PONTIFICIA UNIVERSIDAD CATO´LICA DEL PERU´
ESCUELA DE POSGRADO
IMPLEMENTATION OF A SPECKLE-BASED
SPECTROMETER
TESIS PARA OPTAR EL GRADO DE MAGI´STER EN F´ISICA
AUTOR
DIEGO RENATO UGARTE LA TORRE
ASESOR
FRANCISCO ANTONIO DE ZELA MARTINEZ
JURADO
EDUARDO RUBEN MASSONI KAMIMOTO
EDER RUBE´N SA´NCHEZ ALCA´NTARA
LIMA - PERU´
2016

Agradecimientos
Agradezco la asesor´ıa brindada por el profesor Francisco De Zela, las horas de dis-
cusio´n con el profesor Eduardo Massoni, la ayuda de Yonny Yugra durante la imple-
mentacio´n y calibracio´n del me´todo y al financiamiento de CONCYTEC que hizo posible
la realizacio´n de este trabajo.

Resumen
Existen diversos me´todos para medir la longitud de onda de la luz. Uno de estos
me´todos esta´ basado en la relacio´n que existe entre la longitud de onda y los patrones
de moteado. La implementacio´n de este me´todo consiste en hacer ingresar luz con
un ancho espectral pequen˜o sobre un extremo de una fibra o´ptica multimodal, para
generar patrones de moteado a la salida de la fibra o´ptica. Estos patrones de moteado
se relacionan con las longitudes de onda que contiene la luz que ingresa a la fibra o´ptica.
En el presente trabajo se propone una nueva implementacio´n que consiste en usar una
pantalla de cristal l´ıquido (LCD) en vez de una fibra o´ptica, para obtener patrones de
moteados ma´s estables frente a las vibraciones meca´nicas a las que esta´ expuesto el
arreglo experimental. Los resultados indicaron que es posible implementar el me´todo
usando LCDs. Estos dispositivos ofrecen una mayor resistencia al ruido meca´nico y
mejoran la reproducibilidad de los patrones de moteado generados.
Abstract
There are different methods to measure light wavelength. One of these methods is
based on the relationship between wavelengths and speckle patterns. The original imple-
mentation of this method used a multimode optical fiber to generate speckle patterns.
To test if this method still operates appropriately when it uses a different source of
speckle patterns, this work proposes an implementation in which the multimode optical
fiber is replaced by a liquid crystal display (LCD). Our results show that it is possible to
implement the method with a LCD. The use of a LCD reduces the mechanical instability
and improves the reproducibility of the generated speckle patterns.
Contents
List of Figures
1 Introduction 1
2 Speckle theory 3
2.1 Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.1.1 Huygens-Fresnel principle . . . . . . . . . . . . . . . . . . . . 3
2.1.2 Diffraction from an aperture . . . . . . . . . . . . . . . . . . . 4
2.2 Speckle patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 Speckle statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3 Speckle-based spectroscopy 13
3.1 Wavelength measurement . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2 Spectrum reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.3 Measurement optimization . . . . . . . . . . . . . . . . . . . . . . . . 19
4 Results 21
4.1 Method implementation . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.2 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.3 Spectral composition reconstruction . . . . . . . . . . . . . . . . . . . 26
5 Conclusions 29
6 Bibliography 31
A Statistics of speckle patterns 33
A.1 Random phasors statistics . . . . . . . . . . . . . . . . . . . . . . . . 33
A.2 Intensity and phase statistics . . . . . . . . . . . . . . . . . . . . . . . 38
List of Figures
1 Huygens-Fresnel principle . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Aperture diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3 Plane wave diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . 6
4 Square aperture diffraction pattern . . . . . . . . . . . . . . . . . . . 6
5 Speckle pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
6 Rough surface ilumination . . . . . . . . . . . . . . . . . . . . . . . . 8
7 Random phase range effect . . . . . . . . . . . . . . . . . . . . . . . . 9
8 Correlation function . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
9 Prism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
10 Wavelength dependent speckle patterns . . . . . . . . . . . . . . . . . 14
11 Wavelength measurement . . . . . . . . . . . . . . . . . . . . . . . . . 15
12 Speckle pattern spatial sampling . . . . . . . . . . . . . . . . . . . . . 17
13 LCD representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
14 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
15 Speckle pattern comparison . . . . . . . . . . . . . . . . . . . . . . . 23
16 Speckle pattern intensity distribution . . . . . . . . . . . . . . . . . . 24
17 Calibration speckle patterns . . . . . . . . . . . . . . . . . . . . . . . 25
18 Sample speckle patterns . . . . . . . . . . . . . . . . . . . . . . . . . 26
19 Reconstructed spectra . . . . . . . . . . . . . . . . . . . . . . . . . . 28
Chapter 1
Introduction
Spectrometers have been useful tools in many areas of science, including medicine.
Depending on the specific application different specs are sought in these equipments.
Among different characteristics, the resolving power and the operational bandwidth
have been of special interest. Besides, nowadays’ requirements look for smaller and
more portable apparatus as well.
Classically, diffraction gratings have been used in spectrometers as their main
component. These gratings are dispersive elements that allow to map in a one-to-one
correspondence wavelength information and one dimensional spatial variables [1].
Nowadays there are in the literature novel implementations that take advantage of
the properties of different materials like nanocavities, photonic crystals and optical
fibers [2].
One important feature of the optical fiber spectrometer is a high resolving power
at a relative low implementation cost [1]. This spectrometer maps wavelength infor-
mation in a quite different way as compared to diffraction gratings because optical
fibers allow the generation of random intensity diffraction patterns called speckle
patterns. Speckle patterns are highly dependent on light wavelength and this spec-
trometer uses this feature to achieve a high performance.
On the other hand, optical fiber spectrometers suffer from high mechanical in-
stability. Efforts have been made to offset this disadvantage [3, 4], fixing the optical
1
fiber in order to reduce the vibrational noise. Even though, this solution limits its
possible applications.
In this work an alternative implementation of the optical fiber spectrometer is
proposed. In order to reduce the mechanical instability, a liquid crystal display
(LCD) is used. LCDs acts as spatial light modulators (SLM) and offer a greater
control in the reproducibility of the generated speckle patterns as compared to op-
tical fibers.
The presentation of this work has been organized as follows. In chapter 2 a brief
review of the diffraction theory is given before addressing the speckle phenomenon.
Then, section 2.2 explains how the addition of a random phase term to the diffrac-
tion equation of a square aperture describes the speckle patterns produced with the
LCD and in section 2.3 some statistical properties that characterized speckle pat-
terns are presented.
The spectroscopy method based on speckle patterns is explained in chapter 3.
Section 3.1 shows how the relation between speckle patterns and wavelength is used
as the basis of the method. Section 3.2 describes the calibration process and section
3.3 contains a method to optimize the calculated spectrum. The method implemen-
tation using a LCD is tested and the main results are presented in chapter 4. Finally,
in chapter 5 some conclusion remarks are given.
2
Chapter 2
Speckle theory
This chapter introduces the theory behind the speckle-based spectroscopy. In section
2.1, a qualitative description of light diffraction is presented. Then, the speckle
phenomenon is explained in section 2.2 and finally in section 2.3 the main results
of the speckle statistics are exposed as a way to characterize qualitatively speckle
patterns produced from different light sources.
2.1 Diffraction
2.1.1 Huygens-Fresnel principle
Diffraccion is a term used to describe different phenomena that occur when a propa-
gating wave encounters an obstacle. This topic was addressed by several researchers
such as Huygens, Young, Fresnel, Kirchhoff, Rayleigh and Sommerfeld [5].
One of the first attempts to explain wave propagation through a homogeneous
isotropic medium was made by Christiaan Huygens. He made the following two
assumptions:
1. Each point of a wavefront acts as a source of secondary wavelets. These
wavelets spread out with the same velocity of the wavefront from which they
were originated.
2. The new wavefront at any later time is the same as the tangencial surface of
all the secondary wavelets at that time.
3
These assumptions permitted a better understanding of the propagation of plane
and spherical waves. However, they still could not account for the propagation of
light near edges or through small appertures. It was not until 1818, when Augustin-
Jean Fresnel incorporated his interference principle to Huygens assumptions, that
diffraction effects could also be explained [6].
Figure 1: Huygens-Fresnel principle states that every point of a wave-
front is a point source of a secondary spherical wavelets. The tangent
line to all these wavelets forms a new wavefront.
The addition of Fresnel ideas to Huygens’ work led to the Huygens-Fresnel prin-
ciple (Fig.1). This principle states that “every unobstructed point of a wavefront,
at a given instant, serves as a source of spherical secondary wavelets (with the same
frequency as that of the primary wave). The amplitude of the optical field at any
point beyond is the superposition of all these wavelets (considering theis amplituides
and relative phases)” [7].
2.1.2 Diffraction from an aperture
The Huygens-Fresnel principle can be applied to describe qualitatively the diffrac-
tion produced by a wave when passing through a small aperture Fig.2. According
to this principle, each infinitesimal element of the aperture is represented by a source
point of spherical waves.
4
Figure 2: The field at a point P due to a infinitesimal area element dS
of the aperture. Following Huygens-Fresnel principle, each point of the
aperture acts as a source of spherical waves.
For monochromatic light the field due to an infinitesimal area element at a point
P away from the aperture is expressed as
dE(rp) =
A(r)
|rp − r|e
i(ωt−k|rp−r|+φ(r))dS (2.1)
where r is the position of the infinitesimal element, rp is the position of point P,
A(r) is the field intensity per unit area at point r, ω is the angular frequency, k is
the wavenumber and φ is phase factor. The total field is obtained just integrating
the above expression over all the aperture:
E(rp) =
∫∫
S
A(r)
|rp − r|e
i(ωt−k|rp−r|+φ(r))dS . (2.2)
Finally, when a propagating plane wave is parallel to the aperture (Fig.3), it is
possible to introduce some simplifications to the last expression. For a plane wave,
the field intensity per unit area is constant over the plane, A(r) = Ao. Also, since
5
Figure 3: When a plane wave pass through an aperture, every source
point over the aperture surface produce spherical wavelets with the same
initial phase.
the wave vector is perpendicular to the aperture plane, then all source points will
be in phase with one another allowing to set φ = 0. The resulting equation is
E(rp) =
∫∫
S
Ao
|rp − r|e
i(ωt−k|rp−r|)dS . (2.3)
Evaluating this equation permits to obtain the diffraction pattern generated from
different aperture shapes. In Fig.4 it is shown the diffraction pattern produced by
a square apperture in the Fraunhofer limit 1.
Figure 4: A cross-like diffraction pattern produced by a square aperture.
1The Fraunhofer limit is reached when the relation AλD < 0.01 is fulfilled. Here A is the area of
the aperture, λ is light wavelength and D is the distance between the aperture and the diffraction
pattern [8].
6
2.2 Speckle patterns
Speckle patterns are random intensity distributions formed “when fairly coherent
light is either reflected from a rough surface or propagates through a medium with
random refractive index fluctuations” [9]. One characteristic of these patterns is
to present lots of tiny spots corresponding to zones of maximum constructive and
destructive interference [10] as it is shown in Fig.5.
Figure 5: A speckle pattern simulated in MATLAB 2015b.
In order to study the randomness and other features in the intensity distribution
of a speckle pattern, it is necessary to first find an expression for E, because the
light intensity is proportional to |E|2. Invoking the Huygens-Fresnel principle in
both rough surfaces and the end of optical fibers allows the use of equation 2.2.
However, solving it using experimental data becomes a very difficult task because it
requires to know the quantity |rp − r| for all source points. For example, it would
be necessary to know in detail the topology of a rough surface or the exact field
intensity at the end of an optical fiber.
One solution is to use simulation techniques. There are many methods available
in the literature to simulate speckle patterns [8, 11, 12, 13]. The most well know
example of speckle pattern formation comes from the ilumination of a rough surface
with laser light (Fig.6). Let each point of the surface be a source of spherical waves
following the Huygens-Fresnel principle. For a perfectly flat surface equation 2.3
could be used, but since there is roughness, each source point will be away from P
an extra distance d that depends entirely on its position on the surface.
7
Figure 6: Laser light reflected from a rough surface. Here D is the
distance between the highest and lowest point of the surface and θ is the
angle formed between the line joining a source point and point P and a
line perpendicular to the rough surface.
The corrected expression that accounts for the roughness of the surface is then
E(rp) =
∫∫
S
Ao
|rp − r|e
i[ωt−k(|rp−r|+d(r))]dS
=
∫∫
S
Ao
|rp − r|e
i(ωt−k|rp−r|)eikd(r)dS
=
∫∫
S
Ao
|rp − r|e
i(ωt−k|rp−r|)eiφ(r)dS
(2.4)
where φ(r) = kd(r) = 2pi
d(r)
λ
.
Let D represent the distance between the highest and lowest point in the rough
surface, and θmax be the maximum angle formed between the line joining a source
point and point P and a line perpendicular to the surface. The phase φ(r) will be
bounded by
0 ≤ φ(r) ≤ 2pi
D
λ cos θmax
D ≤ λ cos θmax
0 ≤ φ(r) ≤ 2pi D > λ cos θmax
(2.5)
8
If D is very small compared to the light wavelength used to iluminate the sur-
face (D << λ), φ is almost 0 and there is no contribution from the additional path
length d. On the other hand, when D is in the same order of magnitud of λ or even
greater, φ ranges from 0 to 2pi and makes a significant contribution.
The addition of a uniform distributed random phase term to equation 2.3 repro-
duces a diffraction pattern that gradually resembles a speckle pattern as φ’s range
increases. This means that any element that produces a diffraction pattern can also
generate speckle patterns as long as it introduces a random phase term at each of
the source points. For example, in Fig.7 it is shown how the diffraction pattern
produced by a square aperture becomes a speckle pattern when the range of the
introduced random phase φ is augmented.
Figure 7: Diffraction patterns produced by a square aperture and dif-
ferent random phase ranges. (a) In this case there is no contribution by
the φ term and the well-known diffraction pattern is formed. (b)-(c)
As the phase range increases, the difraccion pattern loses gradually its
original shape. (d) When φ’s range is maximum (0 ≤ φ ≤ 2pi) the
original pattern is completely lost and a speckle pattern is formed.
9
In order to appreciate how φ’s range contribute to the formation of a speckle
pattern a correlation function is introduced:
C(θ) =
〈I(0◦)I(θ)〉
〈I(0◦)〉〈I(θ)〉 (2.6)
where
I(θ) = I(0≤φ≤θ) (2.7)
is the intensity produced when a random phase that ranges from 0 to θ is added.
The value of the correlation function for all the possible ranges of φ is shown in
Fig.8. As the range increases the correlation function quickly decreases and when
the range is at half of its maximum the correlation is below 0.5.
Figure 8: Normalized correlation function C(θ) of the diffraction pattern
produced by a square aperture. As the upper limit θ of the random phase
increases, the resulting pattern decorrelates quickly with the original
diffraction pattern.
10
2.3 Speckle statistics
Due to the random nature of speckle patterns, its properties are best studied using
statistical techniques. These techniques are derived from the statistical tools used
to study phasors because the intensity of a speckle patterns can be seen as a sum
of a large number of complex waves. In the appendix it is demonstrated that the
intensity distribution on a speckle pattern has the following form
PI(I) =
1
〈I〉e
− I〈I〉 (2.8)
where 〈I〉 represents the mean intensity in the speckle pattern and it was assumed
that the speckle pattern was produced with monochromatic light and that the ran-
dom phase φ had its full range of 0 to 2pi.
The moments of this distribution are
〈Iq〉 = 〈I〉qq ! . (2.9)
Using this equation the second moment for the intensity is
〈I2〉 = 2〈I〉2 . (2.10)
In terms of the first and second moments the variance can be calculated:
σ2I = 〈I2〉 − 〈I〉2 (2.11)
= 2〈I〉2 − 〈I〉2 = 〈I〉2 (2.12)
and taking the square root the standard deviation results
σI = 〈I〉 (2.13)
To characterize in a quantitative way the presence of speckle in an intensity
pattern, two important quantities are used: the contrast (C) and its inverse, the
signal-to-noise ratio (S/N). These quantities are defined as
11
C =
σI
〈I〉 S/N =
1
C
=
〈I〉
σI
. (2.14)
Replacing in these definitions the mean intensity and the standard deviation for a
speckle pattern, it is found that for fully developed speckle the constrast and the
signal-to-noise ratio are equal to unity:
C = 1 and S/N = 1 (2.15)
12
Chapter 3
Speckle-based spectroscopy
This chapter presents the speckle-based spectroscopy method in which light’s wave-
length is measured using speckle patterns. This method was first developed by
Brandon Redding and Hui Cao in [1]. They implemented it using a multimode
optical fiber achieving a low spectral resolution (∼ 10−2 nm). Section 3.1 explains
the basis of the speckle-based spectroscopy method. Then, the calibration proce-
dure is described in the section 3.2. Finally, section 3.3 presents the optimization
algorithm used to improve the precision of the resolved spectra.
3.1 Wavelength measurement
Throughout history different methods to measure light’s wavelength have been de-
veloped. Among the first ones are those which relate the spatial dependence of light
intensity with wavelength information. The classic example of these methods is the
use of a prism to produce intensity patterns whose position on a screen depends on
the wavelength of the light source (Fig.9).
To ensure reliability of the method, it is required that the produced intensity
patterns are in one-to-one correspondence with light wavelength. As long as this
requirement is fulfilled, the intensity patterns can adopt any shape. This allows the
use of speckle patterns.
13
Figure 9: Light’s wavelength measurement using a prism. Depending on
the wavelength, an intensity pattern will be formed in a specific position
on the screen.
In chapter 2 the equation 2.4 described the field produced by a speckle pattern.
There it was shown that the uniformly distributed random phase φ was a function
of position r and it was assumed that the light source was monochromatic, i.e.
λ = λ0. When λ is allowed to change, the random phase also becomes a function of
wavelength φ(r,λ) and the speckle pattern now gradually changes as λ does (Fig.10).
Figure 10: Speckle patterns simulated in MATLAB 2015b for different
wavelength values. Since the random phase φ is also a function of λ, a
different pattern is formed in each case.
For a sufficiently large change in lambda (δλ), each speckle pattern will be dif-
ferent from the others. The speckle-based spectroscopy takes advantage of this to
14
stablish the required one-to-one correspondence, in this case between wavelengths
and speckle patterns.
The quantity δλ defines the spectral resolution since it represents the minimum
change in wavelength needed to produce a different speckle pattern. In [1] a spectral
correlation function, C(∆λ), is defined in order to measure how distinguishable two
speckle pattern are when they only differ in a small change in wavelength (∆λ):
C(∆λ) =
〈I(λ)I(λ+∆λ)〉
〈I(λ)〉〈I(λ+∆λ)〉 − 1 . (3.1)
Using this function δλ is defined as twice the value of ∆λ that makes the spectral
correlation function be 50% of it maximum value:
δλ = 2∆λ1/2 ; C(∆λ1/2) =
C(0)
2
. (3.2)
Finally, recording speckle patterns formed by different light’s wavelengths pro-
duces a database of “fingerprints” which can be used to identify by comparison the
wavelength of a unknown light source. In Fig.11 the method is summarized in three
steps.
Figure 11: The speckle-based spectroscopy method takes advantage of
the fact that speckle patterns act as wavelength fingerprints. Step 1:
The calibration database of speckle patterns is built. Step 2: A speckle
pattern is formed with a light source of unknown wavelength. Step 3:
By comparison with the database the unknown wavelength is identified.
15
3.2 Spectrum reconstruction
When a spectrometer is implemented using the speckle-based spectroscopy method,
it needs to be calibrated before usage. The construction of the speckle pattern
database corresponds to this calibration step. In the present method, the database
is represented by a transmission matrix T and its construction will be explained in
this section.
One way of representing the intensity of a speckle pattern produced by different
wavelengths is based on the following equation:
I(r) =
∫
T(r,λ)S(λ)dλ (3.3)
where I(r) is the intensity at a position r in the speckle pattern, T(r,λ) is a transmis-
sion function and S(λ) represents the spectral flux, i.e., the intensity contribution
from each different wavelength [1]. To be able to use this equation experimentally,
first it needs to be discretized. For that purpose two quantities are used: the spec-
tral correlation width, δλ, and spatial correlation width, δl, which is defined as the
average speckle size in a speckle pattern. If I(r) is sampled at positions r1, r2, . . . , rm
(Fig.12) for different light sources whose wavelength are λ1, λ2, . . . , λn, then equa-
tion 3.3 can be written as

Ir1
Ir2
...
Irm
 =

t11 t12 . . . t1n
t21
. . . t2n
...
. . .
...
tm1 tm2 . . . tmn


Sλ1
Sλ2
...
Sλn
 (3.4)
where the values of r differ from one another:
rm = rm−1 + δl (3.5)
and the values of λ similarly:
λn = λn−1 + δλ (3.6)
16
Figure 12: Every speckle pattern is sampled at fixed postions, marked
by “X” in the figure. Each sampling position is separated from one
another by δl, which is equal to the mean speckle size. This assures that
each sampling point is independent of each other.
The transmission matrix T , which is the discretized version of the transmission
function T(r,λ), is a m × n matrix that contains all the information necessary to
operate the spectrometer. In this matix, each column represents the intensity con-
tribution made by a specific monochromatic light of wavelength λ.
To build this matrix the first step is to record different speckle patterns produced
with monochromatic light of various wavelength. In the next step, the sampling
points (r) are selected depending on the quality of the speckle. Finally, the intensity
value at each position (Ir) is used to fill up the transmission matrix column by
column. Once it is built, it can be used to reconstruct the spectral composition of
an input speckle pattern. The spectral composition is represented by the spectral
vector S and it is easily obtained from equation 3.4 as
S = T−1I (3.7)
where the matrix T−1 is the inverse, or more specifically, the left inverse of the
transmission matrix.
17
There are two approaches to obatain the matrix T−1. The first one consists in
making the number of rows and columns of T equal (m = n) and then to calculate
the inverse in the usual way. This can be easily done by selecting an appropiate
number of sampling points. The other one consists in calculating the left pseudoin-
verse (T ′) of the transmission matrix, since in general the number of sampling points
and the number of sampled wavelengths are not the same and the inverse does not
always exist.
From the two approaches described above the second one is prefered because the
use of more sampling points improves the precision of the calculated spectral vector
S, so from here on the term T−1 in equation 3.7 will represent the left pseudoinverse
T ′. One issue when calculating the spectral composition is that despite the number
of sampling points used, the matrix T ′ will not reproduce S correctly due to the
presence of experimental noise. To improve its predictability a truncation method
is proposed in [3]. This method consists in a singular value decomposition (SVD) of
the original transmission matrix prior its inversion. SVD expresses T as the product
of three matrices:
Tm×n = Um×mDm×nV Tn×n (3.8)
where U and V are two orthonormal matrices and D is a diagonal matrix whose
elements are known as the “singular values”. Using the SVD it is possible to invert
T with the following equation:
T ′ = V D′UT . (3.9)
Here, D′ is the psudoinverse of the diagonal matrixD and it is obtained by replacing
its diagonal elements by their reciprocal. The small values in the diagonal are the
most affected by experimental noise but they do not contribute significantly to
the matrix T . However, they do contribute to D′ because the reciprocal of small
diagonal values become big numbers. To avoid the noise amplification, a truncation
step is taken before calculating D′. Only the values of D that are above a threshold
are used to calculate the elements of D′. The new matrix, Dtrunc, is then used in
18
equation 3.9 to obtain an improved matrix T ′:
T ′ = V D′truncUT . (3.10)
Finally, although direct multiplication of the last T ′ and I would now recover
S from an input signal, it is possible to improve the reconstructed spectrum when
the emphasis is in the precision. This is achieved using a minimization algorithm [3]
and will be explained in the next section.
3.3 Measurement optimization
There are two ways in which a spectrometer operates depending on the application.
When the priority is a fast detection process it is enough to use equation 3.10. On
the other hand, when the priority is in the precision it is affordable to expend time
in an optimization process.
The problem of improving the precision of the spectral vector can be seen as a
minimization problem in the following manner. After using T ′ on an input speckle
pattern, the calculated spectral vector, here represented as S′, acts as an initial
guess of the real S. To measure how good this first guess is, the following quantity
is defined:
E = ||I − TS||2 (3.11)
where || · || represents the Euclidean matrix norm defined as the largest singular
value (σmax) of a given matrix,
||A|| = σmax(A) . (3.12)
As E approaches to zero, S′ becomes a better guess of S, so minimizing E becomes
the same as improving the precision of the reconstructed spectral composition.
The minimization of E is performed using a Monte Carlo Energy Minimization
Algorithm. In the first step a new guess, S′′, is built by multiplying all the entries
19
of the initial guess S′ by random numbers that are uniformly distributed between
0.5 and 2.0. Then, the “energy difference” is calculated as follows:
∆E = ||I − TS′||2 − ||I − TS′′||2 (3.13)
In the last step the probabily exp(−∆E/T ) is calculated and used to evaluate if
the new guess is kept (S′ = S′′). Here T is the temperature of the fictitious system
and it starts at an initial temperature T0. In each iteration step the temperature
is reduced by ∆T until it reaches a final temperature Tf (T0 > Tf ). The algorithm
stops when the calculated “energy” is bellow a threshold or when the temperature
reaches Tf . The choice of optimum values for T0, ∆T , Tf , and the energy threshold
varies with each transmission matrix, so they are selected by trial and error until the
desired performance is achieved. After a few cycles of this algorithm the resulting
spectral vector S′ reproduces with much better precision the spectral composition
of the input speckle pattern in comparison with the S obtained by just using the
inverted transmission matrix T ′.
20
Chapter 4
Results
The basic setup of the speckle-based spectroscopy method is composed of three
sections: a light source, a speckle pattern generator and a detection device. The
characteristics of the light source, e.g., spectrum range and coeherence, depend
entirely on the selected speckle pattern generator and the detection device. As for
the speckle pattern generator, an optical fiber has been originally proposed [1, 3, 4].
In this chapter an alternative implementation using a liquid crystal display (SLM) is
proposed. The speckle patterns generated are then compared with the ones obtained
with an optical fiber and finally a transmission matrix T ′ is built using three light
sources of different wavelength in order to test the new implementation.
4.1 Method implementation
In chapter 2 it was shown that equation 2.4 reproduces the speckle pattern gener-
ated by illuminating a rough surface. This system can be simulated with the use
of a liquid crystal display (LCD) that acts as a SLM. One way of representing this
device is as an aperture composed of an array of pixels (Fig.13). Each of these
pixels interacts with light passing through it allowing a modulation in polarization
and phase as a function of the gray level [14, 15].
Projecting on the LCD an image composed of pixels with random gray levels
confer to passing light a different random phase term in each point of the screen.
The gray level scale of 0 to 255 permits a modulation that depends on the input
21
light’s polarization. Since a bigger range in the allowed values of the random phase
helps to the speckle formation (Ch.2 Fig.7), the full grayscale is used.
Figure 13: The LCD is represented as an aperture composed of an array
of pixels. Each pixel can project a color that ranges in the grayscale color
from 0 to 255. Projecting a random configuration of gray levels simulates
a rough surface.
The implementation of the speckle-based spectroscopy method proposed in this
work is represented in Fig.14. Laser light is used as the light source due to its
coherence length. It is followed by a polarizer (P ) that helps to control the angle
at which light beams arrive to the LCD’s screen. In order to take advantage of all
the display projecting area, a lens (L1) is placed before the LCD. After light passes
through the LCD, a speckle pattern is recorded in a CCD camera with the aid of a
second lens (L2).
In Fig.15 a speckle pattern recorded with our implementation is compared with
a speckle pattern produced with the multimode optical fiber implementation. The
LCD used in this work permitted a range of approximately 0 to pi/2 for the random
phase when the pixels have random values in the full grayscale. On the other hand,
for a multimode optical fiber, the range of the random phase is determined by the
fiber length, easily reaching the full 0 to 2pi range.
22
Figure 14: Implementation of the speckle-based spectroscopy method
using a LCD. Laser light first pass through a polarizer (P ), then the
spot is enlarged with a lens (L1) to fully utilize the projecting area of
the LCD. After the speckle pattern is formed, it is focused with a second
lens (L2) in a CCD camera.
Figure 15: Comparison of a speckle pattern produced with a LCD (left)
and a speckle pattern produced with a multimode optical fiber (right).
Since the range of the random phase is determined by the grayscale (0
to 255), the speckle pattern produced with the LCD maintains some
characteristics of the diffraction pattern of a square aperture. On the
other hand, the speckle pattern produced with an optical fiber has a
random phase that ranges from 0 to 2pi.
23
4.2 Calibration
To evaluate if the speckle pattern produced by the LCD behaves properly, the nor-
malized experimental intensity distribution is plotted against the theoretical proba-
bility distribution P(I) corresponding to a speckle pattern (Fig.16). The disagreet-
ment found for lower intensities is a consequence of the camera sensibility. During
the measurements, even when no light was projected on the CCD camera, small
fluctuations still appeared, so pixels with very small intensity values were the more
affected by noise. Nonetheless, the recorded pattern had a contrast of 0.92 and a
signal-to-noise ratio of 1.09 indicating that there were truly speckle patterns.
Figure 16: Intensity distribution of a speckle pattern generated with the
LCD. The disagreement for small intensities is due to the sensibility of
the CCD camera since pixels that should have had zero value fluctuated
in the presence of a light.
After proving that the diffraction patterns produced by the LCD almost behaves
as full developed speckle patterns, the transmission matrix was built. For this
purpose the available laser sources in our laboratory were used (Fig.17). Light of
three different wavelengths produced speckle patterns that were recorded in images
of 600x600 pixels. Due to the small spectral composition, only nine pixels were
24
selected as sampling points. These points were at coordinates (100,100), (100,300),
(100,500), (300,100), and so on.
Figure 17: Speckle patterns used to build the transmission matrix. The
implementation was tested using only three different laser light sources
available in our laboratory.
Using the images in Fig.17 the following transmission matrix was obtained:
T =

0.0784 0.0627 0.0235
0.0627 0.1137 0.0549
0.0275 0.0471 0.0471
0.2980 0.1647 0.0471
0.9333 0.9922 1.0000
0.0863 0.5922 0.0549
0.0824 0.0980 0.0510
0.0784 0.1529 0.1059
0.0980 0.0549 0.0235

Since T is not a square matrix, it was possible only to obtain its left pseudoin-
verse. In this case, the truncation step was unnecessary, because all the entries in
the diagonal matrix D were non-vanishing. The calculated pseudoinverse was (only
two decimals shown):
T ′ =

0.74 0.01 −0.25 3.47 −0.12 −0.87 0.39 −0.44 1.05
0.00 0.18 0.04 −0.23 −0.13 1.87 0.08 0.21 −0.08
−0.67 −0.14 0.25 −3.01 1.22 −1.06 −0.40 0.30 −0.89

25
4.3 Spectral composition reconstruction
Once the left pseudoinverse matrix T ′ was available, the spectrometer was ready to
reconstruct the spectral composition of input samples. This was tested using three
speckle patterns produced with the same light sources used to built the transmission
matrix (Fig.18).
Figure 18: Speckle patterns recorded with the CCD camera. These
patterns were used to test the T ′ matrix.
The input samples were generated with monochromatic light, so the recon-
structed spectral composition vectors were expected to be:
Sblue =

1.0000
0.0000
0.0000
 Sgreen =

0.0000
1.0000
0.0000
 Sred =

0.0000
0.0000
1.0000

The speckle patterns were sampled at the same positions used in the calibration
process and the following intensity vectors were obtained:
Iblue =

0.0235
0.1098
0.0353
0.1373
1.0000
0.0471
0.0314
0.0471
0.0275

Igreen =

0.0039
0.0235
0.0235
0.3333
0.9882
0.1647
0.0314
0.0235
0.0431

Ired =

0.0510
0.2000
0.0863
0.2706
0.9961
0.1529
0.1176
0.0510
0.0235

26
After multiplying these vectors by T ′ the spectral composition S of each input
was recovered. However, they did not reproduce the true spectral composition due
to the deficit in the wavelength sampling and the vibrational noise of the optical
elements:
Sblue =

0.7127
0.0000
0.3461
 Sgreen =

0.0000
0.1135
0.9399
 Sred =

0.1454
0.1531
0.7532

To improve the precision of the recovered S vectors, the optimization process
with the energy minimization algorithm was performed with 10 iterations and a
temperature that varied from T0 = 10 to Tf = 1 with ∆T = 1. The optimized
spectral composition vectors were:
Sblue =

0.9483
0.0000
0.1165
 Sgreen =

0.0000
0.7394
0.2686
 Sred =

0.0384
0.0953
0.8664

After the optimization process, the precision of the reconstructed spectral com-
position vectors was significantly improved. These results are summarized in Fig.19.
27
Figure 19: Reconstruction of the spectral composition of the sampled
speckle patterns. The results before and after the optimization algorithm
are shown in the left and right side respectively. In each case the white
bar represents the expected value for the spectral composition.
28
Chapter 5
Conclusions
The relation between speckle patterns and wavelength is exploited in the speckle-
based spectroscopy. The speckle patterns can be produced in several ways, being
the ilumination of a rough surface the most common technique. In this last case,
the roughness of the surface confers to the light reflected an extra amount of optical
path which translates into an extra phase term. In principle, the added phase could
be calculated if the exact topology of the surface were known, however doing this
would be really impractical. For this reason, the phase term is treated as a random
variable uniformly distributed within a certain range determined by the experimen-
tal conditions and is better described with statistical tools.
The effect of adding random phase terms can be simulated using a LCD. This
device acts as a SLM that produces a modulation in polarization and phase depend-
ing on the graylevel of the projected image in the screen. When an image that has
in each pixel a random level of gray is projected, the diffraction pattern produced
by a square aperture becomes a speckle pattern.
Before using the LCD in the implementation, it was necessary to confirm that the
simulated speckle pattern behaved as a truly one by ploting its distribution. Despite
the fact that small intensity values were affected by noise due to the CCD camera
sensibility, the speckle patterns produced with an LCD resembled true speckle pat-
terns.To calibrate the spectrometer three lasers of different wavelength were used
and nine sampling points were selected in each speckle pattern. This produced a
29
9 × 3 transmission matrix from which it was possible to find its left pseudoinverse
T ′. Then, to test the performance of T ′ three sample speckle patterns generated
with monochromatic light of the same wavelength of the laser light used to calibrate
the method. The recovered vectors S by simple matrix multiplication of T ′ and I
did not reproduce the original spectral composition of each sample and they were
optimized with the energy minimization algorithm.
As stated by the authors who developed the speckle-based spectroscopy method
[1], the major limitation is that speckle patterns do not only depend on the wave-
length of the employed light, but also on the exact manner in which the random
phases are introduced. To be able to use the T ′ matrix, it has to be assured that the
input speckle patterns were produced in the same way that the speckle patterns for
the calibration were. In the case of the original implementation was speckle patterns
are produced with a multimode optical fiber it was necessary to make sure that the
spatial disposition of the fiber remains the same during the calibration and usage of
the spectrometer.
In [4] this issue was addressed by fixing the optical fiber to ensure minimal vari-
ations in the spatial disposition, but this solution limits its possible applications.
In this work the optical fiber was replaced by a LCD. In this implementation the
random phase now depends only on the image projected in the display, making it
easier to reproduce the speckle patterns generated during the calibration process.
However, its usage has only been tested in a simple manner.
The advantage of using an optical fiber is the arbitrary precision that in theory
could be achieved changing the length of the fiber. For the LCD implementation the
precision has not yet been calculated because it required the use of a much greater
number of wavelength samples that were not available during the development of
this work. Obtaining the precision allowed by the LCD will be a necessary step
before aplying this implementation.
30
Chapter 6
Bibliography
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spectrometer. Optics letters. 2012;37(16):3384–3386.
[2] Wan NH, Meng F, Schro¨der T, Shiue RJ, Chen EH, Englund D. High-resolution
optical spectroscopy using multimode interference in a compact tapered fibre.
Nature communications. 2015;6.
[3] Redding B, Popoff SM, Cao H. All-fiber spectrometer based on speckle pattern
reconstruction. Optics express. 2013;21(5):6584–6600.
[4] Redding B, Alam M, Seifert M, Cao H. High-resolution and broadband all-fiber
spectrometers. Optica. 2014;1(3):175–180.
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1976;66(11):1145–1150.
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[11] Duncan DD, Kirkpatrick SJ. Algorithms for simulation of speckle (laser and
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[12] Duncan DD, Kirkpatrick SJ. The copula: a tool for simulating speckle dynam-
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[13] Hamarova´ I, Horva´th P, Sˇmı´d P, Hrabovsky` M. Computer simulation of the
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[14] Moreno I, Velsquez P, Fernndez-Pousa CR, Snchez-Lpez MM, Mateos F. Jones
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32
Appendix A
Statistics of speckle patterns
Some statistical tools developed to study properties of speckle patterns are presented
in this appendix. Here, the probability distribution for the intensity in a speckle
pattern and the expressions for the contrast and the signal to noise ratio used in sec-
tion 2.3 are deduced following step by step the treatment made by Joseph Goodman
in [16].
A.1 Random phasors statistics
First, let us consider the sum of N random phasors:
A = Aejθ = 1√
N
N∑
n=1
an =
1√
N
N∑
n=1
ane
jφn
where A is the resultant phasor with magnitude A and phase θ and an is the nth
component of the sum with magnitude an and phase φn. From this equation the
real and imaginary parts of A can be expressed as:
R = Re{A} = 1√
N
N∑
n=1
an cosφn
I = Im{A} = 1√
N
N∑
n=1
an sinφn
33
Before calculating their statistical properties, two assumptions are made about
the magnitude and phase of each of the n phasor components of the sum:
• The magnitude of every phasor an is a random variable statistically inde-
pendent of its corresponding phase φn. Furthermore, it is also statistically
independent of the magnitude of other phasors am.
• The phase φn of each phasor is uniformly distributed between −pi and pi.
These assumptions permit to obtain easily the expected values for the real and
imaginary part,
E(R) = 〈R〉 = 〈 1√
N
N∑
n=1
an cosφn〉 = 1√
N
N∑
n=1
〈an〉〈cosφn〉 = 0
E(I) = 〈I〉 = 〈 1√
N
N∑
n=1
an sinφn〉 = 1√
N
N∑
n=1
〈an〉〈sinφn〉 = 0
where the following property for the expected value of two indenpendent random
variables was used:
E(XY ) = E(X)E(Y ) .
Since the first moments are zero, the variances are equal to the second moments:
σ2R = E(R
2) = 〈 1√
N
N∑
n=1
an cosφn · 1√
N
N∑
m=1
am cosφm〉
=
1
N
N∑
n=1
N∑
m=1
〈anam cosφn cosφm〉
=
1
N
N∑
n=1
N∑
m=1
〈anam〉〈cosφn cosφm〉
for n 6= m:
〈cosφn cosφm〉 = 〈cosφn〉〈cosφm〉 = 0
34
for n = m:
〈cosφn cosφm〉 = 〈cos2 φn〉 = 1
2pi
∫ pi
−pi
cos2 θdθ
=
1
2pi
∫ pi
−pi
1
2
(1 + cos 2θ)dθ
=
1
2
then,
σ2R =
1
N
N∑
n=1
〈a2n〉
2
.
In a similar way an expression for σ2I is obtained:
σ2I =
1
N
N∑
n=1
〈a2n〉
2
.
The correlation of these two variables is:
ΓR,I = E(RI) =
1
N
N∑
n=1
N∑
m=1
〈anam〉〈cosφn sinφm〉
for n 6= m:
〈cosφn sinφm〉 = 〈cosφn〉〈sinφm〉 = 0
for n = m:
〈cosφn sinφm〉 = 〈cosφn sinφn〉 = 0
so
ΓR,I = 0
showing that they are two uncorrelated random variables.
35
When the number of random phasors N tends to infinity, the central limit the-
orem gives:
p(x) =
1√
2piσx
e
−(x−〈x〉√
2σx
)
and for two independent random variables, the joint probability density function is:
P (x, y) = P (x) · P (y) .
Since σ2R = σ
2
I = σ
2 for R and I, the joint probability is:
PR,I(R, I) = PR(R) · PI(I)
=
1
2piσ2
e−(
R2+I2
2σ2
) .
The joint probability density function of A and φ is related to that of R and I
through:
PA,θ(A, θ) = PR,I(R, I)||J ||
where ||J || represents the Jacobian:
||J || =
∥∥∥∥∥∥
∂R/∂A ∂R/∂θ
∂I/∂A ∂I/∂θ
∥∥∥∥∥∥ .
Using the following relations for R, I, A and θ,
A =
√
R2 + I2 θ = tan−1 (I/R)
R = A cos θ I = A sin θ
the Jacobian gives:
||J || =
∥∥∥∥∥∥cos θ −A sin θsin θ A cos θ
∥∥∥∥∥∥ = A
36
Then,
PA,θ(A, θ) =
A
2piσ2
e−
A2
2σ2 ; A ≥ 0 , −pi ≤ θ ≤ pi
The next step is to find the marginal statistics of A and θ alone:
PA(A) =
∫ pi
−pi
PA,θ(A, θ)dθ =
∫ pi
−pi
A
2piσ2
e−
A2
2σ2 dθ
=
A
σ2
e−
A2
2σ2
This is known as the Rayleigh density distribution:
PA(A) =
A
σ2
e−
A2
2σ2 , A ≥ 0
Integrating in A gives:
Pθ(θ) =
∫ +∞
0
A
2piσ2
e−
A2
2σ2 dA
= − 1
2pi
∫ +∞
0
− A
σ2
e−
A2
2σ2 dA
=
1
2pi
the marginal of θ:
Pθ(θ) =
1
2pi
; −pi ≤ θ ≤ pi .
It is seen that:
PA,θ(A, θ) = PA(A) · Pθ(θ)
which shows that A and θ are statistically independent.
Finally, the moments of the amplitude are:
37
〈Aq〉 =
∫ +∞
0
AqPA(A)dA =
∫ +∞
0
Aq+1
σ2
e−
A2
2σ2 dA .
To solve this integral the gamma function is used. This function is defined as:
Γ(z) =
∫ +∞
0
tz−1e−tdt = 2
∫ +∞
0
t2z−1e−t
2
dt .
Using the gamma function the expression for the amplitude moments is found to be
〈Aq〉 = 2q/2σqΓ(1 + q/2) .
With this equation the moments of A are calulated. The first moment is:
〈A〉 =
√
pi
2
σ〈A2〉 = 2σ2
and the second is:
σ2A = (2−
pi
2
)σ2 ≈ 0.43σ2
A.2 Intensity and phase statistics
In this section the results for the probability density function and moments of the
amplitude A are used to obtain the corresponding probability density function of
the intensity I in a speckle pattern.
Let v be a random variable that is related to a random variable u through a
monotonic transformation v = f(u). Then, two probability density functions will
be related through the following equation:
Pv(v) = Pu(f
−1(v))|du
dv
| .
Replacing with v = I, u = A and I = f(A) = A2 this equation gives
38
PI(I) =
1
2
√
I
PA(
√
I) .
In the case of large N it was found that the probability density function for the
amplitude was
PA(A) =
A
σ2
e−
A2
2σ2 .
Evaluating it with
√
I allows to find the desired expression for the probability density
function of I:
PI(I) =
1
2σ2
e−
I
2σ2 ; I ≥ 0
For this distribution the moments are expressed as:
〈Iq〉 = (2σ2)qq ! .
The expected value of the intensity is:
〈I〉 = 2σ2,
wich can be replaced in the equation for the moments and in the probability density
function to give
〈Iq〉 = 〈I〉qq !
and
PI(I) =
1
〈I〉e
− I〈I〉 .
39