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Issued in December 2020
WHITE PAPER
Broadband radiometric measurement of light-
s with a photodiode
The spectral characteriss of the light-emiode (LED) and the photodiode are well
explained by quantum mechanics. Based on this knowledge, we will demonstrate how our
mul-wavelength LED light source can be measured with a photodiode.
This is a preview version of our white
paper on LED and photodiode
photonic devices
To get the full version, just send us an
e-mail: info@idonus.com
2 LED AND PHOTODIODE PHOTONIC DEVICES idonus.com
1. Introduction
Basic science is primarily a discipline based on
observation. Many fundamental discoveries owe much
to the quality of the underlying observations, i.e. the
precise and accurate measurement of phenomena. A
fascinating story that illustrates this is that of Kepler’s
laws describing the motion of the planets around the
Sun. If Johannes Kepler succeeded to make his
extraordinary discoveries, it is undeniably because he
had extremely accurate data from Tycho Brahe,
especially his meticulous observations of Mars recorded
over a whole decade. Kepler, who had been Brahe’s
assistant, knew that these data were absolutely reliable,
and definitely the most accurate at the time.
The discovery of quantum mechanics is also closely
linked to the development of accurate instrumentation,
particularly that used for the build and characterisation
of blackbody radiation. Here, we are interested in the
spectral characterisation of LEDs. In section 3, we will
present the study of blackbody radiation and see that it
is a very suitable entry point to tackle this subject. Our
approach being made from a historical perspective, this
will lead us to talk about the discovery of the
photoconductive effect in section 4. Section 5 will be
devoted to semiconductor properties and their use in
photonic devices: the photodiode, whose responsivity
model will be presented in section 6; the LED, whose
characteristic electroluminescence spectrum will be
explained in section 7. Finally, in section 8, with the help
of the technical developments presented in previous
sections, we will be able to develop the main idea of this
document: namely, the use of the photodiode for
radiometric measurement of broadband LEDs. Before
that, we will first introduce in section 2 the fundamental
constants that will be used throughout this paper.
2. The seven defining constants of the
International System of units
As early as 1900, Max Planck suggested that the two
constants kB (now known as the Boltzmann constant)
1
and h (the Planck constant) which appear in his equation
of radiation entropy, together with the speed of light in
1
For the sake of consistency with the rest of the document, we have
adapted several historical formulas using today’s most commonly
accepted symbols.
vacuum c and the gravitational constant G,
2
could be
used as
[fundamental constants] to define units for the
length, mass, time and temperature, which are
independent of special bodies or substances, keep
their significance for all times and for all, including
extra-terrestrial and non-human civilisations, and
can therefore be called “natural units of
measurement”.
— Max Planck [1], p. 121.
(quotation translated from
the original German text)
At that time, the centimetre–gram–second (CGS)
system of units was the predominant system used for
scientific purposes. The CGS was superseded by the
metre–kilogram–second (MKS) system, which in turn
was extended (MKSA, the A standing for ampere) and
finally replaced by the International System of Units (SI),
the modern form of the metric system. The SI was
created in 1960 and has become the universal system of
units and the standard measurement language for trade
and science.
Since 1971, the SI consists of seven base units which
are the metre (the unit of length with the symbol m), the
kilogram (mass, kg), the second (time, s), the ampere
(electric current, A), the kelvin (thermodynamic
temperature, K), the mole (amount of substance, mol),
and the candela (luminous intensity, cd). In 2019, the SI
made a decisive step forward. From that date, the
magnitudes of all SI units have been defined by declaring
exact numerical values for seven defining constants (see
Table 1). These defining constants are the speed of light
in vacuum c (defining constant for the meter, c → m), the
Planck constant h (h → kg), the hyperfine transition
frequency of caesium ΔνCs (ΔνCs → s), the elementary
charge e (e → A), the Boltzmann constant kB (kB → K), the
Avogadro constant NA (NA → mol), and the luminous
efficacy Kcd (Kcd → cd). It is quite remarkable that three of
these defining constants happen to be those that had
been advised more than a century before by Planck.
2
Newtonian constant of gravitation, G = 6.67408 × 10-11 m3․kg-1․s-2
(2018 CODATA recommended value).
The spectral power distribution of a light-emitting diode (LED) has a distinctive
asymmetrical gaussian shape. This is the macroscopic expression of finely tuned
properties buried inside the semiconductor. In this white paper, we unveil why and how
a photodiode can be used as a reliable radiometric instrument for the characterisation of
broadband LEDs. The key idea is to use prior knowledge of the centroid wavelength λc of
the emitted light and account for it to calculate the responsivity (λc) of the photodiode.
LED AND PHOTODIODE PHOTONIC DEVICES 3
creative engineering and manufacturing
3. Blackbody radiation and the birth of
quantum mechanics
To understand what motivated Planck to investigate
the radiation entropy mentioned above, let's go back to
the 19th century and examine one object that captivated
Planck and so many other renowned physicists for
several decades. In 1859, Gustav Kirchhoff coined the
term “blackbody” to describe that object: a body that
perfectly absorbs all thermal radiation falling upon it [2].
As is well known, black surfaces absorb light, they also
absorb the greatest amount of thermal radiation. But
there is another phenomenon associated with
absorption which, in scientific terms, can be translated
into the following statement: a surface in thermal
equilibrium has an equal capacity of
absorption and emission of thermal
radiation; this relationship between
absorption and emission is known as
Kirchhoff’s law of thermal radiation.
Thus, a blackbody emits radiations
whose characteristics are
independent of the nature of the
source of radiation and depend solely
on its temperature. To demystify the
blackbody, one can consider that solar
radiation falling on the Earth closely
approaches that of a blackbody in
thermal equilibrium at 5777 K
(≈ 5500 °C), as we shall see later. The
importance of blackbody radiation is
now obvious, as it is crucial for the
understanding of thermal radiation
and its laws.
From a practical perspective, in
order to build a blackbody and be able
to study it, one has to heat a cavity to
a uniform temperature and allow the
radiation to escape through a small
aperture. As Kirchhoff had imagined in
1859, such a black cavity radiator is
very close to an ideal blackbody. Yet,
although simple in appearance, it was
not until the close of the 19th century
that a truly blackbody was designed at
Table 1: The seven defining constants of the International System of Units (SI).
Defining constant
Symbol
Numerical value
Unit
speed of light in vacuum
299 792 458
m․s-1
Planck constant a
6.626 070 15 ×10-34
J․s
hyperfine transition frequency of caesium-133 (133Cs)
9 192 631 770
s-1
elementary charge a
1.602 176 634 ×10-19
C
Boltzmann constant a
1.380 649 ×10-23
J․K-1
Avogadro constant a
6.022 140 76 ×1023
mol-1
luminous efficacy of 540 THz radiation b
683
lm․W-1
a) These numerical values have been fixed to their best estimates, as calculated from the 2017 CODATA
special adjustment.
b) Using the relation and considering a monochromatic radiation of frequency 5.4 ×1014 Hz, we
find 555.17 nm for the corresponding wavelength of the light source (green).
Figure 1: The electrical glowing blackbody designed by O. Lummer and F.
Kurlbaum in 1898 [2]. Current heats the filament located in a tube inside the
cylinder to a fixed temperature, giving rise to blackbody radiation inside that
cylinder. The spectrum of this radiation is observed through the hole found at
one end along the axis of the cylinder. With a current of about 100 A,
temperatures of about 1500 °C (1773 K) could be attained.
4 LED AND PHOTODIODE PHOTONIC DEVICES idonus.com
the Physikalisch – Technische Reichsanstalt (PTR) in
Berlin (see Figure 1). There, blackbody radiation was a
lively research topic for both experimental and
theoretical physicists for two complementary reasons:
1. Practical (metrology) – The search for better
standards (e.g., absolute temperature scales),
and in particular for a reliable standard for the
radiation of light (radiometry).
2. Theoretical (radiation laws) – The construction
of black cavities closely approaching blackbody
radiation opened up a path to investigate the
exact nature of radiation processes.
The apparatuses developed at the PTR meant great
progress for radiation measurements. Experimental
physicists could verify with precision measurements a
law which had been empirically found by Joseph Stefan
in 1879 and theoretically derived by Ludwig Boltzmann in
1884. The Stefan–Boltzmann law states that the radiant
emittance, , of a blackbody is proportional to the
fourth power of its thermodynamic temperature:
[W.m-2] or [kg.s-3]
Eq. 1
The constant of proportionality is called the Stefan–
Boltzmann constant. Today, it can be calculated exactly
from the SI defining constants introduced in Table 1:
[W.m-2.K-4] or [kg.s-3.K-4]
Eq. 2
5.670 374 419 … ×10-8 W.m-2.K-4
They could also conduct an experimental proof of Wien's
displacement law that had been discovered by Wilhelm
Wien. Wien's displacement law states that the blackbody
radiation curve for different temperatures will peak at
wavelengths that are inversely proportional to the
temperature. When considering the spectral radiance of
blackbody radiation per unit wavelength (), it is
found that it peaks at the wavelength:
[m]
Eq. 3
The constant of proportionality is called Wien's
displacement constant. It can be calculated exactly by
solving a transcendental equation. Using the SI defining
constants introduced in Table 1, an approximate value is:
[m.K]
Eq. 4
2.897 771 955 … ×10-3 m.K
More importantly, they were also able to test Wien
distribution law of thermal radiation (now known as
Wien’s approximation). Using our current knowledge,
3
3
Of course, photon energy () didn’t appear in this form in
Wien’s original equation since it was Planck who introduced energy
this law may be written in terms of the spectral energy
density as a function of frequency :
[J.m-3.Hz-1]
or [kg.m-1.s-1]
Eq. 5
Alternatively, Wien’s approximation can be written in
terms of the spectral energy density as a function of
wavelength :
[J.m-3.m-1]
or [kg.m-2.s-2]
Eq. 6
The decisive contribution of the PTR team came from the
tremendous refinement of their measurements at longer
wavelengths. Indeed, the investigations revealed
significant deviations from Wien’s theoretical radiation
at longer wavelengths. Conversely, a law that had been
proposed earlier by John W. Rayleigh proved valid on
long wavelengths but failed dramatically on short
wavelengths (a divergence that would be coined the
“ultraviolet catastrophe” by Paul Ehrenfest in 1911). This
law, today known as the Rayleigh–Jeans law, can be
obtained using only arguments from “classical” physics:
[J.m-3.Hz-1]
or [kg.m-1.s-1]
Eq. 7
[J.m-3.m-1]
or [kg.m-2.s-2]
Eq. 8
These contradictory results were presented by H. Rubens
and F. Kurlbaum to the Prussian Academy in October
1900 and published one year after [3].
This work is considered to be the turning point in
theoretical research on blackbody radiations. Indeed, it
turns out that Rubens was a friend of Planck, as reported
by the science historian A. Pais [4]. In the course of a
conversation, Rubens mentioned to Planck that he had
found to be proportional to for large
wavelengths, i.e. in the infrared. In fact, it didn’t take
Planck long time to find a solution satisfying both Eq. 6 at
short wavelengths and Eq. 8 at long wavelengths.
Through interpolation, he found:
[J.m-3.m-1]
or [kg.m-2.s-2]
Eq. 9
for the spectral energy density as a function of
wavelength. Or, when expressed in terms of frequency
instead of wavelength, using Eq. 5 and Eq. 7:
[J.m-3.Hz-1]
or [kg.m-1.s-1]
Eq. 10
quanta (see Appendix) and Einstein who introduced the concept of
light quanta (see section 4).
LED AND PHOTODIODE PHOTONIC DEVICES 5
creative engineering and manufacturing
These two equations appear exactly in this form in
Planck’s famous paper “On the law of distribution of
energy in the normal spectrum” published in 1901 [5].
Planck’s discovery is by no means limited to an
interpolation of experimental data. This was actually the
starting point of the most heroic period of his life.
Blackbody radiation involved an inescapable break with
classical physics. As a physicist, he had to find a rational
explanation. His law had to derive from a fundamental
principle. He succeeded to give a physical explanation,
but in order to do so he had to make the following
hypothesis: radiation energy is found in the form of
discrete energy elements – i.e., quantized energy – that
are proportional to the frequency :
[J] or [kg.m.s-2]
Eq. 11
For further details, readers will find in the Appendix a
brief overview of the masterful demonstration that led
Planck to the postulate of energy quanta.
To conclude this section on blackbody radiation, we
show in Figure 2 the solar radiation spectrum as
4
The Sun as seen from Earth has an average apparent angular
diameter of 2 0.5334°. The corresponding solid angle is
, expressed in steradian [sr].
compared to a 5777 K blackbody (about 5500 °C). The
ASTM E-490 solar spectral irradiance is based on a
collection of data recorded above the atmosphere. Its
integrated spectral irradiance has been made to conform
to the value of the solar constant accepted by the space
community, which is 1366.1 W/m². The spectral
irradiance is calculated from the spectral energy
density given in Eq. 6:
[W/(m2.m)]
Eq. 12
with 6.807×10-5 sr, the solid angle of the Sun as
seen from Earth.
4
Although the Sun is not a perfect
blackbody, we can see a relatively good correspondence
with the 5777 K blackbody. According to Wien’s
displacement law (Eq. 3), the peak wavelength of the
5777 K blackbody is around 502 nm:
(5777 K)
5.016 ×10-7 m
502 nm
Figure 2: Solar radiation spectrum is compared with a 5777 K blackbody (≈ 5500 °C). ASTM E-490 represents solar
spectral irradiance above the atmosphere. For this temperature of the blackbody, Wien’s displacement law predicts a
peak wavelength around 500 nm.
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
0 250 500 750 1000 1250 1500 1750 2000 2250 2500
Spectral irradiance Ee,λ(λ,T), W/(m².nm)
λ (nm)
ASTM E-490 AM0 Standard Spectra
E_e,λ (λ,T = 5777 K)
Wien's displacement law
Peak wavelength: 502 nm (T=5777 K)
14 LED AND PHOTODIODE PHOTONIC DEVICES idonus.com
of LEDs using a simple photodiode. The proof can be
summarised in two main points:
1. The electroluminescence spectrum of a LED can
be approximated by a sum of gaussian curves
from which a centroid wavelength can be
derived;
2. By appropriate choice of a photodiode (i.e, with a
linearizable responsivity throughout the
entire bandwidth of the LED), its output signal
gives a correct radiometric measurement of the
LED provided scaling by its responsivity at
centroid wavelength, .
This is detailed in section 8 and we could have limited this
paper to that section. However, we wanted to take the
investigation on LEDs spectra a step further and explain
the origin of their asymmetric gaussian-like shape.
One thing leading to another, what we did out of
scientific curiosity eventually led us to dig in history of
science, with blackbody radiation as the starting point. As
we have stressed it in section 3, it was indeed the need
for reliable standard for the radiation of light that
prompted standardisation institutes to take an interest
in blackbody radiation at the end of the 19th century. In
light of what we have seen throughout this paper, the
blackbody is in many ways essential to the understanding
of quantum photonic devices, and LEDs in particular.
Since we have developed the subject from a historical
perspective, this has led us to mention several renowned
scientists, including Nobel Prize laureates from the 20th
century. It is therefore logical that we conclude this
paper by recalling that the Nobel Prize in Physics 2014
was awarded jointly to Isamu Akasaki, Hiroshi Amano
and Shuji Nakamura “for the invention of efficient blue
light-emitting diodes which has enabled bright and
energy-saving white light sources.” The Nobel committee
was not mistaken: LED technology has a bright future
ahead of it!
Appendix: Planck’s steps to the discovery of
quantum theory
Planck wanted to interpret Eq. 10 which he had
discovered empirically. His original derivation of that
equation made him the discoverer of quantum theory
[21]. To appreciate the importance of his work, we shall
outline the three steps he took.
1. Classical electromagnetic theory
First, he established the relation:
Eq. 45
6
Planck chose the symbol , which is the first letter of
“Wahrscheinlichkeit”, the German word for probability.
between the energy density of the equilibrium
radiation at temperature and the average energy
of a resonator of frequency and temperature . He
completed this proof on the basis of classical
electromagnetic theory. Comparing Eq. 10 and Eq. 45,
he could then find :
Eq. 46
2. Thermodynamics and entropy
Planck was a convinced promoter of entropy. In the
second step, he determined the entropy, , of the
resonators by integration of . From Eq. 46,
he evaluated as a function of for a fixed
frequency . He obtained:
Eq. 47
3. Statistical thermodynamics
The third step was the revolutionary one. To complete
this ultimate stage, he drew heavily on Boltzmann’s
work on statistics and entropy [22], [23]. To this end,
he considered a system of resonators vibrating at
frequency . The total energy of these oscillators is
, to which corresponds a total entropy
. In an “act of desperation”, as he would
qualify it later, he then made the ad hoc assumption
that the total energy was made up of finite energy
elements , such that , with a large
integer.
He followed one of Boltzmann’s ideas according to
whom entropy, apart from an additive constant, is
proportional to the logarithm of the number of
“complexions” that constitute the equilibrium state of
the system. Although Boltzmann never wrote down
the equation, Planck formulated it as follows:
Eq. 48
Then, he calculated the number of “complexions”
(or permutations in combinatorics)
6
for a discrete
system consisting of energy elements that are
distributed between resonators:
Eq. 49
Applying Stirling’s formula,
7
he found:
Eq. 50
7
LED AND PHOTODIODE PHOTONIC DEVICES 15
creative engineering and manufacturing
Finally, using and , he
obtained:
Eq. 51
Since entropy only depends on according to
Wien’s displacement law, it follows from the
comparison of Eq. 47 and Eq. 51 that
Eq. 52
This is how quantum theory was born. Planck has
been quite criticized for his audacity on this third step,
especially for his use of Eq. 49 for which he had no
justification, except that it was giving him the result
he was looking for...
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Long Wave-Length Emitted by Black Bodies at Different
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348, 1901.
[4]
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[5]
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© 2020 idonus sàrl. All rights reserved.
About idonus
Founded in 2004, idonus is a Swiss company that
develops and manufactures special equipment for the
MEMS and watchmaking industries. Our product
portfolio includes UV-LED exposure systems for
photolithography, IR microscope for wafer inspection,
vapor phase chemical etcher for silicon-based devices.
Since 2016, we also provide ion implantation services
and machines for the surface treatment of materials.
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T: +41 32 724 44 40
info@idonus.com
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