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**Quantifying Photometric Spectral Mismatch Uncertainties in LED Measurements**

Richard Young Optronic Laboratories Kathleen Muray INPHORA Carolyn Jones CJ Enterprises

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Introduction We can see mis-matches at low response better on a logarithmic plot. Ideally, photometer response should match the photopic curve

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**Introduction They often deviate in the Blue Photometer 2 Photometer 3**

The highest response and best fit are normally around 555 nm And in the Red Photometers use filter/detector combinations to approximate photopic response This approximation can sometimes be quite good, but is never perfect. This plot shows 3 photometers.

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Introduction If the photometer is calibrated with a white light source, such as illuminant A: Correct measurements will only be made if the test source is also illuminant A. The errors in measuring other sources depend on: The accuracy of matching the photometer response to the photopic curve. The difference between the test source and illuminant A.

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**Introduction If the photometer response is very close to photopic:**

There is little error, relaxing the need for similarity between calibration and test sources. If the test source is very close to illuminant A: There will be little error, relaxing the accuracy requirements of the photometer response.

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Introduction However, an LED is so different from illuminant A that the photometer needs to match the photopic response curve very closely. A “goodness of fit” parameter, f1’, has been used for many years as the selection parameter for photometers. It is intended to apply to white light sources and DOES NOT WORK for LEDs (with the possible exception of white LEDs).

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**The calculation requires the photometer relative response.**

Introduction To remind you how f1’ is defined: Where: Illuminant A Publication CIE : Methods of characterizing illuminance meters and luminance meters: Performance, characteristics and specifications The calculation requires the photometer relative response.

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**LEDs are generally narrow band, and are very unlike illuminant A**

Introduction Especially in the Blue And in the Red Measurements of LEDs can therefore have large errors associated with white light calibrations. LEDs are generally narrow band, and are very unlike illuminant A

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Introduction If the relative spectral distribution of the LED and photometer response are known, the measured photopic value can be corrected for the difference between the calibration source and the LED. This is called the spectral mismatch correction factor, F (also known as color correction factor in some older documents). When the calibration source is illuminant A, the spectral mismatch factor is given the symbol F*.

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**Spectral Mismatch Factors**

We can therefore calculate the spectral mismatch factors for Photometer 1. Here are the spectral distributions for a range of LEDs

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**Spectral Mismatch Factors**

LED measurements using this photometer, can be multiplied by the appropriate F* to give corrected results.

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**Spectral Mismatch Factors**

Can we calculate the spectral mismatch factors without measuring a whole range of LEDs? Although spectral distributions of LEDs are often asymmetric, they have a similarity of shape that might be reproduced by calculation. To keep the calculation simple and relevant, it should be based on information readily available: peak wavelength and full-width-at-half-maximum (FWHM).

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**Spectral Mismatch Factors**

Using a Gaussian curve within the FWHM limits and an exponential curve outside, the LED spectrum is represented reasonably well.

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**Spectral Mismatch Factors**

Mathematically, for lL l lH [lL is the lower and lH is the upper FWHM limit, lp is the peak wavelength]

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**Spectral Mismatch Factors**

For l < lL and lH > l [lL is the lower and lH is the upper FWHM limit, lp is the peak wavelength]

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**Spectral Mismatch Factors**

…and here are the predicted F* values using the modelled LED spectra (shown in red). So, here are the F* factors calculated from real LED spectra again…

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**Spectral Mismatch Factors**

The agreement between real and modelled LED spectral calculations means we can express F* as a smooth curve rather than individual points. We don’t have to do all those LED spectral measurements. We can express F* for different FWHM values at each peak wavelength. And then something interesting happens…

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**Spectral Mismatch Factors**

And other ranges where F* hardly changes at all There are wavelength ranges where F* changes rapidly And other places where F* changes rapidly with FWHM We see that the F* curve has places where FWHM hardly matters

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**Spectral Mismatch Factors**

LEDs differ in peak wavelength and FWHM, so if we want to describe how F* changes for real LEDs: We must include a wavelength component We must include a FWHM component

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**Spectral Mismatch Factors**

The mathematical model for the LED spectra works for this photometer, but does it work for all?

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**Spectral Mismatch Factors**

It seems to work even better for Photometer 2 than it did for Photometer 1.

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**Spectral Mismatch Factors**

This is because the mathematical model is symmetric and the LED spectrum is not. These LEDs are narrow band and highly asymmetric, combined with a poor photopic fit of the detector However, it still matches the general shape of the F* curve, which is all that is required in this paper. Photometer 3 shows some differences as the F* value increases

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**Spectral Mismatch Factors**

The point of this presentation is not to replace LED spectral measurement in the calculation of spectral mismatch factors. Though it seems to do a good job of this. The point is, when testing LEDs in a production environment, there are small changes in peak wavelength and FWHM between devices of the same type. And measuring the spectrum, or even peak wavelength, to get a new F* for each device is not practical.

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**Spectral Mismatch Factors**

At this point it is worth noting that if a calibrated LED is used to calibrate the photometer rather than a white light source, the photometer will already read correctly for that LED. It is equivalent to calibrating and applying the F* factor in one process. All other LEDs will still need a spectral mismatch factor, F, to correct the measurement result. And that includes the production devices.

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**Spectral Mismatch Factors**

Let us take a closer look at some of these F* values. Magnify

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**Spectral Mismatch Factors**

The size of the error depends on how different the wavelength is and how quickly the F* factor changes in that region. This means that measurements of LEDs that have a slightly different wavelength still have an associated error When we apply the F* factor, we are effectively offsetting the curve at one wavelength

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fLED We can define a “goodness of fit” parameter, like f1’ but specifically applying to LEDs – fLED. The fLED parameter is “the average absolute spectral mismatch error over a wavelength region relative to the central wavelength of that region.” NOTE: It is NOT a correction factor to be applied, but it IS an indicator of the suitability and quality of the photometer for measurement of any single color LED.

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fLED There is one value of fLED for each wavelength and FWHM, but because we can effectively model the LED spectral distribution, it can be easily calculated from the photometer response. fLED has two components. Errors introduced by measuring LEDs at different wavelengths to the calibration –wLED. Errors introduced by measuring LEDs at different FWHMs to the calibration – hLED.

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wLED Mathematically, the F* value for an LED at the central wavelength, c, is: Where s() is the photometer response and ScLED() is the LED spectral distribution.

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wLED Similarly, the F* value for an LED at some other wavelength, p, is: Where s() is the photometer response and SpLED() is the LED spectral distribution.

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wLED The error when measuring an LED at wavelength p using the Fc* value at wavelength c is: NOTE: This equation no longer contains a reference to the calibration source, so it does not matter if it was calibrated with white light or a calibrated LED. p,c depends only on the photometer and the LED spectral distributions. If the modelled spectral distributions are used, it is purely a photometer characteristic.

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**Where p1 and p2 are the wavelength limits of the region**

wLED Recall the definition of fLED: “the average absolute spectral mismatch error over a wavelength region relative to the central wavelength of that region.” We can now define wLED in mathematical terms: Where p1 and p2 are the wavelength limits of the region

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**wLED So wLED can be calculated for any central wavelength and FWHM.**

It should be shown as wLED(c,FWHM) to reflect this. Since it is independent of calibration source, a full photometer response curve is not required. 3 FWHMs around the central wavelength should be sufficient. The photometer response does need to be done at 1nm intervals or smaller for good results.

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wLED We still need to define the wavelength “region” in order to calculate wLED(c,FWHM). Based on data for over 900 LEDs in 63 batches, covering most of the visible range, we propose ± 5 nm around the central wavelength.

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**wLED The first stage is to calculate p,c over the region.**

This is the result for photometer 1 at 20 nm FWHM.

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**wLED The next stage is to calculate wLED values.**

These results show that wLED varies strongly with FWHM.

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**hLED Using similar reasoning to wLED calculations**

The error when measuring an LED at FWHM h using the FH* value at FWHM H, both at peak wavelength c is:

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**Where h1 and h2 are ± 5 nm limits around the central FWHM value, H**

hLED We can define hLED in similar mathematical terms to wLED: Where h1 and h2 are ± 5 nm limits around the central FWHM value, H

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**Like wLED, hLED is strongly dependent on FWHM.**

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**hLED So now we have the two components:**

wLED(c,H) gives the error for peak wavelength change. hLED (c,H) gives the error for FWHM change. We can combine them to give the general error indicator, fLED(c,H):

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**You can see that high hLED is generally close to a low wLED.**

fLED You can see that high hLED is generally close to a low wLED. This means there are wavelengths where the photometer error is more sensitive to LED peak wavelength shifts and others where it is more sensitive to FWHM changes. We add hLED Here is an example of wLED And finally fLED.

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fLED Giving large errors with wavelength changes Where the photometer response crosses the photopic curve, their slopes are very different But high and low contributions offset one another for changes in FWHM. This is the photometer response graph shown earlier but rescaled.

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**fLED fLED(c,H) values can aid in the design of photometers.**

It provides instant feedback on the performance of the photometer for LED measurements. It shows that it is the slope of the response rather than the absolute value that is important. It does not require spectral data over the full visible region. Photometer 4, specially designed for blue LEDs, can now be added to our list.

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**fLED Photometer 4 is confirmed as generally the best for blue LEDS.**

But photometer 1 is best at 430 nm.

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**fLED Photometer 3: f1’ = 2.51% Photometer 3 is the worst**

Values of fLED(c,H) show the suitability for LED measurement, but bear no relation to the f1’ value. Photometer 3: f1’ = 2.51% Photometer 3 is the worst At 40 nm FWHM Photometer 4 is the best for blue LEDS even at 430 nm Photometer 1: f1’ = 6.35% Photometer 2: f1’ = 1.98%

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fLED A 3-D plot shows the variations of fLED(c,H). The value is color coded to show iso-value lines. Seen from above, this is a map.

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**We can overlay a plot of FWHM vs. wavelength for some modern LEDS**

fLED – Photometer 1 We can overlay a plot of FWHM vs. wavelength for some modern LEDS These would be measured with <1% fLED. These would be measured with <2% fLED.

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**fLED – Photometer 2 Photometer 2 has <1% fLED for most LEDs.**

But offers no significant improvement for these LEDs.

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**Photometer 3 also has a wide range of <1% fLED.**

fLED – Photometer 3 Photometer 3 also has a wide range of <1% fLED. But up to 7% fLED for these LEDs.

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**And has fLED<3% even for these LEDs.**

fLED – Photometer 4 Photometer 4 data has a limited wavelength range, but <1% fLED extends further into the blue region than the others. And has fLED<3% even for these LEDs.

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**Calibration LEDs shown in black**

fLED The “central” LED in each batch was used to calibrate the photometers for the measurement of all other LEDs in the batch. Each batch included similar LEDs in terms of peak and FWHM, regardless of manufacturer To test the validity and usefulness of fLED, several batches of LEDs were measured. Calibration LEDs shown in black

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**But the extent is not ± 5 nm like fLED.**

The smaller the spread in wavelengths, the lower the batch error. We can scale the errors to a ± 5 nm region to compare directly with fLED. The spectra of each of these LEDs is known, so we can calculate the error in measurement and hence the standard deviation for each batch But the extent is not ± 5 nm like fLED.

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**The blue line represents equivalence.**

fLED The blue line represents equivalence.

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**fLED fLED and and LEDs: fLED is specific to LED measurement.**

fLED is based on variations in spectral mismatch factors. fLED reflects actual measurement procedures. fLED agrees with results. fLED applies to all LEDs and photometers investigated and is robust enough for future developments.

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**fLED fLED and manufacturers:**

fLED helps in design of better photometers. fLED does not require any more measurements than is currently done for calculation of f1’. fLED can be calculated from limited range data – it does not require the full visible range. fLED should be calculated from data at 1 nm or smaller intervals.

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fLED fLED and users: fLED provides a much better selection criterion than f1’. fLED is a property of the photometer, eliminating confusion on calibration requirements. fLED allows for optimization of photometer selection across all the user’s LED requirements. fLED gives an indication of errors in measurement. Advances in quality of photometers and better selection will reduce uncertainties in measurement.

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**Acknowledgements Thanks to NIST and Lumileds.**

For the great quantity and quality of data provided by them. Thanks to all the members of CIE TC2-45 and TC2-46. For their useful input and discussions. Special thanks to Yoshi Ohno, NIST. For all his help.

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