# Analysis of optical absorption and non-linearity of chromium (III) Oxide

## Abstract

Chromium (III) Oxide is an inorganic compound having the formula. We have studied the linear and non-linear optical properties and measured the energy band gap of nanoparticles. A graph has been plotted between *absorbance and energy* to find the properties of optical absorption. Also, the *tauc plot* gives the value of energy band gap (~3.45eV). To study the non-linearity of the sample, we have performed closed and open aperture *Z-Scan* technique and obtained required graphs. *Z-Scan* refers to a single beam technique through which we can effectively measure the non-linear refractive index and non-linear absorption coefficient of different materials.

**Keywords****:** optical absorption, tauc plot, non-linearity, Z-Scan

## Abbreviations

NLO | Non-Linear Optics |

SA | Saturable Absorption |

TPA | Two-photon Absorption |

SA | Saturable Absorption |

RSA | Reverse Saturable Absorption |

OA | Open Aperture |

CA | Closed Aperture |

## INTRODUCTION

## Background

Metal oxide nanostructures are of great interest of research in recent times. Among them, chromium oxide (*Cr _{2}O_{3}*) is one of the most important wide band gap p-type semiconductor transition metal-oxide material. To study the linear optical properties, optical absorption is taken into consideration. Absorption is a process in which the outer electrons of atoms or molecules absorb radiant energy and undergo transitions to high energy levels. In this process, the spectrum obtained due to optical absorption can be analyzed to get the energy band gap of the material. Again, non-linearity is studied by means of Z-Scan technique. The Z-Scan technique is performed by translating a sample through the beam waist of a focused beam and then measuring the power transmitted through the sample. Non-linear Optics came into being with the advent of LASERs in early 1960's. It has now reached level of maturity and proves to be an exciting field of research that is growing day by day and finds its major application in emerging photonic technology.

## Objectives of the Project

The primary objective of this project is to study the linear and non-linear optical properties of the sample and its potential applications.

The primary objectives are listed as follows:

➜ To find out the energy band gap of the sample

➜ To measure the nonlinear refractive index and nonlinear absorption coefficient of the sample using Z-Scan technique

## Scope

$C{r}_{2}{O}_{3}$ has wide variety of applications such as: catalysts, hydrogen storage, wear resistance materials, dye and pigment, advanced colorants, digital recording system, black matrix films, solar energy application, coating materials for thermal protection and electrochromic material.

## THEORY

## Non-Linear Optics

Non-linear Optics is the branch of optics that describes the behaviour of light in non linear media, i.e., media in which the polarization density (**P**) responds non linearly (change in output is not proportional to change in input) to the electric field (E) of the light. The non linearity is typically observed only at very high light intensities (values of atomic electric fields~ 10^{8} V/m) such as those provided by LASER.

Though many predictions were made about the optical non linearity in earlier years, the birth of non linear optics is identified with the discovery of 2nd Harmonic generation by Peter Franken and co-workers in 1961 at the University of Michigan. In their experiment, red light from a Ruby Laser was frequency doubled into the ultraviolet. This came shortly after the discovery of Laser by Theodore Harold Maiman in 1960.

## Meaning of optical non-linearity

In linear optics, the induced polarization depends linearly on the electric field strength that can be described as

P(t)= ϵ_{0} χ^{(1)} E(t)

where, χ^{(1)} = linear susceptibility

ϵ_{0} = permittivity of free space.

In non linear optics, the polarization is described as a power series in the field strength E(t) as

P(t)= ϵ_{0} [ χ(^{1)} E(t) + χ^{(2)} E^{2}(t) + χ^{(3)} E^{3}(t) + ……….]

= P^{(1)}(t) + P^{(2)}(t) + P^{(3)}(t)+………………

where, χ^{(2)} , χ^{(3)} = 2nd and 3rd order non linear optical susceptibilities respectively.

## 2nd order NLO processes

2nd Order Non-linear Optical interactions can occur only in non-centrosymmetric crystals, i.e. in crystals that do not display inversion symmetry. Liquids, gases, amorphous solids such as glass cannot display 2nd Order NLO processes for this reason.

2.1.2.1 2nd Harmonic Generation (SHG)

SHG or frequency doubling is the generation of light with a doubled frequency. Here, two photons are destroyed creating a single photon of twice the frequency of the original photons.

Let us consider a laser beam with electric field intensity E(t) is incident upon a crystal having non zero 2nd order susceptibility χ^{(2)}

E(t)=Ee^{(-i}^{ωt)}+E^{∗} e^{iωt}

where E^{∗}e^{iωt }= complex conjugate of the 1^{st} term

Here, ** **P^{(2)} (t)= ε_{0} χ^{(2)} (Ee^{(-iωt)}+c.c.)^{2}

where P^{(2)}(t) = 2nd order non-linear polarization

** **P^{(2)}(t)= ε_{0} χ^{(2)} (2EE^{∗}+E^{2} e^{(-2iωt)}+E^{∗}^{2} e^{2iωt })

** **** **=2ε_{0} χ^{(2)} EE^{∗}+(ε_{0} χ^{(2)} E^{2} e^{(-2iωt)}+ε_{0} χ^{(2)} E^{∗2 }e^{2iωt})

** **P^{(2)}(t) = 2ε_{0} χ^{(2)} EE^{∗}+(ε_{0} χ^{(2)} E^{2} e^{(-2iωt)}+ε_{0} χ^{(2)} E^{∗2} e^{2iωt})

Thus the second order polarization consists of a contribution at zero frequency (1st term) and a contribution at frequency 2ω (2nd term).

2.1.2.2 Sum frequency generation

The complex amplitude of the non-linear polarization contributing to SFG is

P(ω_{1}+ω_{2} )= 2χ^{(2)} ε_{0} E_{1} E_{2}

In SFG, two input photons with frequencies ω_{1} and ω_{2} results into a photon of frequency ω_{3}.

It is a parametric process, i.e. here the photons satisfy energy conservation.

Its application includes generation of red light (red laser), e.g. by mixing the outputs of a 1064nm Nd:YAG laser and a 1535 nm fiber laser resulting in an output at 628 nm.

2.1.2.3 Difference frequency generation

The complex amplitude of the non-linear polarization describing the DFG process is

P(ω_{1}-ω_{2} )=2χ^{(2)} ε_{0} E_{1} E*_{2}

Thus, the difference frequency component is obtained as one of the outcomes when two beams interact in a medium with a non zero χ^{(2)}.

Here, every photon that is created at the difference frequency ω_{3 }= ω_{1}-ω_{2} , a photon at a higher input frequency(ω_{1}) must be destroyed producing a photon at a lower frequency ω_{2}. As the lower frequency field is amplified by this process, DFG is also called as optical parametric amplification. One of its application is the generation of light around 3.3 μm by mixing lasers of 1570 nm and 1064 nm.

## 3rd order NLO processes

The 3rd order contribution to the optical polarisation is given as

P^{(3)}(t) = ϵ_{0} χ^{(3)} E^{(3)} (t)

The generalised electric field is taken as-

E(t)=E_{1} e^{(-iω}_{1}^{ t)}+E_{2} e^{(-iω}_{2}^{ t)}+E_{3} e^{(-iω}_{3 }^{t)}+c.c.

Then, P^{(3)}(t) will contain 44 different frequency components, when the positive and negative frequencies are considered to be distinct.

For simplicity, let us consider the applied field to be monochromatic and given by-

E(t)= ϵ cosωt

Hence, P^{(3)}(t)=1/4ϵ_{0} χ^{(3)} ε^{3}cos3ωt + 3/4 ϵ_{0} χ^{(3)} ε^{3} cosωt

2.1.3.1 3rd harmonic generation

We have, P^{(3)}(t)= 1/4 ϵ_{0} χ^{(3)} ε^{3} cos3ωt+3/4 ϵ_{0} χ^{(3)} ε^{3} cosωt

The first term describes the response at frequency 3ω that is created by an applied field at frequency ω. This term leads to the 3rd Harmonic Generation. Here, 3 photons of frequency ω are destroyed forming one photon of frequency 3ω.

## Non-Linear Absorption

When light interacts with matter, it can get affected by absorption, refraction or scattering based on the intensity of light as well as the property of the material. If the intensity of light is strong enough to invoke the non-linear terms in the electric ploarization, the transmission will be changed. In non-linear absorption, the change in transmission is studied as a function of input intensity. Here, we will discuss the major NLO absorption processes namely two photon absorption(TPA), saturable absorption(SA), reverse saturable absorption(RSA).

## Two photon absorption (TPA)

The process of transition of a system from the ground state to a higher level by the simultaneous absorption of two photons from an incident radiation field is termed as two photon absorption. The state to which the transition happens is approximately resonant at 2ω. TPA process is proportional to the square of input intensity. A schematic representation of two-photon absorption is shown in fig 1.

The propagation of laser light through the system describing optical loss is given by

$\frac{\operatorname dI}{\operatorname dz}=\;-\alpha I-\beta I^2$

where α is the linear absorption coefficient and β is the two-photon absortion coefficient.

β value characterizes the material and is related to TPA cross section σ_{2} by the equation

${\sigma}_{2}=\frac{\u0127\omega \beta}{N}$

where N is the number density of the molecules in the system and ω is the incident radiation frequency.

## Saturable absorption (SA)

Saturable absorption is a property of materials where the absorption of light decreases with increasing light intensity. This occurs when atoms or molecules are excited with sufficiently high intensity at a rate such that there is insufficient time for them to decay back to the ground state before it gets depleted. Under steady state condition the kinetic model describing the SA is given by

$\frac{\operatorname dN}{\operatorname dt}=\frac{\sigma I}{h\nu}(N_g-N)-\frac N\tau$

where N is the concentration of excited state molecules, N_{g} is the undepleted ground state concentration, σ is the ground state concentration, hν is the photon energy and τ is the life time of the excited state population. Assuming α = σ(N_{g} − N), the saturation can be described as

$\alpha ={\alpha}_{0}\frac{1}{1+{\displaystyle \frac{I}{{I}_{s}}}}$

where, I_{s}= $\frac{h\nu}{\tau\sigma}$is the saturation intensity, I is the intensity and α_{0} = σN_{g} is the coefficient.

## Reverse saturable absorption (RSA)

When a material is incident upon by laser beam, the molecules in the medium can get excited to a higher energy level 2. It is also possible that the molecule in the excited state 1 can further absorb a photon to make a transition to a second excited state 3. This process is known as RSA. The possibility of this process depends on the number of molecules N_{2} at the first excited state 2, the incident intensity I and the excited state absorption cross-section σ_{23}. Also, N_{2} is related to N_{1} and I by the relation,

$N_2\propto\sigma_{12}N_1I$ (1)

where σ_{12} is the cross section of the transition from the ground state to state 2. Under the steady state condition, the intensity change of the laser beam in the non-linear medium along its propagation direction is expressed as-

$\frac{\operatorname dI}{\operatorname dz}=\;-\sigma_{12}(N_1-N_2)I\;-\;\sigma_{23}N_2I$

$\frac{\operatorname dI}{\operatorname dz}=\;-\sigma_{12}(N_1-N_2)I\;-\;b\sigma_{12}\sigma_{23}N_0I^2$

or $\frac{\operatorname dI}{\operatorname dz}=\;-\alpha_0I-\beta^/I^2$

where b is a proportionality constant and the linear absorption coefficient α_{0} and nonlinear absorption coefficient β^{/} are defined as

$\begin{array}{l}\alpha_0=\sigma_{12}N_0\\\beta^/=b\sigma_{12}\sigma_{23}N_0\end{array}$

A schematic representation of RSA is shown in the figure below:

## EXPERIMENTAL SECTION

The sample of *Cr _{2}O_{3}* is prepared using hydrothermal technique

^{(2)}. We have performed ultraviolet visible infrared (UV-Vis-NIR) spectroscopy for finding out the optical absorption. Nonlinear optical properties of

*Cr*was studied using open and closed aperture Z-scan technique. NLO absorption properties were studied using open aperture technique whereas nonlinear refraction was studied using the closed aperture Z-scan technique where a fraction of transmitted light is made to fall on the photodetector. The fraction of transmitted light can be controlled using the aperture before the detector. A short description of the experiment processes are given below.

_{2}O_{3}## Optical Absorption Spectroscopy

UV-Vis-NIR spectroscopic technique measures the absorption of radiation when it interacts with a sample as a function of wavelenth in the region 190nm to 2000nm. Absorption of the ultra-violet radiations results in the excitation of the electrons from the ground state to higher energy state. The energy of the UV radiation that are absorbed is equal to the energy difference between the ground state and the higher energy states ($\u2206E=h\nu $). Usually, the most favoured transition is from highest occupied molecular orbital (HOMO) to lowest unoccupied molecular orbital (LUMO). The absorption response of a sample obeys the Beer-Lambert law which states that- When a beam of monochromatic light is passed through a solution of an absorbing substance, the rate of decrease of intensity of radiation with thickness of the absorbing solution is proportional to the incident radiation as well as the concentration of the solution. Its expression is given by-

$A=\mathrm{log}\left(\frac{{I}_{0}}{I}\right)=ECL$

where I_{0 }is the incident intensity, E is the molar absorptivity, L is the path length of the sample and C is the molar concentration of the solute. From the law, it is clear that if the number of molecules capable of absorbing a particular wavelength is larger, higher would be the extent of absorption.

## Tauc Plot

A** **Tauc plot** **is used to determine the optical bandgap, or Tauc gap. ^{}The Tauc gap is often used to characterize practical optical properties of different materials

Typically, a Tauc plot shows the quantity *hν* (the energy of the light) on the abscissa and the quantity *(αhν) ^{1/r}* on the ordinate, where

*α*is the absorption coefficient of the material. The value of the exponent

*r*denotes the nature of the transition.

r = 1/2 for direct allowed transitions

r = 3/2 for direct forbidden transitions.

r = 2 for indirect allowed transitions

r = 3 for indirect forbidden transitions.

** **Z-Scan

Z-Scan is a single beam technique used to determine the magnitude of non-linear absorption and magnitude & sign of the non-linear refraction. For measuring the real part of the non-linear refractive index, the Z-Scan setup is used in its closed aperture form and to measure the imaginary part of the non-linear refractive index or the non-linear absorption coefficient, open aperture setup is taken.

## Closed aperture Z-Scan

To examine the effect of translation of the sample, along the beam path, a Zn-Se(Zinc Selenide) sample is taken. In closed aperture form, an aperture is placed to prevent some of the light from reaching the detector. A lens focus a laser to a certain point, and after this point, the beam automatically defocuses. A further distance apart, an aperture is placed with a detector behind it. When the sample is far away from the focus, the optical field intensity on the material is very low. Accordingly the non-linear contribution to the refractive index η_2 I is also very low. So, there is no effect on the transmitted beam. As the sample is brought closer to the focus, the beam irradiance increases. The beam narrowing occurs at the aperture. So, the measured transmittance increases. As the sample passes the focal plane towards the +z, the beam gets diverged. Beam broadening at the aperture reduces the transmittance. Whereas, a null is there at the focus. A schematic diagram of closed aperture Z-Scan is shown below-

## Open aperture Z-Scan

It is similar to closed aperture Z-Scan with the aperture removed. When the aperture is removed then all the light can reach the detector. Now, the transmittance depends on only the absorption non-linearities. There is no effect of non-linear refraction. The transmittance obtained in this case is symmetrical with respect to the focus. Thus, by performing the open aperture Z-Scan, the non linear absorption coefficient Δα can be obtained. A schematic diagram of open aperture Z-Scan is shown below

## RESULTS AND DISCUSSION

## Optical Analysis

For linear optical analysis, the *Cr _{2}O_{3}* nanocrystals are dissolved in 1-methyl 2-pyrrolidone and the experiment is performed in the UV-Vis-NIR spectrophotometer.

*Cr*nanoparticles show characteristic absorption peak at ~326nm as shown in Fig. 5.

_{2}O_{3}A plot between absorption and energy is also shown in fig. 6.

The spectrum obtained due to optical absorption can be analyzed to get the energy band gap of the material. The spectrum shows the general trend of absorption i.e the absorbance of the material decreases with decreasing frequency of incident radiation. The optical band gap ‘E_{g}’ is calculated using the following well known Tauc’s relation

(αhν)^{2} = A(hν − E_{g})^{}

where, A is a constant, α is the absorption coefficient and n is a constant for a given transition which is equal to $\frac12$for direct band gap. The direct band gap ‘E_{g}’ is determined by extrapolating the straight portion of the plotted graph to the energy axis at α = 0, as shown in figure 7.

From the graph, the band-gap energy of *Cr _{2}O_{3}* nanoparticles is found to be ~3.45 eV, which is in agreement with the reported value.

## Z-Scan Results

For studying NLO absorption and NLO refractive index, we perform open aperture and closed aperture Z-Scan technique. We dissolve the sample in ethanol and sonicate for appropriate time.

In open aperture Z-Scan, the sample is excited using 7 ns pulsed laser centred at 532 nm from the second harmonics of Nd:YAG laser. The normalized Z-Scan peak shape of the sample *Cr _{2}O_{3}* at peak intensity 4.4 μJ is shown in figure 8.

From the graph of open aperture Z-Scan, we found reverse saturable absorption (RSA) in case of *Cr _{2}O_{3 }*nanocrystals.

The laser propagation equation through the material is given by the following equation

$\frac{dI}{dz}=-{\alpha}_{tot}I$

where $\alpha_{tot}=\frac{\alpha_l}{1+{\displaystyle\frac I{I_{sat}}}}+\alpha_{nl}I$

Here, α_{l} , α_{nl} and I* _{sat}* are the linear absorption coefficient, nonlinear absorption coefficient and saturable intensity respectively.

Now, to further study third order NLO properties such as non-linear refraction, we perform closed aperture Z-scan technique on the same experimental setup. Here, we close the aperture such that only a fraction of the transmitted light can fall on the aperture.

The graph of closed aperture Z-Scan is shown in figure 9.

There is nonlinear refraction as well as nonlinear absorption in close aperture Z-scan trace. To extract only nonlinear refraction only we will divide the closed aperture Z-scan trace by open aperture Z-scan trace and then take the peak-valley difference (T_{p-v}). The graph of CA/OA trace is shown in figure 10.

The non-linear refractive index is given by the following equation-

${n}_{2}=\frac{\u2206\Phi}{k{I}_{0}{L}_{eff}}$

where k , *I _{0}* are wave vector and irradiance at focus respectively L

_{eff}is the effective propagation length where sample thickness is L and $\u2206\Phi =\frac{\u2206{T}_{p-v}}{0.406(1-S{)}^{0.27}}$.

## CONCLUSION

We have studied the linear and non-linear optical properties of *Cr _{2}O_{3}* nanoparticles making use of UV-Vis-NIR Spectrophotometer and Z-Scan technique respectively. From the optical absorption graph we found out the energy band gap of

*Cr*nanocrystals which comes out to be ~3.45eV. Further, using open aperture Z-Scan technique, we came to the evidence that reverse saturable absorption (RSA) occurs in

_{2}O_{3}*Cr*nanoparticles and from closed aperture we will get the non-linear refractive index. In future we will try fitting of the graphs to calculate the exact numerical values of the above mentioned quantities.

_{2}O_{3}## ACKNOWLEDGEMENTS

First of all, I would like to thank the Science Academies' for selecting me in this Summer Research Fellowship Program 2019. I convey my sincere gratitude to Prof. Kumaran Nair Valsala Devi Adarsh Sir for selecting me as his intern and for his invaluable guidance and support throughout this period. His constant motivation to work hard and sincerely has inspired me a lot. I would also take the oppurtunity to thank the research fellows of K.V. Adarsh lab, Mr. Dipendranath Mandal, Dr. Rajesh Kumar Yadav, Mr. Anirban Mondal and Ms. Riyanka Karmakar and others for their most valuable advice and help during this period. I especially thank Ms. Rasna Baruah, my fellow intern and friend for her relentless support. All this would not have been possible without the recommendation of Dr. Paban Kumar Sahariah (Department of Physics, Cotton University). I convey my sincere thanks to him for his encouragement from the very beginning. I express my gratitude to IISER, Bhopal and its administration as well as to my new friends I made here for a wonderful time. Authorcafe has provided a fantastic platform for writing the report, thanks for that. Last but not the least, I owe my deepest gratitude to my family and my friends for their unconditional love and persistent support.

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_{2}O_{3}nanostructures: Evaluation of its dielectric properties, AIP Advances 4, 027121 (2014) - R. W. Boyd, S. Choudhary: Tutorial on nonlinear optics
- Mansoor Sheik-Bahae, Ali A. Said, Tai-Huei Wei, David J. Hagan, E. W. Van Stryland: Sensitive Measurement of Optical Nonlinearities Using a Single Beam. IEEE Journal of Quantum Electronics. Vol. 26. No. 4, April 1990.
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