UNIT -I
LASERS AND WAVE OPTICS
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LASERS
Introduction:
Laser is one of the outstanding inventions of the 20th century. LASER stands for Light Amplification by Stimulated Emission of Radiation. . It is artificial light sources that differ vastly form the traditional light source. Laser have widespread applications in our everyday life and in technological devices ranging from CD players, DVD players, barcode readers in supermarket, laser printers, eye surgery equipments, dental drills, optical communication system etc.
It is device which produces a highly directional, coherent, monochromatic, polarized and intense beam of light that depends upon the phenomenon of stimulated emission.
Interaction of light radiation with matter / Three quantum processes:
To understand the working principal of LASER it is required to understand the three quantum processes that takes place in material when it is exposed to radiation. A material is composed of identical atoms each of which has characterized by discrete allowed energy levels. Atoms are characterized by many energy levels but for the
sake of simplicity, consider only two energy levels E1 and E2, E1 is lower energy state (ground state) and E2 is
higher energy state (exited state). As the atoms of the material are identical, energy level E1 and E2 are common to all atoms within material.
The incident radiation may be viewed as a stream of photons, each photon carrying an energy hv. If hv = E2 - E1, then interaction of radiation with atom leads to the following three transition. They are absorption, spontaneous emission and stimulated emission. Transition or Quantum jump is the process of transfer of atom from one energy state to another energy state.
1. Absorption / Induced Absorption / Stimulated Absorption :
An atom residing in the lower energy level E1 may absorb the incident photon of energy E2 –
E1= hv and jump to the excited state E2 . This transition is known as stimulated absorption or induced absorption or simply as absorption. Corresponding to each transition made by an atom one photon is disappears from the incident beam. It may be represented as A + hv A* Where A is an atom in the lower state and A* is atom in excited state.
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Fig. Absorption Process
The no of absorption transition take place is depended on N1 no of atom in lower energy level E1 and photon density Q. The number of absorption transitions occurring in the material at any instant will be proportional to the number of atoms at the energy level E1 and the photon density in incident beam. The number of atoms Nab excited during time Δt is therefore given by
Nab = B12 N1Q Δt
Where B12 is the constant of proportionality, known as the Einstein coefficient for induced absorption
2. Spontaneous Emission: Excited energy state with higher energy is inherently unstable. To achieve minimum potential energy condition, atoms are always tends to remain in the lower energy level.
The excited atom in the state E2 may return to lower state E1 on its own out of natural tendency to attain minimum potential energy condition by releasing excess energy in the form of photon . This type of process in which photon emission occurs without any external impetus is called spontaneous emission.
We may write the process as
Where hv is energy of incident photon, A is an atom in the lower state and A* is an excited atom.
Fig. Spontaneous Emission process
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The no of spontaneous transition occurring during the time ∆t is depended on N2 number of atom in excited energy level E2 . Thus the no of spontaneous transition in time ∆t is given by
Nsp = A21 N2∆t
Where A21 is the constant of proportionality which is known as Einstein coefficient for Spontaneous Emission. The important features of this process are: (1) It cannot be controlled from outside. (2) The light emitted due to this process is incoherent i.e. the photons have different phases, planes of polarization and direction of propagations. (3) The light spreads in all direction around the source. The light intensity goes on decreasing rapidly with distance from the source
3. Stimulated Emission / Induced Emission :
A photon of energy hv=E2-E1 can induce / trigger the excited atom to make a downward transition releasing the excess energy in the form of photon. The phenomenon of forced emission of photon by an excited atom due to the action of an external agency is called stimulated emission. It is also known as induced emission. It is represented as
The existence of this mechanism was predicted by Einstein in 1916.
Fig. Stimulated Emission process
The no of Stimulated transition occurring in the material in time ∆t is depended on N2 no of atom in excited energy level E2 and photon density Q. Thus the no of Stimulated transition in time ∆t is given by
Nst = B21 N2 Q ∆t
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Where B21 is the constant of proportionality which is known as Einstein coefficient for Stimulated Emission. The phenomenon of stimulated emission is to be maximized for laser operation because it produces coherent photons and also multiplies it.
Characteristics of stimulated emission:
1) The photon induced in this process propagates in the same direction as that of incident photon. 2) The induced photon has features identical to that of the incident photon. It has the same frequency, phase and plane of polarization as that of the incident photon. 3) The outstanding feature of this process is the multiplication of photons. For one photon interacting with an excited atom, there are two photons emerging. The two photons traveling in the same direction interact with two more excited atoms and generate two more photons and produce a total of four photons. These four photons in turn stimulate four excited atoms and generate eight photons, and so on. The number of photons builds up in an avalanche like manner, as shown in Fig. below.
Fig: Multiplication of stimulated photons into an avalanche.
Distinction between Spontaneous and Stimulated Emission:
Sr. Spontaneous emission Stimulated emission No
1. Spontaneous emission is a random and Not a random process. probabilistic process.
2. Not controllable from outside. Controllable from outside.
3. Photons are emitted uniformly in all The photons emitted in the process travel in the directions from an assembly of atoms. As same direction as that of incident photon, so light a result, the light is non-directional. produced by the process is essentially directional.
4. Photons of slightly different frequencies The frequency of emitted photon is nearly equal are generated. As a result, the light is not to that of incident photon. As a result the light is
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monochromatic. nearly monochromatic.
5. The photons emitted by this process have The photons emitted by this process are all in not any correlation in their phases. phase and therefore, the light is coherent. Therefore, the light produced by this process is incoherent.
6. In this process multiplication of photons In this process multiplication of photons take does not take place. Hence light place. Hence light amplification occurs. (One amplification does not occur. stimulating photon causes emission of two more photons. These two produce four photons, which in turn generate eight photons and so on. Thus, if there are N excited atoms, 2N photons will be produce.)
7. The net intensity of the generated light is As all the photons are in phase, they given by constructively interfere and produce an intensity
2 IT.=N I IT =N I
Where N is the number of atoms emitting photons and I is the intensity of each photon.
8. The planes of polarization of the photons The planes of polarization are identical for all are oriented randomly. Hence, light from photons. Consequently, light is polarized. the source is unpolarized.
Active Medium A medium in which light gets amplified is called active medium. The medium may be solid, liquid or gas.
Active Centers Out of the different atoms in the medium only small fraction of the atoms are responsible for stimulated emission and consequent light amplification. They are called active centers. The remaining bulk of the medium supports the active centers.
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Population and thermal equilibrium The number of active atoms occupying an energy state is called as population of that state.
Let N1 and N2 be the population of lower energy level E1 and upper energy level E2 . In thermal
equilibrium the population of energy level E1 and E2 is given by Boltzmann factor
N 2 (E2 E1 ) exp N1 kT
The negative exponent in this equation indicates that N2 N1 at equilibrium. It means more atoms
are in lower energy levels E1. This state is called as “Normal State” or “Thermal Equilibrium state”.
Conditions for Light Amplification: Light amplification requires that stimulated emission occur almost exclusively. In practice, absorption and spontaneous emission always occur together with stimulated emission. The laser operation is achieved when stimulated emission exceeds than absorption and spontaneous emission.
1) Condition for Stimulated emission to dominate spontaneous emission process
Stimulated transition B N Q B 21 2 21 Q ------(1) spontaneous transition A21 N 2 A21
Eq. (1) shows that in order to increase stimulated transition the photon density Q is must be larger.
2) Condition for Stimulated emission to dominate Absorption transitions process
Stimulated Transition B N Q N 21 2 2 ------(2) Absorption Transition B12 N1 Q N1
Where B21 = B12 as the probability of Stimulated Emission must be equal to probability of absorption transition. The eq. (2) indicate that stimulated transition will be larger than absorption, only
when N2>N1 A medium amplifies light only when above two conditions are satisfied.
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Population Inversion / Inverted State / Negative Temperature State :
Population inversion is the condition of the material in which population of upper energy level N2 far exceeds the population of lower energy level N1 , i.e. N2 >> N1.
Therefore, a non equilibrium state is to be produced (required) in which the population of the upper energy level exceeds to a large extent than the population of lower energy level. When this situation occurs the population distribution between the energy level E1 and E2 is said to be inverted, and medium is said to have gone into the state of population inversion.
The condition N2 N1 is called population inversion. It is also referred to as a condition of negative temperature as the ratio N2 / N1 could be larger than unity only if the temperature were negative in the
N2 (E2 E1) boltzmann‟s factor exp . Therefore, it is known as negative Temperature State. N1 KT In practice population inversion may be obtained at normal temperature also
Metastable state An atom can be excited to a higher level by supplying energy to it. Normally excited atoms have short lifetimes and release their energy in 10-8sec or 10-9sec through spontaneous emission. To establish the condition of population inversion, the excited atoms are required to wait at the upper energy level till a large number of atoms accumulate at the excited level. Such an opportunity would be provided by metastable states. Atoms excited to a metastable state remain excited for an appreciable time, which is of the order of 10-6 to 10-3 s.
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Metastable state can be defined as the intermediate state between ground state and excited state where excited atom can remain for a longer time i.e 10-6 to 10-3 s than life time of normal excited state 10-8sec or 10-9sec.
There could be no population inversion and hence no laser action , if metastable state does not exist.
Pumping / Pump : The process of supplying energy to the medium to transfer it into the state of population inversion is known as pumping. Three types (Techniques/ Methods) of pumping: 1. Optical pumping: In optical pumping a light source such as flash discharge tube is used to supply energy. Most often this energy comes in the form of short pulses of light. This mechanism is used in solid state laser e.g. Ruby lasers. 2. Electric Discharge: In this process electric field is used for ionization of gas medium which excites the ground state atoms to the excited state. This is used in gas lasers especially He-Ne laser. 3. Direct Conversion: In this process Electrical energy is directly converted into light energy. This is used in semiconductor diode lasers.
The Principal Pumping Schemes: There are two important pumping schemes . they are known as three level and four level pumping schemes. Atoms in general are characterized by large number of energy level. Among them only three or four levels are suitable for pumping process The transition between two level that generates the stimulated emission is called Lasing transitions. The lower level is called as lower lasing level and upper level is called upper lasing level. The uppermost level is called as Pumping level. Upward arrow implies – Pumping Transition Downward arrow implies – Lasing Transition
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1. Three level pumping scheme:
Let us consider an atomic system has three energy level as shown in figure . E1 is Ground State. E2 is
Metastable state and E3 is Excited /Pumping State.
When light photon of energy hν= (E3- E1 ) is incident on medium , the atoms in ground state are readily
excited to pumping level E3 . The pumping level E3 is not stable state, thus atoms do not stay at the E3
Level and take a downward transition either to E1 or E2 through non radiative or Spontaneous transition
.The probability of spontaneous transition E3 to E1 is less than E3 to E2. Since E2 is metastable state,
atom get trapped in the state E2 the pumping continuous and after short time there will be a large
accumulation of atoms at the level E2 when more than half of the ground state atoms accumulate at E2 ,
the population inversion condition is achieved between two state E2 and E1. Now the spontaneously
emitted photon of energy hv=E2-E1 can trigger stimulated emission of atom at E2.
Disadvantages:
1) In this scheme terminal state of laser transition is simultaneously the ground state. To achieve population inversion more than half of ground state atoms must be pumped to upper state. Therefore a very high pump power is required in this pumping scheme. 2) The three level schemes produce, light only in Pulses. Once stimulated emission commences, the metastable state is quickly emptied and the population of ground state increases rapidly ; as a result population inversion ends, one has to wait till the population inversion is reestablished. Thus the three level laser operates in pulsed mode.
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2. Four level Pumping Scheme:
Let us consider an atomic system has four energy level as shown in figure . E1 is Ground State.
E3 is Metastable state and E2 and E4 are the Excited State.
Pumping frequency lifts the active center (active atoms) from ground state E1 to uppermost level E4.
From pumping level E4 , atom rapidly fall to the metastable state E3 . The population at this state grows
rapidly while the level E2 is virtually empty. Therefore population inversion is achieved between the
state E2 and E3. A photon of energy hv= (E3-E2) can start a chain of stimulated emission , bringing the
atom into the the state E2 . Atoms from E2 undergo non radiative transition subsequently to the ground
state E1 and will be available once again to participate in the process.
The lower lasing level E2 is virtually empty. Therefore less pump power is sufficient to achieve population inversion. Four level lasers operate in continuous wave (CW) mode.
3. Two level Pumping Scheme: Two level pumping scheme is not suitable for obtaining population inversion . the time span ∆t for
which atom have to stay at the upper level E2 must be longer for achieving population inversion condition. According to Heisenberg Uncertainty Principle, ∆E. ∆t ≥ h/2π
∆t will be longer , if ∆E is smaller that is E2 is narrow. If ∆E is smaller the pumping efficiency is smaller as a consequence of which less number of atoms are excited. Through a sharp energy level
supports the population inversion , enough population cannot be excited to level E2 in view of small energy ∆E. The result is that upward transition would be accompanied by premature downward
stimulated transition and population in level E2 would not accumulate to the required extent.
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Optical Resonator or Cavity Resonator or Optical resonant cavity or Fabry perot resonator : A pair of optically plane parallel mirrors enclosing active medium in between them is called as optical resonator. One of the mirrors is 100% reflecting while the other is made semitransparent. The mirrors are set normal to the optic axis of the active material. This structure is also known as Fabry- Perot resonator
Action Optical Resonator/ Lasing action:
Fig 1 shows the active medium enclosed in optical resonator. The resulting laser action which is shown in fig (2) consists of following steps,
Step 1: Pumping: Let us assume that the active centers are initially in the ground state. Through the suitable pumping mechanism the material is taken into a state of population inversion as shown in fig. 2(a)
Step 2: Population inversion: The life time of atom at the excited state is extremely small of the order of 10-9 Sec. Therefore the atom drop spontaneously from the excited state to the metastable state. As the life time of metastable state is comparatively longer (10-3 to 10-6 Sec), the atoms go on accumulating at the metastable state. As soon as the number of atoms in metastable state exceeds that of ground state, the medium goes into the state of population inversion.
Step 3: Spontaneous emission / stimulated emission: Some of the excited atoms at the metastable state may emit photon can trigger many stimulated transitions. The photon emitted in a direction other than the axial direction which will pass through the sides of the medium and all lost forever.
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Fig (2): Action Optical Resonator
Step 4: Optical feedback and Light amplification: A majority of photons traveling along the axis cause stimulated emission and are reflected back on reaching the end mirror. They travel towards the opposite mirror and on their way stimulated more and more atoms and build up the photon strength fig 2(d). The photon that strike the opposite mirror are reflected once more into the medium [fig 2(e)]. as the photon are reflected back and forth between the mirrors stimulated emission sharply increases and provide positive feedback of light into the medium so that stimulated emission acts as sustained and the medium operates as an oscillator.
Step 5: Light Oscillations: At each reflection at the front end mirror, light is partially transmitted through it. The transmitted component constitutes a loss of energy from the resonator. when the losses at the mirror and within the medium balance the gain; a steady and strong laser beam will emerge from the front end mirror fig.2(f)
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Role of Optical resonator 1. The primary function of optical resonator is to provide positive feedback of photons into the medium so that stimulated emission is sustained and laser acts as a generator of light. 2. In order to make the stimulated emission dominate spontaneous emission, a high photon density is required to be present in the active medium. The optical resonator builds up the photon density to a very high value through repeated reflections of photons and confines them within the medium 3. The optical resonator selects the direction in which the light is to be amplified; the direction being the optical axis of the pair of mirror. Thus optical cavity makes the laser beam highly directional. 4. Optical cavity selects and amplifies only certain frequencies causing the laser output to be highly monochromatic.
Condition of steady state Oscillation / condition of resonance:
The wave propagation within the cavity resonator should take on standing wave pattern so that a continuous increase in the wave amplitude occurs the condition for forming standing wave within the resonator is that the optical path length traveled by a wave between the consecutive reflection should be an integral multiple of wavelength, thus
2L = m λ (m= 1, 2, 3……..) L = m λ/2 Where L is distance between mirrors, λ is wavelength of light within the material.
Because of its relative length as compared to the wavelength of light ,the resonator may simultaneously support several standing wave of slightly different wavelength. They are called as longitudinal cavity mode. Each mode has distinct frequency given by
νm = mc / 2L
Where m is called as mode number , νm are the frequency which are sustained in the cavity and hence the emerging beam is restricted as invisible border having wide range of frequency.
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Types of Laser:
The lasers are normally classified on the basis of material (Active Medium ) used are as follows
1) Solid state laser: E.g. Ruby Laser, Nd: YAG Laser 2) Gas laser:
He-Ne Laser, CO2 Laser 3) Liquid Dye laser: E.g. Liquid dye laser an organic dissolved in liquid such as alcohol; benzene etc. acts the active centre 4) Solid state Diode laser: E.g. Semiconductor diode laser
On the basis of nature of output, lasers can be operated in
1. Continuous Wave Mode (CW) 2. Pulsed Mode (PM)
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RUBY LASER:
Historically the first working laser was the Ruby laser. It belongs to the class of solid-state lasers. Ruby is
basically Al203 crystal containing about 0.05% of chromium atoms. Cr3 ions are the actual active centers, while aluminum and oxygen atoms are host medium. Chromium ions have absorption bands in the blue and green regions
Construction:
Ruby is taken in the form of cylindrical rod of about 4 cm long and 1cm in diameter. Its end faces are made perfectly flat and parallel to each other. One of them is completely silvered to achieve 100 % reflection and other one is partially silvered to make it semitransparent. Thus, two silvered faces constitutes the mirrors of fabry perot resonator.
Ruby rod is surrounded by a helical photographic flash lamp filled with xenon. It produces flashes of white light , whenever activated by the power supply. The system is cooled with help of coolant around the ruby rod.
Working: The energy levels of the chromium ions in the crystal are as shown in the figure.
Ruby Laser uses three level pumping scheme. Cr3+ ions are the active centers while aluminium and oxygen atoms are inert. The xenon flash lamp generates an intense of white light lasting for few milliseconds.
3+ The Cr ions are excited to the energy bands E3 and E3‟ by the green and blue components of white light. The
-9 3+ normal excited level E3 is highly unstable, the life time of E3 is 10 sec . From there the Cr ions undergo
non-radiative transitions and quickly drop to the metastable level E2. .
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-3 -6 E2 is metastable state having life time of the order of 10 sec to 10 sec. The metastable state E2 has a lifetime of approximately 1000 times more than the lifetime of E3 level. Therefore Cr 3+ ions accumulate at 3+ E2 level. When more than half of the Cr ion population accumulates at E2 level, the population inversion is 3+ established between E2 and E1 levels. A chance photon emitted spontaneously by a Cr ion initiates a chain of stimulated emissions by other Cr 3+ ions in the metastable state. Red photons of wavelength 6943 AO travelling along the axis of the ruby rod are repeatedly reflected at the end mirrors and light amplification takes place. A strong intense beam of red light (λ=6943AO) emerges out of the front-end mirror.
Disadvantages:
The xenon flash lamp lasts for few milliseconds. However laser does not operate throughout this
period. Once stimulated emission starts, the mestastable state E2 gets depopulated very rapidly and lasing ceases. The laser becomes inactive till the population inversion is once again established. Therefore the output of the laser is not continuous but occurs in the form of pulses of micro seconds duration. It requires high pumping power as more than half the active atoms are to be lifted to excited state from the ground state to achieve population inversion. The efficiency of ruby laser is less as only the green and blue component of the white light is utilized while the rest of the components of incident light are unused.
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He-Ne Laser:
He-Ne laser was first successful gas laser built by Ali javan, W.Bennett and D.Herriot in 1961. , Neon atoms are the active centers while Helium atoms helps in exciting Neon atoms.
Construction:
The He-Ne laser consists of a long and narrow discharge tube having length about 50cm and diameter 1 cm. The tube is filled with the mixture of He and Ne in the ratio 10:1 respectively. Electrodes are provided in the discharge tube to produce discharge in the gas. They are connected to a high voltage power supply. The tube is hermetically sealed by inclined windows arranged at its two ends. On the axis of tube, two mirrors are arranged externally which forms the optical resonator. Out of which, one mirror is completely reflecting and the other mirror is partially reflecting. The distance between the two mirrors is adjusted such that mλ/2 and supports the standing wave pattern.
Working:
Helium neon laser employs a four level pumping scheme. When the power is switched on, the electric field ionizes some of the atoms in the mixture of helium and neon gas. Due to electric field the electrons and ions are accelerated towards the anode and cathode respectively. Since the electrons have a smaller mass, they acquire a higher velocity. The energetic electrons excites helium atoms through
atomic collisions to the excited state F2 and F3 which lie at at 19.81 eV and 20.61 eV above the ground level respectively . These two state are metastable states and hence the helium atoms excited to these levels spend a sufficient long amount of time before getting de-excited. With the passage of current
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through the discharge tube more and more helium atoms are accumulated in the excited state . These excited helium atom can return to the ground state by transferring their energy to the Neon atom through collision. Such an energy transfer can take place when the two colliding atoms have identical energy state. Such an energy transfer is called as resonant energy transfer.
Fig.b) Energy level of Helium and Neon atom and transition between the levels
The E6 level which is at 20.66 eV and E4 level which is at 18.7 eV of neon atom nearly coincide in
energy with the F3 and F2 levels of helium atom. Therefore resonant transfer of energy can occur easily. The kinetic energy of Helium atoms provide the additional 0.05 eV for exciting the neon atoms to level. This is the main pumping mechanism in He-Ne laser. Neon atoms are the active centers and role of He atoms is to excite neon atoms and cause population inversion. This results
in the excitation of neon atoms to the levels E4 and E6 and de-excitation of the helium atoms to the ground level. Thus, discharge through the gas mixture continuously populates the neon excited energy
levels E4 and E6 . This helps to create a state of population inversion between the levels E6 and E5,
E6 and E3 ,E4 and E3. Consequently three laser transitions can occurs.
0 i) E6 E3 Transitions: this transition generates a laser beam of red colour of wavelength 6328A . 0 ii) E6 E5 Transitions: It produces far infrared laser beam at a wavelength of 33900 A (3.39μm) 0 iii) E4 E3 Transitions: It produces infrared laser beam at a wavelength of 11500 A . (1.15μm)
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From the level E3 the neon atoms drop to E2 level spontaneously. The E2 level is a metastable state and neon atoms tends to accumulate in this level. This disturbs the condition of population inversion.
Hence, to drop the neon atoms from E2, they are made to collide with wall of tube and get de-excited to ground state. Thus to increase the probability of atomic collision with the walls, the discharge tube is made narrow. Semiconductor Diode Laser:
A semiconductor diode laser is a specially fabricated pn junction device that emits coherent light when it is forward biased. R N Hall and his coworkers made the first semiconductor laser in 1962.
Construction: A schematic diagram of semiconductor diode laser is shown in fig. The diode is extremely small in size with sides of the order of 1mm. the junction lies in a horizontal plane through the center. The top and bottom faces are metalized and two electrodes are provided to pass current through the diode. The front and rear faces are polished parallel to each other and perpendicular to the plane of the junction. The polished faces constitute the Fabry-Perot resonator. The other two opposite faces are roughened to prevent lasing action in that direction. The active region consists of a layer of about 1µm thickness.
Working:
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Fig. (a) Heavily doped pn junction without bias and (b) Heavily doped pn junction forward biased above threshold value .
The energy band diagram is shown in figure. The donor and acceptor levels get broaden due to heavy doping. Hence the Fermi levels in p and n regions get shifted in valence band and conduction band respectively. At thermal equilibrium, the Fermi levels in conduction band and valence band are at same level. The electrons remain accumulated below Fermi level in conduction band and holes above the Fermi level in valence band as shown in figure.
When the junction is forward biased, the energy levels shift and electron and holes are injected into the depletion region. Due to this, the width of region decreases. At low forward biased voltage, the electron-hole concentration is low and diode works as LED.
However at larger forward biased voltage, the carrier concentration reaches to very high value. In upper levels, electron concentration is more and lower levels, hole concentration is also large. Thus, the condition of population inversion is achieved in semiconductor diode.
At this state, the few recombination of electrons with holes result in spontaneous emission. The photons that propagate in the junction plane induce stimulated recombination of electron hole pair and intense laser beam of wavelength 9000 A0 is emitted
Advantages:
The semiconductor diode laser is simple, compact and highly efficient It requires very little power and little auxiliary equipment.
Limitations:
This laser gives more divergent beam having an angular spread of the order of 50 to 150. They are less monochromatic Highly temperature sensitive
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Characteristics of Laser: 1. Coherence: A conventional light source such as incandescent lamp or natural source such as sun produces incoherent light since they emit random wavelength light waves with no common phase relationship.
On other hand the wave emitted by a laser source will be in phase and are of same frequency. Therefore light generated by a laser is highly coherent. The coherence length is determined by the relation.
2 lcoh Is typically of the order of a few kilometers. in case of laser whereas the coherence length of light radiation by conventional monochromatic source is of the order of few millimeters or centimeters.
2. Directionality: Conventional light source emits light in all direction but laser emits (spreads) light in one directional as the photons traveling along the optical axis of the system are only selected and augmented with the help of optical resonator. The width of laser beam is very narrow and can travel to long distance without spreading Hence it can be focused sharply.
3. Divergence: Light from conventional sources spread out in the form of spherical wave front and hence it is highly divergent on the other hand light from a laser propagates in the form of plane waves. The small divergence that exist is due to the diffraction of the beam of the exit mirror.
4. Intensity: The intensity of light from the conventional sources decreases rapidly with distance as it spreads in the form of spherical waves. Laser emits light in the form of a narrow beam which propagates in the form of plane waves. As the energy is concentrated in a very narrow region and stays constant with distance.
5. Monochromaticity: A light source is said to be monochromatic , if it emits light of single frequency or single wavelength. The light from normal monochromatic sources spreads over a wavelength of the order 100 A0 to 1000 A0 . The laser light is highly monochromatic. The spread is of the order of a few angstrom units (<10A0) only. That is the line width associated with laser beams is extremely narrow.
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Coherence :
Coherence is an important property of light. Coherent light means that the light waves maintain constant phase difference over a period of time. It refers to the constant phase difference between a wave or waves at a point or different points in space with time.
Types of Coherence:
1. Temporal Coherence: i) If the wave maintains constant phase difference at a given time and certain time later then the waves are said to be temporally coherent. ii) It refers to the continuity and uniformity of a wave along (parallel to) the direction of propagation of wave. iii) It is the characteristic of one wave. iv) It is related to finite bandwidth of the source. v) It is also called longitudinal coherence.
2. Spatial coherence: i) A wave exhibits spatial coherence, if the phase difference between any two fixed points , in a plane perpendicular to the wave propagation does not vary with time. ii) It refers to the continuity and uniformity of a wave in a direction perpendicular to the direction of propagation of wave. iii) It is the characteristic of more than one waves travelling along different paths. iv) It is related to finite size of the disturbance. v) It is also called transverse coherence or lateral coherence.
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Coherence length : Let us consider the wave train generated by an atom at a particular instant. The middle of the wave train appears fairly sinusoidal for some number of oscillations whereas at its end occur abrupt change of frequency and phase. Defn : The length of wave train over which it may be assumed to have a fairly sinusoidal character and predictable phase is known as coherence length.
It is denoted by lcoh. The coherence length may also be defined as the product of number of waves contained in the wave train N and wavelength λ. Thus,
lcoh = Nλ Coherence time: The time interval over which the phase of the wave train can be predicted reliably is called coherence time.
It is denoted by tcoh or τ
lcoh = c ∆t
As tcoh = ∆t
lcoh = c tcoh ------(1)
Relation between Coherence length and frequency band width: According to Fourier analysis, the frequency bandwidth is given by 1 (2) t Where ∆t is the average lifetime of the excited state of atom In eqn (2) ∆t is the time during which a wave train is emitted by atom and corresponds to the coherence
time tcoh. 1 1 t tcoh Using eqn(1) in eqn (2) c l coh c l (3) coh
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Relation between Coherence length and wavelength band width: The frequency and wavelength of light wave is given by 푐 푣 = ------(4) 휆 Where λ is wavelength. On Differentiating eqn (4) on both side c 2 c c 2 (Using Eq. (3)) lcoh 2 l coh
Where ∆λ is called as natural line width or band width.
Application of Lasers:
A] Engineering Application: 1) Welding: Welding is the joining of two or more pieces into a single unit. The laser beam heats the edges of two plates to their melting point and causes them to fuse together where they are in contact. The main advantage of laser welding is that it is a contact less process and hence no possibility of introduction of impurities into the joint. 2) Drilling: the principal underlying drilling is the vaporization of the material at the focus of the beam. With laser, one can drill holes as small as 10 mm in diameter. 3) Surveying : Civil engineers use laser for surveying process 4) Laser beam can be used for cutting, molding the metals. 5) Electronics Industry: electronics industry used lasers in the manufacture of electronic component and integrated circuits. 6) Laser beam can be used in fiber optics communication.
B] Scientific Applications: 1] The distance between the earth and moon has been precisely measured by using laser beam.
2] It is used for tracking of the satellites and space craft tracking means the determination of a path of a moving object.
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C] Medical Applications:- 1] In medicine it is used for diagnosis as in endoscopy in which inner parts of the body are examined with the help of laser using an optical probe. 2] Affected part of the body can be burnt out by irradiating with laser Hence it is used in treatment of cancer and ulcers. 3] Laser is useful in painless surgery specially that of an eye. 4] Dentists use the laser beam to destroy germs in tooth cavities without pain It is also used to seal cracks in the enamel of the tooth.
D] Military Applications:- 1] Laser can serve as a user weapon since laser beam is highly directional and energetic. 2] It can be used as death ray which could destroy anything near or far.
E] Miscellaneous Applications:- 1] The most exciting application of laser is in holography in which three dimensional view of the object is obtained. 2] Holography is also used for data storage in computers. 3] Laser is used for recording audio and video information on discs. It also used in laser printers.
WAVE OPTICS
Interference: The phenomenon of redistribution of light due to superposition of light waves from two or more coherent sources is known as Interference. Types of Interference
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Constructive Interference (Maxima) If the optical difference between two rays is integral number of wavelength, then the rays meet each other in phase then the reflected waves interfere constructively at the point of their meeting. Optical Path difference, ∆ = n λ ( n = 0,1,2,3,……) Destructive Interference (Minima) If the optical difference between two rays is odd integral number of wavelength, then the rays meet each other in out of phase then the reflected waves interfere destructively at the point of their meeting. Optical Path difference = ∆ = (2n + 1) λ /2 ( n = 0,1,2,3……)
Geometrical path:
The shortest distance L between two points is known as the geometrical path. It is independent of the medium.
Optical path:
The distance experienced by light waves in a medium is releted to the R. I. of the medium. It is called optical path ∆. It is related to the geometric path L through the relation ∆ = µ x geometric path relation ∆ = µ L
Thin Film:
A film is said to be thin when its thickness is of the order of one wavelength of visible light which is taken to be 5500 A0. Thus a film of thickness in the range of 0.5µ to 10 µm may be considered as a thin film. A thin film may be a thin sheet of transparent material such as glass, mica, an air film enclosed between two transparent plates or a soap bubble.
1) Interference due to reflected light (Plane parallel film)
Consider a plane parallel of uniform thickness „t‟ as shown in the figure. Let „µ‟ be the refractive index of the material of the film. The film is surrounded by air on both sides. Let a plane parallel monochromatic light beam is incident on the upper surface of the film. Let ray AB be incident on the upper surface of the thin film at an angle of incidence „i‟.
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The ray AB is partially reflected along BC and is partially transmitted into the film along BF. The transmitted ray BF makes an angle „r‟ with the normal and meets the lower bounding surface. At F, it is partly reflected back into the film along DF, while a major portion refracts into the outer medium along FK. The reflected ray FD refracts at the outer surface and emerges out along DE which is parallel to ray BC. Both the rays BC and BFDE are travelling along the same direction and are obtained from the same incident ray AB. Therefore they are coherent and produces interference. The condition for interference depends on the optical path difference between the rays BC and BFDE.
Actual path difference :
∆a = 2µ t cos r
Correction on account of phase change at reflection : When a ray is reflected at the rarer medium, a path change of λ/2 ( or a phase change of π) occurs for the ray BC due to abrupt change of π rad in the phase of wave BC. By including the change in path difference due to reflection, the true path difference is 훌 ∆ = ퟐ흁풕풄풐풔풓 − t ퟐ
i) Condition for Maxima (Brightness) When the optical difference between two rays is integral number of wavelength, then the rays meet each other in phase then the reflected waves interfere constructively. ∆ = n λ ( n = 0,1,2,3,……) 훌 ퟐ흁풕풄풐풔풓 − = n λ ퟐ
훌 훌 ퟐ흁풕풄풐풔풓 = n λ+ ∴ ퟐ흁풕풄풐풔풓 = (2n+ퟏ) ퟐ ퟐ
ii) Condition for Minima (Darkness) When the optical difference between two rays is odd integral number of wavelength, then the rays meet each other in out of phase then the reflected waves interfere destructively. 훌 ∆ = (2n + 1) ( n = 0,1,2,3……) ퟐ 훌 훌 ퟐ흁풕풄풐풔풓 − = (2n + 1) ퟐ ퟐ 훌 훌 ퟐ흁풕풄풐풔풓 = (2n + 1) + ퟐ ퟐ Department Of Applied Physics, ACET BE-Second Semester – Advanced Physics Page 28
∴ ퟐ흁풕풄풐풔풓 = (n +1) λ = n λ ( n +1 is replaced by n for simplicity of expression)
In both the expressions n is called as order of interference. Since the rays reflected from the surface of thin film are parallel to each other they do not intersect at finite distances.
2) Variable thickness or wedge shaped thin film:
A thin film having zero thickness at one end and progressively increasing to particular thickness at the other end is called a wedge / or wedge shape thin film.
Thin wedge of air film is formed by two glass slides resting on each other at one end and separated by a thin spacer at the other end. Arrangement for observing the interference pattern in a wedge shaped film is as shown in fig.
Fig. Experimental arrangement
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Fig. Ray diagram
When a monochromatic ray of light AB be incident on the wedge from above, it gets reflected from glass to air boundary i.e. at top surface of the air film along BC. Part of incident light AB is transmitted through the air film and gets reflected partly at the air to glass boundary along DE. Thus two rays BC and DE are coherent because they are derived from same ray AB through division of amplitude. These rays interfere if the thickness of the film (t) is of the order of wavelength of light (t ≈ λ). So for small thickness ray BC & DE interfere & produce darkness depending on the phase difference.
Optical Path difference between two ray BC & DE is
흀 횫 = ퟐ흁풕 퐜퐨퐬 풓 − ……… (1) ퟐ
휆 be the gain of half wave due to abrupt jump of π-radian of the wave reflected from bottom boundary i.e. 2 air to glass.
Condition of Maxima (Brightness):
흀 ퟐ흁풕 퐜퐨퐬 풓 − = 풏흀 ퟐ
흀 So ퟐ흁풕 퐜퐨퐬 풓 = (ퟐ풏 + ퟏ) ퟐ The waves interfere constructively.
Condition of Maxima (Darkness):
흀 흀 ퟐ흁풕 퐜퐨퐬 풓 − = (ퟐ풏 + ퟏ) ퟐ ퟐ So, ퟐ흁풕 퐜퐨퐬 풓 = 풏흀
The fringes are bright and dark when monochromatic light is incident; otherwise coloured fringes are observed with white light.
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a) Expression for fringe width in wedge shaped thin film:-
When a monochromatic light is incident on the wedge shaped thin film alternate dark and bright fringes are observed.
Let nth dark fringe occurs at A. For the dark fringe condition
2µt1 cos r = nλ …………………………………(1)
If normal incidence light is considered cos r = 1
And thickness of the wedge at A is considered to be t1.
Then equation (1) becomes 2µt1 = nλ ………………………………….(2)
Let the next dark fringe occurs at C where the thickness of wedge is t2.
Then at the C 2µt2 = (n+1) λ ………………………………………….(3)
Subtracting, equan. (3) From (2), we get
2µ (t2 - t1) = λ ………………………………………….(4)
In ∆ ABC, BC = t2 − t1, A = θ, AB = β
Using BC = t2 − t1, in eq. (4), we get
2µ BC = λ …………………………………………………………...……(5)
Also in ∆ ABC, BC = AB tan θ
BC = β θ tan θ …………………………….…(6) ( since AB = β )
Using eq. (6) in (5),
2µ (β tan θ) =λ
λ ∴ 훽 = 2휇 tan 휃
Since θ is very very small, tan θ = θ
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λ ∴ 훽 = …………………………………… (7) 2휇휃
Above equation represent the expression for fringe width in wedge shaped thin film.
In this equation µ, λ and θ all are constant hence β is constant for a given wedge angle θ. So the interference fringes are equidistant from one another. As wedge angle θ increases fringes get closer because β decreases. When θ decreases β increases. b) Fringe at apex is dark:-
At the apex thickness of air film is very small i.e. t<< λ or t=0
흀 훌 Therefore at the apex, optical path difference ∆= ퟐ흁풕 퐜퐨퐬 풓 − = ퟎ − ퟐ ퟐ 훌 ∆= − ퟐ It implies that the interfering ray will be 1800 out of phase and interfere destructively at the apex. Hence wedge fringe pattern always begin with dark fringe. c) Fringes are straight and parallel:
Each of the fringes is produced by the interference of ray reflected from section of wedge having the same thickness. The locus of point having the same thickness lie along lines parallel to the apex line of the wedge. Therefore the fringes are straight and parallel. d) Fringes are of equal thickness:
As each maxima and minima is a locus of point of same thickness, the fringes are called as fringes of equal thickness. e) Fringes are localized:
Fringes are said to be localized as they are located at the top surface of film. f) Determination of wedge angle θ:
Experimentally wedge angle θ can be determined with help of travelling microscope. Position of dark fringes at two distant point say Q &R located at distances x1 and x2 respectively from the apex are noted. Let t1 and t2 be the thickness of the wedge at Q & R respectively.
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Condition for dark fringe at Q is,
2µt1cos r = nλ
For normal incidence cos r =1
푡 But 푡푎푛휃 1 푥1
푡1 = 푥1푡푎푛휃 = 푥1휃 for very small θ
2µ푥1휃 = 푛휆 ------(1)
Similarly for dark fringe at R,
2µ푡2 = (푛 + 푁)휆
2µ푥2휃 = (푛 + 푁)휆 ------(2)
Where, N is the no. of dark fringes between Q & R
Subtracting equa. (1) From equ. (2) We get
2µ(푥2 − 푥1)휃 = 푁휆
퐍훌 휽 = 2µ(푥2−푥1)
If the film is of air, µ=1.
퐍훌 휽 = 2(푥2−푥1) g) Spacer thickness: Let the spacer thickness t be used to from the air wedge, kept at distance l from
apex O
풕 풕풂풏휽 = 풍
푡 = 푙 푡푎푛휃 ≈ 푙휃
l = length of the air wedge
풍퐍훌 ∴ 풕 = 2(푥2−푥1)
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Newton ring:
Newton‟s ring is observed in the air film enclosed in between the curved surface of convex lens and flat glass plate.
Newton‟s ring is example of fringes of equal thickness. They are formed when light is reflected from a plano- convex lens of long focal length placed in contact with a glass plate. Thin film of air is formed in between the plate and the lens. The thickness of the air film varies from zero at the point of contact to some value t.
If the lens plate system is illuminated with a monochromatic light falling at normal to the plane surface of the lens, concentric bight and dark interference rings are observed in reflected light. These circular fringes are called Newton‟s rings. When a ray AB is incident on the system it gets partially reflected at the bottom curved surface of the lens (ray 1) and part of the transmitted ray is partially reflected (ray2)from the top surface of the plane glass sheet. The ray1 and ray2 are derived from the same incident ray by division of amplitude, and are the thus coherent.
Ray 1 undergoes no π change (because reflection from glass to air boundary), but ray 2 acquires phase change of π upon reflection air to glass boundary. The conditions of brightness or darkness depend on the path difference between two reflected rays which in turn depends upon thickness of thin film. The conditions for interference of the rays are identical on circumference with radius OJ.
The fringes are bright and dark when monochromatic light is incident, colored fringes are formed when white light is incident. Interference does not take place between the ray reflected from the surfaces of lens and glass plate owing to their thickness.
Conditions:-
1) For constructive interference / Bright rings: 2µt cos r − = nλ 2
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2µt cos r = (2n+1) ------(for bright ring) 2 2) For destructive interference / dark rings: Simillary, we get 2µt cos r = n λ ------(for dark ring)
For normal incidence, cos r = 1
2t = (2n+1) ------(for bright ring)
2t = n λ ------(for dark ring)
Properties/ Main features:-
1) Central spot is dark:
Newton‟s rings are formed as result of interference between light waves reflected from the top and bottom surface of a thin air film enclosed between a plano-convex lens and a plane glass plate. The occurrence of brightness or darkness depends on the optical path difference arising between the reflected rays. The optical path difference is given by ∆=2t-λ/2 At the point of contact O of the lens and glass plate as shown in figure, the thickness of air film is negligibly small compared to a wavelength of light.
t≈0 ∆=λ/2 The wave reflected from the lower surface of the air film suffers a phase change of rad while the wave reflected from the upper surface of the film does not suffer such change. Thus the superposing waves are out of step λ/2 which is equivalent to a phase difference of 1800 (or π rad ). Thus, the two interfering waves at the centre are opposite in phase and produce a dark spot.
2) Fringes are of equal thickness:
Maxima and minima depends on thickness of air film. Reflected light wave have maximum intensity for those thickness for which path difference is ∆ = n λ and minimum intensity for those thickness for which ∆=
(2n+1) . Thus each maxima and minima is a locus of points of contact film thickness. Hence fringes are
known as fringes of equal thickness. Department Of Applied Physics, ACET BE-Second Semester – Advanced Physics Page 35
3) Circular Fringes : In Newton‟s ring arrangement, a thin air film is enclosed between plano- convex lens and a glass plate. The thickness of the air film at the point of contact is zero and gradually increases as we move outward. The locus of points where the air film has the same thickness then fall on circle whose center is the point of contact. Thus the thickness of air film is constant at point at points on any circle having the point of lens- glass plate contact as the center. The fringes are therefore circular. 4) Fringes are localized in the film: When the film is illuminated with a parallel light beam, the reflected rays are not parallel. They interfere nearer to the top surface of the air film and appear to diverge from these when viewed from the top. The fringes are seen near the upper surface of film and hence are said to be localized in the film. 5) Radii of the dark rings: Let R be the radius of curvature of the lens. Let a dark fringes be located at point Q The thickness of air film at Q is t th Let the radius of n dark fringe at Q is rn
In ∆ NEP, (PN)2 = (PE)2 + (EN)2
2 2 2 R = rn + (R – t)
2 2 rn = 2Rt – t As R >> t, 2Rt >> t2
2 rn = 2Rt But, condition for dark fringes at point Q is 2µt cos r = nλ Since cos r = 1 and for air µ=1 2 t = nλ
2 rn = nλR
푟푛 = 푛휆푅 ------for dark ring The diameter of the dark ring is given by
Dn = 2 rn = 2 푛휆푅
Radii of the different dark ring can be found by substituting n= 0,1,2,3,…….
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r0 = 0, 푟1 = 휆푅, 푟2 = 2휆푅 Radii (diameter) of the dark ring are proportional to the square root of the natural numbers.
6) Radii of bright fringes/ rings : The radius of 0th dark ring is given by 2 rn = 2Rt = 2t.R
2 rn = (2n+1) R (since for bright ring 2t= (2n+1) ) 2
푟푛 = (2n + 1) 푅
퐷푛 = 2 (2n + 1) 푅
Radii of the different bright ring can be found by substituting n= 0,1,2,,…….
푟0 = R
푟1 = 3R
Radii (diameter) of the bright ring are proportional to the square root of the odd natural numbers.
7) Lens of larger radius of curvature is used:
The radius of nth dark ring is given by
푟푛 = 푛휆푅
From above equation it is seen that the radius of dark ring is proportional to square root of radius of curvature of the lens.
i.e. 푟푛 α 푅
It implies that greater the radius of curvature of the lens, the larger would be the diameter of the ring. Therefore less error in measurement of diameter
8) Rings gets closer away from the centre: It is observed that two consecutive fringes away from the centre are closer to each other and hence crowded. The diameter of dark ring is proportional to the square root of the natural numbers while the diameter of bright ring is proportional to the square root of the odd numbers.
From equa. 퐷푛 = 2 푛휆푅 is observed that the order of rings (n) increases; the diameter does not increases in a same proportion with the result that the rings gets closer and closer away from the centre.
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9) Transmitted light : The rings are exactly complementary to the reflected ring system so that central spot is bright. Contrast of the fringes is less. 10) White light : Produced rings are colored near the point of contact.
APPLICATIONS OF INTERFERENCE:
1. Determination of wave length of light. 푫ퟐ −푫ퟐ 흀 = 풏+풑 풏 ퟒ풑푹
where, λ is incident monochromatic light p is no. of rings R is Radius of curvature of plano convex lens th Dm+p is the diameter for (m+p) ring th Dm is the diameter for (m) ring
2. For measuring refractive indices of liquids and gases.
Determination of Refractive Index of Liquid using Newton‟s Ring:
ퟐ ퟐ 푫풏+풑 − 푫풏 µ = 푨풊풓 푨풊풓 푫ퟐ − 푫ퟐ 풏+풑 푳풊풒 풏 푳풊풒
3. For measuring small displacements. 4. Testing of lens surface. 5. Testing of Flatness/ Smoothness of Surface:
The smoothness of a surface can be quickly inspected visually by keeping an optical flat on the component at an angle and illuminating it with a monochromatic light. The air wedge formed between the component and optical flat produces straight and equidistant fringes if the component surface is smooth.
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If the fringes are curved towards the contact edge, the surface is concave and if the fringes curve away, it is convex as shown in fig.
6. Anti-Reflection Coating:
A thin transparent coatings of optical thickness of one quarter wavelength given on a surface in order to suppress the reflection from the surface is called as Anti–reflection film coating.
Principle:
A thin film coating can act as an AR coating if the waves reflected from its top and bottom surface are exactly 180° out of phase and the waves have equal amplitudes.
Conditions for antireflection coating: i. Amplitude condition : The reflected waves will have equal amplitudes, if the refractive index of the coating material is nearly equal to the sqyare root of the refractive index of the substrate material, thus
µ풇풊풍풎 ≈ µ풔풖풃풔풕풓풂풕풆 ii. Phase condition: The waves reflected from the top & bottom surface (ray1 & ray2) of the thin film should be 180º out of phase with each other.
The minimum thickness of the thin film coating required for the purpose can be obtained from the phase condition. From the above fig., the optical path difference between the reflected wave is
λ λ ∆ = 2µ 푡푐표푠푟 − − 푓 2 2
∆ = 2µ푓푡푐표푠푟 휆 휆 The first corressponds to phase change at the top surface and the second corresponds to change which 2 2 occurs at the bottom surface because 휇푓 < 휇푠푢푏 . The two waves interfere destructively if
훌 ∆ = (2n + 1) ퟐ
훌 2µ 푡푐표푠푟 = (2n+ퟏ) 푓 ퟐ
For normal incidence cos r = 1
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훌 2µ 푡 = (2n+ퟏ) 푓 ퟐ
When m=0 the thickness of the film will be minimum
훌 2µ 푡 = 푓 푚푖푛 ퟐ
훌 µ 푡 = 푓 푚푖푛 4
Thus the optical thickness (µftmin ) of the AR coating should be of one quarter wavelength.
훌 푡푚푖푛 = 4µ푓
This is expression for minimum thickness of antireflection coating.
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