in the entire area, but this error is ignored because it is usually small.)



Figure 2: Calculation of light energy handled by micro mirror S

If the energy received by the micro mirror of the above figure p(n) is considered as the total radiation from the light source, the ratio of the solid angle received by the micro mirror to the total solid angle can be calculated.


N: Number of divisions in the radial direction (Divisions that appears as radial in the above figure)

4. Method to calculate the luminous intensity distribution on the screen

As a method of evaluating the luminous intensity distribution on the screen, depending on the accuracy required, a large number of virtual sensors are virtually placed on the screen. Usually 20 (total 400) along the horizontal and vertical directions is sufficient.

And then whether the light source image made on the screen by the micro mirror contain their respective sensors is determined. If present, the irradiation power density at that time is added to the sensor. This operation is performed for all the micro mirrors. By doing this, the irradiation power density from the respective micro mirrors gets added to the respective sensors, and illumination distribution data as a whole can be obtained.

When the light source is spherical or even rod-shaped but is arranged coaxially on the optical axis, and the mirror surface is a rotating body without distortion, then the luminous intensity distribution is of a simple rotation target shape, and even if the sensors are not distributed on the surface, one row arrangement from the center is sufficient. Surface arrangement is necessary only when when the heat distribution is expected to be of a complicated form such as the light source is not spherical in shape, or is displaced away from the optical axis, or the mirror is of a distorted shape.

In real situations, cases requiring illuminance distribution evaluation on surface is rare. One row arrangement mostly meets the requirements, even for a somewhat complex luminous intensity distribution, the requirements can be met with one row horizontal and one row vertical arrangement.

5. Example and concept of luminous intensity distribution based on a ellipsoidal mirror (for point condensing)

Figure 3: Luminous intensity distribution with a ellipsoidal mirror (spheroidal surface)

* In the case of a point light source, the light is condensed to a point, but real light sources have an area and are not points. This is an example of a spherical light source. When more light is reflected at the back of the mirror, more is the magnification with which the light source image is projected onto the screen. The magnification of light that is reflected at the mirror edge (near the opening edge) is small and a small light source image is created. Then, if all the light source images are synthesized, a luminous intensity distribution that is high at the center with an encompassing circle is obtained as shown in the above figure.

Diameter of the light source image on the screen = Diameter of light source x (L2/L1) ----- Refer to the explanation of the preceding section.

6. Example and concept of luminous intensity distribution based on a parabolic mirror (for parabolic → parallel light)

Figure 4: Luminous intensity distribution based on a parabolic mirror (paraboloid of revolution - parabola)

* In case of a point light source, the light distribution of the irradiation onto the screen is flat and equal to the mirror opening diameter, but never turns into parallel light as real light sources have an area, and also a flat distribution is not obtained. The light distribution is always bright in the center and dark in the periphery.

* When more light is reflected at the back of the mirror, the light source image is projected with a higher magnification rate onto the screen and a larger light source image is created.

* The magnification of light that is reflected at the mirror edge (near the opening edge) is small and a small light source image is created. Then, if all the light source images are synthesized, a luminous intensity distribution with a peak at the center is obtained as shown in the above figure.

7. Example and concept of mirror structure with uniform luminous intensity distribution

* Mirrors that have been designed such that the edges of the light source image created with micro mirrors are at the edge of the entire light distribution range and a fairly flat luminous intensity distribution is obtained with a very good light utilization.

8. How to determine the coordinates for the mirrors?

The opening diameter of the mirrors is determined by the application in most cases, first this has to be decided. If this is determined then the position of the light source can be determined. Usually the distance from the opening is maintained at approximately half the opening diameter. Generally deeper the position of the light source, higher the utilization. However, manufacturing the mirror becomes difficult, and other problems also arise.

The projection magnification of micro mirrors will be small in the case of a deep mirror, and the irradiation spot diameter can be made small when condensing to a point with an ellipsoidal mirror. However, this does not necessarily mean that plenty of light will be collected in a narrow range, and the power density peak value of the center is almost the same. This should be considered as: The peak shaped curve (close to a trapezoid with a shallow mirror) of the light distribution has collapsed and the tip section has become small (shaped like Mount Fuji).

If the opening diameter and light source position are determined, the angle gets determined from the mirror that is closest to the edge of the opening. The method to determine is first determine the angle at which light is reflected from the light source to the target point on the screen and then determine the coordinates. If the coordinates of the first micro mirror is determined, then the coordinates of the next micro mirror is determined such that it is continuous with the first mirror. This process is repeated up to the back of the mirror.

The method to determine the target point is a trial-and-error method at present. That is, a position considered appropriate is set as the target point, and the shape of the whole mirror is designed for the time being. Then the luminous intensity distribution that will be obtained is simulated, and fine tuning is performed if the light distribution is not as required. This process can be fully automated using a computer, but this not performed as it is not simple because the causal relationship is somewhat complicated (there are multiple solutions for the mirror shape to obtain a certain light distribution) and we have not felt that this is necessary.

9. Evaluation based on prototype

The mirrors are cut out in the form of a rotating polyhedron from the coordinates determined in the above mentioned design using a NC lathe. Light distribution can then be evaluated by finishing to a mirror surface with buffing and a polishing agent.

- Reference -

The following is a design example in which the mirror was created with the mirror design method using computer software. A mirror with outer diameter [dia]1.18 in ([dia]30 mm) was used with the design objective to heat the range [dia]1.18 in ([dia]30 mm) at a distance of 1.18 in (30 mm).

The figure below shows a simulation of the results of the mirror design based on the above input, and has been designed to obtain a flat light distribution.

When irradiation is carried out to obtain the required light distribution over a certain range, the method to direct the light path to the target, or the method to direct the light path to the target after crossing the center of the optical axis can be used. The mirror becomes deeper in case of the latter, and light utilization improves. However, if the irradiation range is changed, the radiation range changes significantly. However, when used for illumination, the irradiation distance is usually very long (several feet), and the difference is less in such cases, and the crossing method is generally used.

The crossing method has been used in the following design example.

This is the mirror fabrication data based on this design. Can be fabricated if x(n) and y(n) data is available.

Information required for the design work. The below figure shows the light distribution extent of each micro mirror.

Light distribution simulation when the screen is moved by 0.787 in (20 mm). The changes in light distribution can be simulated easily when the position of the screen or the light source is changed.

The following is Windows version.