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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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[2]

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foundation of quantum theory,” Centaurus, vol. 43, no. 3-

4, pp. 240-259, 2001.

[3]

H. Rubens and F. Kurlbaum, “On the Heat Radiation of

Long Wave-Length Emitted by Black Bodies at Different

Temperatures,” Astrophysical Journal, vol. 14, pp. 335-

348, 1901.

[4]

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Albert Einstein, Oxford University Press, 1982.

[5]

M. Planck, “Über das Gesetz der Energieverteilung im

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553-563, 1901.

[6]

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des Lichtes betreffenden heuristischen Gesichtspunkt,”

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[7]

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