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Chapter 1 Optical Resonator
Fundamentals of Photonics 2018/9/11
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What is an optical resonator?
An optical resonator, the optical counterpart of an electronic resonant circuit, confines and stores light at certain resonance frequencies. It may be viewed as an optical transmission system incorporating feedback; light circulates or is repeatedly reflected within the system, without escaping. (a) (b) (c) (d) Fundamentals of Photonics 2018/9/11
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Fundamentals of Photonics
Contents: 1.1 Brief review of matrix optics 1.2 Planar Mirror Resonators Resonator Modes The Resonator as a Spectrum Analyzer Two- and Three-Dimensional Resonators 1.3 Spherical-Mirror Resonators Ray confinement 1.4 Gaussian waves and its characteristics The Gaussian beam Transmission through optical components Gaussian Modes Resonance Frequencies Hermite-Gaussian Modes Finite Apertures and Diffraction Loss Fundamentals of Photonics 2018/9/11
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1.1 Brief review of Matrix optics
Light propagation in a optical system, can use a matrix M, whose elements are A, B, C, D, characterizes the optical system Completely ( known as the ray-transfer matrix.) to describe the rays transmission in the optical components. One can use two parameters: y: the high q: the angle above z axis Fundamentals of Photonics 2018/9/11
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Fundamentals of Photonics
For the paraxial rays y2 y1 y2,q2 -q2 y1,q1 q1 q Along z upward angle is positive, and downward is negative Fundamentals of Photonics 2018/9/11
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Fundamentals of Photonics
Free-Space Propagation Refraction at a Planar Boundary Refraction at a Spherical Boundary Transmission Through a Thin Lens Reflection from a Planar Mirror Reflection from a Spherical Mirror Fundamentals of Photonics 2018/9/11
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Fundamentals of Photonics
A Set of Parallel Transparent Plates Matrices of Cascaded Optical Components Fundamentals of Photonics 2018/9/11
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Periodic Optical Systems
The reflection of light between two parallel mirrors forming an optical resonator is a periodic optical system is a cascade of identical unit system. Difference Equation for the Ray Position A periodic system is composed of a cascade of identical unit systems (stages), each with a ray-transfer matrix (A, B, C, D). A ray enters the system with initial position y, and slope 8,. To determine the position and slope (y,,,, 0,) of the ray at the exit of the mth stage, we apply the ABCD matrix m times, Fundamentals of Photonics 2018/9/11
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Fundamentals of Photonics
From these equation, we have And then: linear differential equations where Fundamentals of Photonics 2018/9/11
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Fundamentals of Photonics
If we assumed: So that, we have If we defined We have then A general solution may be constructed from the two solutions with positive and negative signs by forming their linear combination. The sum of the two exponential functions can always be written as a harmonic (circular) function Fundamentals of Photonics 2018/9/11
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Fundamentals of Photonics
If F=1, then Condition for a Harmonic Trajectory: if ym be harmonic, the f=cos-1b must be real, We have condition or The bound therefore provides a condition of stability (boundedness) of the ray trajectory If, instead, |b| > 1, f is then imaginary and the solution is a hyperbolic function (cosh or sinh), which increases without bound. A harmonic solution ensures that y, is bounded for all m, with a maximum value of ymax. The bound |b|< 1 therefore provides a condition of stability (boundedness) of the ray trajectory. Fundamentals of Photonics 2018/9/11
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Condition for a Periodic Trajectory
Unstable b>1 Stable and periodic Stable nonperiodic The harmonic function is periodic in m, if it is possible to find an integer s such that ym+s = ym, for all m. The smallest such integer is the period. The necessary and sufficient condition for a periodic trajectory is: sf = 2pq, where q is an integer Fundamentals of Photonics 2018/9/11
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Fundamentals of Photonics
EXERCISE : A Periodic Set of Pairs of Different Lenses. Examine the trajectories of paraxial rays through a periodic system composed of a set of lenses with alternating focal lengths f1 and f2 as shown in Fig. Show that the ray trajectory is bounded (stable) if Fundamentals of Photonics 2018/9/11
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Fundamentals of Photonics
Home works 4 X 4 Ray-Transfer Matrix for Skewed Rays. Matrix methods may be generalized to describe skewed paraxial rays in circularly symmetric systems, and to astigmatic (non-circularly symmetric) systems. A ray crossing the plane z = 0 is generally characterized by four variables-the coordinates (x, y) of its position in the plane, and the angles (e,, ey) that its projections in the x-z and y-z planes make with the z axis. The emerging ray is also characterized by four variables linearly related to the initial four variables. The optical system may then be characterized completely, within the paraxial approximation, by a 4 X 4 matrix. (a) Determine the 4 x 4 ray-transfer matrix of a distance d in free space. (b) Determine the 4 X 4 ray-transfer matrix of a thin cylindrical lens with focal length f oriented in the y direction. The cylindrical lens has focal length f for rays in the y-z plane, and no focusing power for rays in the x-z plane. Fundamentals of Photonics 2018/9/11
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Fundamentals of Photonics
1.2 Planar Mirror Resonators This simple one-dimensional resonator is known as a Fabry-Perot etalon. Resonator Modes Resonator Modes as Standing Waves A monochromatic wave of frequency v has a wavefunction as Represents the transverse component of electric field. The complex amplitude U(r) satisfies the Helmholtz equation; Where k =2pv/c called wavenumber, c speed of light in the medium Fundamentals of Photonics 2018/9/11 15
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Fundamentals of Photonics
the modes of a resonator must be the solution of Helmholtz equation with the boundary conditions: d So that the general solution is standing wave: With boundary condition, we have Resonance frequencies Fundamentals of Photonics 2018/9/11 16
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Resonator Modes as Traveling Waves
The resonance wavelength is: The length of the resonator, d = q lq /2, is an integer number of half wavelength Attention: Where n is the refractive index in the resonator Resonator Modes as Traveling Waves A mode of the resonator: is a self-reproducing wave, i.e., a wave that reproduces itself after a single round trip , The phase shift imparted by a single round trip of propagation (a distance 2d) must therefore be a multiple of 2p. q= 1,2,3,… Fundamentals of Photonics 2018/9/11 17
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Fundamentals of Photonics
Density of Modes (1D) The density of modes M(v), which is the number of modes per unit frequency per unit length of the resonator, is For 1D resonator The number of modes in a resonator of length d within the frequency interval v is: This represents the number of degrees of freedom for the optical waves existing in the resonator, i.e., the number of independent ways in which these waves may be arranged. Fundamentals of Photonics 2018/9/11 18
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Losses and Resonance Spectral Width
The magnitude ratio of two consecutive phasors is the round-trip amplitude attenuation factor r introduced by the two mirror reflections and by absorption in the medium. Thus: U1 U2 U3 U0 Mirror 1 Mirror 2 So that, the sum of the sequential reflective light with field of finally, we have Finesse of the resonator Fundamentals of Photonics 2018/9/11 19
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Fundamentals of Photonics
The resonance spectral peak has a full width of half maximum (FWHM): Due to We have where Fundamentals of Photonics 2018/9/11 20
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Fundamentals of Photonics
Spectral response of Fabry-Perot Resonator The intensity I is a periodic function of j with period 2p. The dependence of I on n, which is the spectral response of the resonator, has a similar periodic behavior since j = 4pnd/c is proportional to n. This resonance profile: The maximum I = Imax, is achieved at the resonance frequencies whereas the minimum value The FWHM of the resonance peak is Fundamentals of Photonics 2018/9/11 21
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Sources of Resonator Loss
Absorption and scattering loss during the round trip: exp (-2asd) Imperfect reflectance of the mirror: R1, R2 Defineding that we get: ar is an effective overall distributed-loss coefficient, which is used generally in the system design and analysis Fundamentals of Photonics 2018/9/11 22
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Fundamentals of Photonics
If the reflectance of the mirrors is very high, approach to 1, so that The above formula can approximate as The finesse F can be expressed as a function of the effective loss coefficient ar, Because ard<<1, so that exp(-ard)=1-ard, we have: The finesse is inversely proportional to the loss factor ard Fundamentals of Photonics 2018/9/11 23
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Photon Lifetime of Resonator
The relationship between the resonance linewidth and the resonator loss may be viewed as a manifestation of the time-frequency uncertainty relation. Form the linewidth of the resonator, we have Because ar is the loss per unit length, car is the loss per unit time, so that we can Defining the characteristic decay time as the resonator lifetime or photon lifetime The resonance line broadening is seen to be governed by the decay of optical energy arising from resonator losses Fundamentals of Photonics 2018/9/11 24
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Fundamentals of Photonics
The Quality Factor Q The quality factor Q is often used to characterize electrical resonance circuits and microwave resonators, for optical resonators, the Q factor may be determined by percentage of that stored energy to the loss energy per cycle: Large Q factors are associated with low-loss resonators For a resonator of loss at the rate car (per unit time), which is equivalent to the rate car /n0 (per cycle), so that The quality factor is related to the resonator lifetime (photon lifetime) The quality factor is related to the finesse of the resonator by Fundamentals of Photonics 2018/9/11 25
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Fundamentals of Photonics
In summary, three parameters are convenient for characterizing the losses in an optical resonator: the finesse F the loss coefficient ar (cm-1), photon lifetime tp = 1/car, (seconds). In addition, the quality factor Q can also be used for this purpose Fundamentals of Photonics 2018/9/11 26
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B. The Resonator as a Spectrum Analyzer
Transmission of a plane wave across a planar-mirror resonator (Fabry-Perot etalon) U1 U2 U0 Mirror 1 Mirror 2 t1 t2 r1 r2 Where: The change of the length of the cavity will change the resonance frequency Fundamentals of Photonics 2018/9/11 27
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C. Two- and Three-Dimensional Resonators
Two-Dimensional Resonators Mode density Determine an approximate expression for the number of modes in a two-dimensional resonator with frequencies lying between 0 and n, assuming that 2pn/c >> p/d, i.e. d >>l/2, and allowing for two orthogonal polarizations per mode number. Fundamentals of Photonics 2018/9/11 28
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Three-Dimensional Resonators
Physical space resonator Wave vector space Mode density The number of modes lying in the frequency interval between 0 and v corresponds to the number of points lying in the volume of the positive octant of a sphere of radius k in the k diagram Fundamentals of Photonics 2018/9/11 29
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1.2 Optical resonators and stable condition
A. Ray Confinement of spherical resonators The rule of the sign: concave mirror (R < 0), convex (R > 0). The planar-mirror resonator is R1 = R2=∞ d z The matrix-optics methods introduced which are valid only for paraxial rays, are used to study the trajectories of rays as they travel inside the resonator Fundamentals of Photonics 2018/9/11 30
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C. Stable condition of the resonator
For paraxial rays, where all angles are small, the relation between (ym+1, qm+1) and (ym, qm) is linear and can be written in the matrix form d z -q 1 q 0 q 2 R1 R2 y0 y1 y2 reflection from a mirror of radius R1 reflection from a mirror of radius R2 Attention here: we just take general case spherical so doesn’t take the sign propagation a distance d through free space Fundamentals of Photonics 2018/9/11 31
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Fundamentals of Photonics
It the way is harmonic, we need f =cos-1b must be real, that is for g1=1+d/R1; g2=1+d/R2 Fundamentals of Photonics 2018/9/11 32
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resonator is in conditionally stable, there will be:
In summary, the confinement condition for paraxial rays in a spherical-mirror resonator, constructed of mirrors of radii R1,R2 seperated by a distance d, is 0≤g1g2≤1, where g1=1+d/R1 and g2=1+d/R2 For the concave R is negative, for the convex R is positive Fundamentals of Photonics 2018/9/11 33
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Stable and unstable resonators
Planar (R1= R2=∞) Symmetrical resonators a b c d 1 -1 e Non stable b. Symmetrical confocal (R1= R2=-d) stable c. Symmetrical concentric (R1= R2=-d/2) d. confocal/planar (R1= -d,R2=∞) Non stable e. concave/convex (R1<0,R2>0) d/(-R) = 0, 1, and 2, corresponding to planar, confocal, and concentric resonators Fundamentals of Photonics 2018/9/11 34
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The stable properties of optical resonators
Planar (R1= R2=∞) Crystal state resonators b. Symmetrical confocal (R1= R2=-d) Stable c. Symmetrical concentric (R1= R2=-d/2) unstable Fundamentals of Photonics 2018/9/11 35
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Fundamentals of Photonics
Unstable resonators Unstable cavity corresponds to the high loss a. Biconvex resonator d b. plan-convex resonator c. Some cases in plan-concave resonator When R2<d, unstable d R1 d. Some cases in concave-convex resonator When R1<d and R1+R2=R1-|R2|>d e. Some cases in biconcave resonator Fundamentals of Photonics 2018/9/11 36
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1.2 Gaussian waves and its characteristics
A. Gaussian beam Helmholtz equation Normally, a plan wave (in z direction) will be When amplitude is not constant the wave is An axis symmetric wave in the amplitude z frequency Wave vector Fundamentals of Photonics 2018/9/11
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Paraxial Helmholtz equation
Substitute the U into the Helmholtz equation we have: where One simple solution is spherical wave: The other solution is Gaussian wave: where z0 is Rayleigh range q parameter Fundamentals of Photonics 2018/9/11
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Fundamentals of Photonics
Electric field of Gaussian wave propagates in z dirextion Beam width at z Waist width Radii of wave front at z Phase factor Fundamentals of Photonics 2018/9/11
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Fundamentals of Photonics
Gaussian beam at z=0 where E Beam width: will be minimum wave front: -W0 W0 at z=0, the wave front of Gaussian beam is a plan surface, but the electric field is Gaussian form W0 is the waist half width Fundamentals of Photonics 2018/9/11
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Fundamentals of Photonics
B. The characteristics of Gaussian beam z Beam radius Gaussian beam is a axis symmetrical wave, at z=0 phase is plan and the intensity is Gaussian form, at the other z, it is Gaussian spherical wave. Fundamentals of Photonics 2018/9/11
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Intensity of Gaussian beam
z=0 z=z0 z=2z0 The normalized beam intensity as a function of the radial distance at different axial distances Fundamentals of Photonics 2018/9/11
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On the beam axis (r = 0) the intensity
1 0.5 The normalized beam intensity I/I0 at points on the beam axis (p=0) as a function of z Fundamentals of Photonics 2018/9/11
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Fundamentals of Photonics
Power of the Gaussian beam The power of Gaussian beam is calculated by the integration of the optical intensity over a transverse plane So that we can express the intensity of the beam by the power The ratio of the power carried within a circle of radius r. in the transverse plane at position z to the total power is Fundamentals of Photonics 2018/9/11
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Fundamentals of Photonics
Beam Radius W(z) Beam waist 2W0 W0 q0 z -z0 z0 The beam radius W(z) has its minimum value W0 at the waist (z=0) reaches at z=±z0 and increases linearly with z for large z. Beam Divergence Fundamentals of Photonics 2018/9/11
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The characteristics of divergence angle
Let’s define f=z0 as the confocal parameter of Gaussian beam The physical means of f :the half distance between two section of width Fundamentals of Photonics 2018/9/11
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Depth of Focus Since the beam has its minimum width at z = 0, it achieves its best focus at the plane z = 0. In either direction, the beam gradually grows “out of focus.” The axial distance within which the beam radius lies within a factor 20.5 of its minimum value (i.e., its area lies within a factor of 2 of its minimum) is known as the depth of focus or confocal parameter wo 20.5wo 2zo z The depth of focus of a Gaussian beam. Fundamentals of Photonics 2018/9/11
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Fundamentals of Photonics
Phase of Gaussian beam The phase of the Gaussian beam is, On the beam axis (p = 0) the phase Phase of plan wave an excess delay of the wavefront in comparison with a plane wave or a spherical wave The excess delay is –p/2 at z=-∞, and p/2 at z= ∞ The total accumulated excess retardation as the wave travels from z = -∞ to z =∞is p. This phenomenon is known as the Guoy effect. Fundamentals of Photonics 2018/9/11
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Fundamentals of Photonics
Wavefront Confocal field and its equal phase front Fundamentals of Photonics 2018/9/11
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Parameters Required to Characterize a Gaussian Beam
How many parameters are required to describe a plane wave, a spherical wave, and a Gaussian beam? The plane wave is completely specified by its complex amplitude and direction. The spherical wave is specified by its amplitude and the location of its origin. The Gaussian beam is characterized by more parameters- its peak amplitude [the parameter A, its direction (the beam axis), the location of its waist, and one additional parameter: the waist radius W0 or the Rayleigh range zo, q-parameter q(z) is sufficient for characterizing a Gaussian beam of known peak amplitude and beam axis If the complex number q(z) = z + iz0, is known the real part of q(z) z is the beam waist place the imaginary parts of q(z) z0 is the Rayleigh range Fundamentals of Photonics 2018/9/11
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Fundamentals of Photonics
Parameters required to describe a Gaussian beam The Gaussian beam is characterized by : its peak amplitude [the parameter A, its direction (the beam axis), the location of its waist, and one additional parameter: the waist radius W0 or the Rayleigh range zo q parameter is an sufficient factor to characteristic a Gaussian beam of known peak amplitude and beam axis If the complex number q(z) = z + iz0, is known, the distance z to the beam waist and the Rayleigh range z0. are readily identified as the real and imaginary parts of q(z). Fundamentals of Photonics 2018/9/11
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Fundamentals of Photonics
C. TRANSMISSION THROUGH OPTICAL COMPONENTS a). Transmission Through a Thin Lens Notes: R is positive since the wavefront of the incident beam is diverging and R’ is negative since the wavefront of the transmitted beam is converging. Fundamentals of Photonics 2018/9/11
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Fundamentals of Photonics
In the thin lens transform, we have If we know W0, Z1, f, we can get The minus sign is due to the waist lies to the right of the lens. Fundamentals of Photonics 2018/9/11
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Fundamentals of Photonics
because Waist radius The beam waist is magnified by M, the beam depth of focus is magnified by M2, and the angular divergence is minified by the factor M. Waist location Depth of focus Divergence angle , magnification where Fundamentals of Photonics 2018/9/11
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Fundamentals of Photonics
Limit of Ray Optics Consider the limiting case in which (z - f) >>zo, so that the lens is well outside the depth of focus of the incident beam, The beam may then be approximated by a spherical wave, thus 2W0 2W0’ z z’ The magnification factor Mr is that based on ray optics. provides that M < Mr, the maximum magnification attainable is the ray-optics magnification Mr. Fundamentals of Photonics 2018/9/11
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Fundamentals of Photonics
b). Beam Shaping Beam Focusing If a lens is placed at the waist of a Gaussian beam, so z=0, then If the depth of focus of the incident beam 2z0, is much longer than the focal length f of the lens, then W0’= ( f/zo)Wo. Using z0 =pW02/l, we obtain The transmitted beam is then focused at the lens’ focal plane as would be expected for parallel rays incident on a lens. This occurs because the incident Gaussian beam is well approximated by a plane wave at its waist. The spot size expected from ray optics is zero Fundamentals of Photonics 2018/9/11
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Fundamentals of Photonics
Focus of Gaussian beam For given f, changes as when decreases as z decreases reaches minimum, and M<1, for f>0, it is focal effect when , increases as z increases when the bigger z, smaller f, better focus when reaches maximum, when , it will be focus Fundamentals of Photonics 2018/9/11
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Fundamentals of Photonics
In laser scanning, laser printing, and laser fusion, it is desirable to generate the smallest possible spot size, this may be achieved by use of the shortest possible wavelength, the thickest incident beam, and the shortest focal length. Since the lens should intercept the incident beam, its diameter D must be at least 2W0. Assuming that D = 2Wo, the diameter of the focused spot is given by where F# is the F-number of the lens. A microscope objective with small F-number is often used. Fundamentals of Photonics 2018/9/11
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Fundamentals of Photonics
Beam collimate locations of the waists of the incident and transmitted beams, z and z’ are The beam is collimated by making the location of the new waist z’ as distant as possible from the lens. This is achieved by using the smallest ratio z0/f (short depth of focus and long focal length). Fundamentals of Photonics 2018/9/11
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Fundamentals of Photonics
Beam expanding A Gaussian beam is expanded and collimated using two lenses of focal lengths f1 and f2, Assuming that f1<< z and z - f1>> z0, determine the optimal distance d between the lenses such that the distance z’ to the waist of the final beam is as large as possible. overall magnification M = W0’/Wo Fundamentals of Photonics 2018/9/11
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Fundamentals of Photonics
C). Reflection from a Spherical Mirror Reflection of a Gaussian beam of curvature R1 from a mirror of curvature R: f = -R/2. R > 0 for convex mirrors and R < 0 for concave mirrors, Fundamentals of Photonics 2018/9/11
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Fundamentals of Photonics
If the mirror is planar, i.e., R =∞, then R2= R1, so that the mirror reverses the direction of the beam without altering its curvature If R1= ∞, i.e., the beam waist lies on the mirror, then R2= R/2. If the mirror is concave (R < 0), R2 < 0, so that the reflected beam acquires a negative curvature and the wavefronts converge. The mirror then focuses the beam to a smaller spot size. If R1= -R, i.e., the incident beam has the same curvature as the mirror, then R2= R. The wavefronts of both the incident and reflected waves coincide with the mirror and the wave retraces its path. This is expected since the wavefront normals are also normal to the mirror, so that the mirror reflects the wave back onto itself. the mirror is concave (R < 0); the incident wave is diverging (R1 > 0) and the reflected wave is converging (R2< 0). Fundamentals of Photonics 2018/9/11
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Fundamentals of Photonics
d). Transmission Through an Arbitrary Optical System An optical system is completely characterized by the matrix M of elements (A, B, C, D) ray-transfer matrix relating the position and inclination of the transmitted ray to those of the incident ray The q-parameters, q1 and q2, of the incident and transmitted Gaussian beams at the input and output planes of a par-axial optical system described by the (A, B, C, D) matrix are related by Fundamentals of Photonics 2018/9/11
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Fundamentals of Photonics
ABCD law The q-parameters, q1 and q2, of the incident and transmitted Gaussian beams at the input and output planes of a par-axial optical system described by the (A, B, C, D) matrix are related by Because the q parameter identifies the width W and curvature R of the Gaussian beam, this simple law, called the ABCD law Invariance of the ABCD Law to Cascading If the ABCD law is applicable to each of two optical systems with matrices Mi =(Ai, Bi, Ci, Di), i = 1,2,…, it must also apply to a system comprising their cascade (a system with matrix M = M1M2). Fundamentals of Photonics 2018/9/11
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C. HERMITE - GAUSSIAN BEAMS
The self-reproducing waves exist in the resonator, and resonating inside of spherical mirrors, plan mirror or some other form paraboloidal wavefront mirror, are called the modes of the resonator Hermite - Gaussian Beam Complex Amplitude where is known as the Hermite-Gaussian function of order l, and Al,m is a constant Hermite-Gaussian beam of order (I, m). The Hermite-Gaussian beam of order (0, 0) is the Gaussian beam. Fundamentals of Photonics 2018/9/11
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G1(u) = 2u exp( -u2/2) is an odd function,
H0(u) = 1, the Hermite-Gaussian function of order O, the Gaussian function. G1(u) = 2u exp( -u2/2) is an odd function, G2(u) = (4u2 - 2) exp( -u2/2) is even, G3(u) = (8u3 - 12u)exp( -u2/2) is odd, Fundamentals of Photonics 2018/9/11
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Fundamentals of Photonics
Intensity Distribution The optical intensity of the (I, m) Hermite-Gaussian beam is Fundamentals of Photonics 2018/9/11
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Fundamentals of Photonics
C. Gaussian Modes Gaussian beams are modes of the spherical-mirror resonator; Gaussian beams provide solutions of the Helmholtz equation under the boundary conditions imposed by the spherical-mirror resonator z Beam radius a Gaussian beam is a circularly symmetric wave whose energy is confined about its axis (the z axis) and whose wavefront normals are paraxial rays Fundamentals of Photonics 2018/9/11
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Fundamentals of Photonics
Gaussian beam intensity: The Rayleigh range z0 where z0 is the distance called Rayleigh range, at which the beam wavefronts are most curved or we usually called confocal prrameter Beam width minimum value W0 at the beam waist (z = 0). The radius of curvature Beam waist Fundamentals of Photonics 2018/9/11
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Gaussian Mode of a Symmetrical Spherical-Mirror Resonator
z z1 z2 Fundamentals of Photonics 2018/9/11
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Fundamentals of Photonics
the beam radii at the mirrors An imaginary value of z0 signifies that the Gaussian beam is in fact a paraboloidal wave, which is an unconfined solution, so that z0 must be real. it is not difficult to show that the condition z02 > 0 is equivalent to Fundamentals of Photonics 2018/9/11
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Gaussian Mode of a Symmetrical Spherical-Mirror Resonator
Symmetrical resonators with concave mirrors that is R1 = R2= -/RI so that z1 = -d/2, z2 = d/2. Thus the beam center lies at the center The confinement condition becomes Fundamentals of Photonics 2018/9/11
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Fundamentals of Photonics
Given a resonator of fixed mirror separation d, we now examine the effect of increasing mirror curvature (increasing d/lRI) on the beam radius at the waist W0, and at the mirrors Wl = W2. As d/lRI increases, W0 decreases until it vanishes for the concentric resonator (d/lR| = 2); at this point W1 = W2 = ∞ The radius of the beam at the mirrors has its minimum value, WI = W2= (ld/p)1/2, when d/lRI = 1 Fundamentals of Photonics 2018/9/11
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C. Resonance Frequencies
The phase of a Gaussian beam, At the locations of the mirrors z1 and z2 on the optical axia (x2+y2=0), we have, where As the traveling wave completes a round trip between the two mirrors, therefore, its phase changes by For the resonance, the phase must be in condition If we consider the plane wave resonance frequency We have Fundamentals of Photonics 2018/9/11
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Fundamentals of Photonics
Spherical-Mirror Resonator Resonance Frequencies (Gaussian Modes) The frequency spacing of adjacent modes is VF = c/2d, which is the same result as that obtained for the planar-mirror resonator. For spherical-mirror resonators, this frequency spacing is independent of the curvatures of the mirrors. The second term in the fomula, which does depend on the mirror curvatures, simply represents a displacement of all resonance frequencies. Fundamentals of Photonics 2018/9/11
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D. Hermite - Gaussian Modes
The resolution for Helmholtz equation An entire family of solutions, the Hermite-Gaussian family, exists. Although a Hermite-Gaussian beam of order (I, m) has the same wavefronts as a Gaussian beam, its amplitude distribution differs . It follows that the entire family of Hermite-Gaussian beams represents modes of the spherical-mirror resonator Spherical mirror resonator Resonance Frequencies (Hermite -Gaussian Modes) Fundamentals of Photonics 2018/9/11
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Fundamentals of Photonics
longitudinal or axial modes: different q and same indices (l, m) the intensity will be the same transverse modes: The indices (I, m) label different means different spatial intensity dependences Longitudinal modes corresponding to a given transverse mode (I, m) have resonance frequencies spaced by vF = c/2d, i.e., vI,m,q – vI’,m’,q = vF. Transverse modes, for which the sum of the indices l+ m is the same, have the same resonance frequencies. Two transverse modes (I, m), (I’, m’) corresponding mode q frequencies spaced Fundamentals of Photonics 2018/9/11
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*E. Finite Apertures and Diffraction Loss
Since the resonator mirrors are of finite extent, a portion of the optical power escapes from the resonator on each pass. An estimate of the power loss may be determined by calculating the fractional power of the beam that is not intercepted by the mirror. That is the finite apertures effect and this effect will cause diffraction loss. For example: If the Gaussian beam with radius W and the mirror is circular with radius a and a= 2W, each time there is a small fraction, exp( - 2a2/ W2) = 3.35 x10-4, of the beam power escapes on each pass. Higher-order transverse modes suffer greater losses since they have greater spatial extent in the transverse plane. In the resonator, the mirror transmission and any aperture limitation will induce loss The aperture induce loss is due to diffraction loss, and the loss depend mainly on the diameters of laser beam, the aperture place and its diameter We can used Fresnel number N to represent the relation between the size of light beam and the aperture, and use N to represent the loss of resonator. Fundamentals of Photonics 2018/9/11
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Fundamentals of Photonics
Diffraction loss The Fresnel number NF Attention: the W here is the beam width in the mirror, a is the dia. of mirror Physical meaning:the ratio of the accepting angle (a/d) (form one mirror to the other of the resonator )to diffractive angle of the beam (l/a) . The higher Fresnel number corresponds to a smaller loss Fundamentals of Photonics 2018/9/11
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Fundamentals of Photonics
N is the maximum number of trip that light will propagate in side resonator without escape. 1/N represent each round trip the ratio of diffraction loss to the total energy Symmetric confocal resonator For general stable concave mirror resonator, the Fresnel number for two mirrors are: Fundamentals of Photonics 2018/9/11
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Home work 2 The light from a Nd:YAG laser at wavelength 1.06 mm is a Gaussian beam of 1 W optical power and beam divergence 2q0= 1 mrad. Determine the beam waist radius, the depth of focus, the maximum intensity, and the intensity on the beam axis at a distance z = 100 cm from the beam waist. Beam Focusing. An argon-ion laser produces a Gaussian beam of wavelength l = 488 nm and waist radius w0 = 0.5 mm. Design a single-lens optical system for focusing the light to a spot of diameter 100 pm. What is the shortest focal-length lens that may be used? Spot Size. A Gaussian beam of Rayleigh range z0 = 50 cm and wavelength l=488nm is converted into a Gaussian beam of waist radius W0’ using a lens of focal length f = 5 cm at a distance z from its waist. Write a computer program to plot W0’ as a function of z. Verify that in the limit z - f >>z0 , the relations (as follows) hold; and in the limit z << z0 holds. Beam Refraction. A Gaussian beam is incident from air (n = 1) into a medium with a planar boundary and refractive index n = 1.5. The beam axis is normal to the boundary and the beam waist lies at the boundary. Sketch the transmitted beam. If the angular divergence of the beam in air is 1 mrad, what is the angular divergence in the medium? Fundamentals of Photonics 2018/9/11
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Resonance Frequencies of a Resonator with an Etalon. (a) Determine the spacing between adjacent resonance frequencies in a resonator constructed of two parallel planar mirrors separated by a distance d = 15 cm in air (n = 1).(b) A transparent plate of thickness d1= 2.5 cm and refractive index n = 1.5 is placed inside the resonator and is tilted slightly to prevent light reflected from the plate from reaching the mirrors. Determine the spacing between the resonance frequencies of the resonator. Mirrorless Resonators. Semiconductor lasers are often fabricated from crystals whose surfaces are cleaved along crystal planes. These surfaces act as reflectors and therefore serve as the resonator mirrors. Consider a crystal with refractive index n = 3.6 placed in air (n = 1). The light reflects between two parallel surfaces separated by the distance d = 0.2 mm. Determine the spacing between resonance frequencies vF, the overall distributed loss coefficient ar, the finesse F, and the spectral width dv. Assume that the loss coefficient (as= 1 cm-1). Fundamentals of Photonics 2018/9/11
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b). Parameter q transform for thin lens
If put the original point at the waist of beam, the parameter q will be The real part is concern with wave front R(z), the imaginary part is corresponding to beam radii w(z) For q0 is the value at R(0)→∞ Finally we have: Fundamentals of Photonics 2018/9/11
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Parameter q in the thin lens transform
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continue Real part Imaginary part Fundamentals of Photonics 2018/9/11
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Special cases when z1→∞, then z2=F when incident beam waist is in the far infinite the outlet beam waist is located at the focal plan when z1=f, then z2=F, the incident beam waist is in the object focal plan, the outlet beam waist will be in the imaging focal plan when , then Fundamentals of Photonics 2018/9/11
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Beam waist radius Waist location Depth of focus Divergence Magnification , Fundamentals of Photonics 2018/9/11
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CHAPTER 2 PHOTONS AND ATOMS
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Max Planck Nobel prize 1918 Albert Einstein Nobel prize 1921 Fundamentals of Photonics 2018/9/11
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1. Atoms, Molecules, and Solids
Atomic Physics’ starting point: the old Bohr model Schrodinger equation Time independent Schrodinger equation Fundamentals of Photonics 2018/9/11
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-on discrete stationary states -radiative transitions quantum jumps between levels -The old Bohr model -energy levels For molecular oscillation: Fundamentals of Photonics 2018/9/11
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Vibrational and Rotational Energy Levels of Molecules ev ev N2 CO2 050 0.4 0.4 200 040 Energy 0.3 q=1 001 0.3 9.6um 030 10.6um 0.2 100 0.2 020 0.1 010 0.1 q=0 000 Asymmetric stretch Symmetric stretch Bending Figure Lowest vibrational energy levels of the N2 and CO2 molecules (the zero of energy is chosen at q=0). The transitions marked by arrows represent energy exchanges corresponding to photons of wavelengths 10.6um and 9.6um, as indicated. These transitions are used in CO2 lasers. Fundamentals of Photonics 2018/9/11
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Electron Energy Levels of Atoms and Molecules
He Ne 21 2p55s 1s2s 1S0 3.39um 2p54p 19 18 17 16 21 20 20 1s2s 3S1 2p54s 632.8nm Energy 19 2p53p 18 17 2p53s 16 Odd parity Even parity Figure Some energy levels of He and Ne atoms. The He transitions marked by arrows correspond to photons of wavelengths 3.39m and 632.8nm, as indicated. These transitions are used in He-Ne lasers. Fundamentals of Photonics 2018/9/11
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Electron Energy Levels in Solids
Isolated atoms and molecules exhibit discrete energy levels, as shown in Figs to For solids, however, the atoms, ions, or molecules in close proximity to each other and cannot therefore be considered as simple collections of isolated atoms; rather, they must be treated as a many-body system. Vacuum level Eg Eg 3p 3s Energy 2p 2s 1s Isolated atom Semi-conductor Metal Insulator Figure Broadening of the discrete energy levels of an isolated atom into bands for solis-state materials. Fundamentals of Photonics 2018/9/11
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Quantum-well Structure
Multi-layers of different semiconductor materials Energy 120 20 40 60 80 100 Distance nm Conduction band Valence band GaAs AlGaAs Figure Quantized energies in a single-crystal AlGaAs/GaAs multiquantum-well structure. The well widths can be arbitrary (as shown) or periodic. Fundamentals of Photonics 2018/9/11
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2. Occupation of energy levels in thermal equilibrium
Boltzmann distribution Fundamentals of Photonics 2018/9/11
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Em Em E3 E3 E2 E2 E1 E1 P(Em) Energy levels Occupation Fundamentals of Photonics 2018/9/11
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Fermi-Dirac Distribution
Pauli exclusion principle Fundamentals of Photonics 2018/9/11
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Boltzmann P(Em) Ef Fermi-Dirac f(E) 1/2 1 Figure The Fermi-Dirac distribution f(E) is well approxiamated by the Boltzmann distribution P(Em) when E>>Ef. Fundamentals of Photonics 2018/9/11
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3.Interactions of photons with atoms
Semi-classical view of atom excitations e Ze Atom in ground state e Ze Atom in excited state Fundamentals of Photonics 2018/9/11
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Spontaneous Emission hn Figure Spontaneous emission of a photon into the mode of frequencyn by an atomic transition from energy level 2 to level 1. The photon energy hn=E2-E1 点击查看flash动画 transition cross-section. Fundamentals of Photonics 2018/9/11
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Decay of the number of the excited atoms N(t) N(0) t 1/Psp Figure Spontaneous emission into a single mode causes the number of excited atoms to decrease exponentially with time constant 1/Psp Fundamentals of Photonics 2018/9/11
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Absorption Figure Absorption of a photon hn leads to an upwoard transition of the atom from energy level 1 to energy level 2. hn 点击查看flash动画 Fundamentals of Photonics 2018/9/11
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Stimulated Emission ground state excited state Super-excited state hn Energy 点击查看flash动画 When a photon enters, it “knocks” an electron from the inverted population down to the ground state, thus creating a new photon. This amplification process is called stimulated emission. Fundamentals of Photonics 2018/9/11
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Lineshape Function Transition Strength Lineshape function Fundamentals of Photonics 2018/9/11
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Figure The transition cross section s(n) and the lineshape function g(n) Fundamentals of Photonics 2018/9/11
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Total Spontaneous Emission into All Modes
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Relation between the Transition Cross Section and the Spontaneous Lifetime Fundamentals of Photonics 2018/9/11
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Stimulated Emission and Absorption
Transitions Induced by Monochromatic Light Transitions Induced by Broadband Light Stimulated Emission and Absorption Fundamentals of Photonics 2018/9/11
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(4.4-17) Fundamentals of Photonics 2018/9/11
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Einstein Coefficients
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Lineshape Broadening Life-time broadening Caused by the time uncertainty of the occupation of the energy level Fundamentals of Photonics 2018/9/11
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Collision Broadening Figure A sinewave interrupted at the rate fcol by random phase jumps has a Lorentzian spectrum of width △n=fcol/p Fundamentals of Photonics 2018/9/11
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Inhomogeneous Broadening
Doppler effect 2 3 4 1 Direction of observation Figure The radiated frequency is dependent on the direction of atomic motion relative to the direction of observation. Radiation from atom 1 has higher frequency than that from atoms 3 and 4. Radiation from atom 2 has lower frequency. Fundamentals of Photonics 2018/9/11
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P(v) u0 u Velocity v Figure The velocity distribution and average lineshape function of a Doppler-broadened atomic system. Fundamentals of Photonics 2018/9/11
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3. Thermal light Thermal Equilibrium Between Photons and Atoms Fundamentals of Photonics 2018/9/11
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The average number of photons in a mode of frequency n Fundamentals of Photonics 2018/9/11
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Black-body Radiation Fundamentals of Photonics 2018/9/11
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kT v Figure Semilogarithmic plot of the average energy E of an electromagnetic mode in thermal equilibrium at temperature T as a function of the mode frequency n. At T=300K, KBT/h = 6.25THz, which corrsponds to a wavelength of 48um. Fundamentals of Photonics 2018/9/11
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Spontaneous emissionl 返回 Fundamentals of Photonics 2018/9/11
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Absorption 返回 Fundamentals of Photonics 2018/9/11
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Stimulated Emission 返回 Fundamentals of Photonics 2018/9/11
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Chapter 4 Laser Fundamentals of Photonics 2018/9/11
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LASERS In 1958 Arthur Schawlow, together with Charles Townes, showed how to extend the principle of the maser to the optical region. He shared the 1981 Nobel Prize with Nicolaas Bloembergen. Maiman demonstrated the first successful operation of the ruby laser in 1960. Fundamentals of Photonics 2018/9/11
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LASERS an oscillator is an amplifier with positive feedback Two conditions for an oscillation: 1. Gain greater than loss: net gain 2. Phase shift in a round trip is a multiple of 2π Fundamentals of Photonics 2018/9/11
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Stable condition: gain = loss If the initical amplifier gain is greater than the loss, oscillation may initiate. The amplifier then satuates whereupon its gain decreases. A steady-state condition is reached when the gain just equals the loss. Steady-state power Gain Loss Power An oscillator comprises: ◆ An amplifier with a gain-saturation mechanism ◆ A feedback system ◆ A frequency-selection mechanism ◆ An output coupling scheme Fundamentals of Photonics 2018/9/11
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Light amplifier with positive feedback
Gain medium (e.g. 3-level system w population inversion) When the gain exceeds the roundtrip losses, the system goes into oscillation + g + Fundamentals of Photonics 2018/9/11
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LASERS Light Amplification through Stimualted Emission Radiation
点击查看flash动画 Amplified once Gain medium (e.g. 3-level system w population inversion) Initial photon Reflected Amplified twice Output Reflected Amplified Again Partially reflecting Mirror Light Amplification through Stimualted Emission Radiation Fundamentals of Photonics 2018/9/11
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Partially transmitting mirror Laser output
Active medium d Partially transmitting mirror Laser output A laser consists of an optical amplifier (employing an active medium) placed within an optical resonator. The output is extracted through a partially transmitting mirror. Fundamentals of Photonics 2018/9/11
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Optical amplification and feedback
★ Gain medium The laser amplifier is a distributed-gain device characterized by its gain coefficient Small signal Gain Coefficient Saturated Gain Coefficient Fundamentals of Photonics 2018/9/11
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Phase-shift Coefficient (Lorentzian Lineshape) Figure Spectral dependence of the gain and phase-shift coefficients for an optical amplifier with Lorentzian lineshape function Fundamentals of Photonics 2018/9/11
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Optical Feedback-Optical Resonator
Feedback and Loss: The optical resonator A Fabry-Perot resonator, comprising two mirrors separated by a distance d, contains the medium (refractive index n). Travel through the medium introduces a phase shift per unit length equal to the wavenumber In traveling a round trip through a resonator of length d, the photon-flux density is reduced by the factor R1R2exp(-2asd). The overall loss in one round trip can therefore be described by a total effective distributed loss coefficient ar, where Fundamentals of Photonics 2018/9/11
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Loss coefficient Photon lifetime ar represents the total loss of energy (or number of photons) per unit length, arc represents the loss of photons per second Fundamentals of Photonics 2018/9/11
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Resonator response Resonator modes are separated by the frequency and have linewidths Fundamentals of Photonics 2018/9/11
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Conditions for laser oscillation
Gain condition: Laser threshold Threshold Gain Condition where or Threshold Population Difference Fundamentals of Photonics 2018/9/11
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For a Lorentzian lineshape function, as
If the transition is limited by lifetime broadening with a decay time tsp As a numerical example, if l0=1 mm, tp=1 ns, and the refractive index n=1, we obtain Nt=2.1×107 cm-3 Fundamentals of Photonics 2018/9/11
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Conditions for laser oscillation(2) Phase condition: Laser Frequencies Frequency Pulling or Fundamentals of Photonics 2018/9/11
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Laser Frequencies Fundamentals of Photonics 2018/9/11
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The laser oscillation frequencies fall near the cold-resonator modes; they are pulled slightly toward the atomic resonance central frequency n0. Fundamentals of Photonics 2018/9/11
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Characteristics of the laser output
Internal Photon-Flux Density Gain Clamping Fundamentals of Photonics 2018/9/11
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Laser turn-on Time Steady state ar Loss coefficient g(u) Gain coefficient Photon-flux density Determination of the steady-state laser photon-flux density f. The smaller the loss, the greater the value of f. Fundamentals of Photonics 2018/9/11
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Steady Photon Density Since and
Steady-State Laser Internal Photon-Flux Density Fundamentals of Photonics 2018/9/11
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Steady-state values of the population difference N, and the laser internal photon-flux density f, as functions of N0 (the population difference in the absence of radiation; N0, increases with the pumping rate R). Output photon-flux density Optical Intensity of Laser Output Fundamentals of Photonics 2018/9/11
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Optimization of the output photon-flux density From We obtain When, use the approximation Then Fundamentals of Photonics 2018/9/11
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Spectral Distribution
Determined both by the atomic lineshape and by the resonant modes Number of Possible Laser Modes Linewidth ? Schawlow-Townes limit Fundamentals of Photonics 2018/9/11
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ar Loss Gain B Allowed modes n1n2……nM Resonator modes nF Figure (a) Laser oscillation can occur only at frequencies for which the gain coefficient is greater than the loss coefficient (stippled region). (b) Oscillation can occur only within dn of the resonator modal frequencies (which are represented as lines for simplicity of illustration ). Fundamentals of Photonics 2018/9/11
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Homogeneously Broadened Medium
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Inhomogeneously Broadened Medium 点击查看flash动画 Frequency n ar (a) (b) Figure (a) Laser oscillation occurs in an inhomogeneously broadened medium by each mode independently burning a hole in the overrall spectral gain profile. (b) Spectrum of a typical inhomogeneously broadened multimode gas laser. Fundamentals of Photonics 2018/9/11
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Hole burning in a Doppler-broadened medium
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Hole burning in a Doppler-broadened medium
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Spatial distribution and polarization
Spherical mirror Laser intensity x,y The laser output for the (0,0) tansverse mode of a spherical-mirror resonator takes the form of a Gaussian beam. Fundamentals of Photonics 2018/9/11
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TEM0,0 a11 a00 B11 B00 n (1,1) modes (0,0) modes Laser output TEM1,1 n Figure The gains and losses for two transverse modes, say (0,0) and (1,1), usually differ because of their different spatial distributions. Fundamentals of Photonics 2018/9/11
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Two Issues: Polarization, Unstable Resonators Fundamentals of Photonics 2018/9/11
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Mode Selection Selection of 1. Laser Line 2. Transverse Mode 3. Polarization 4. Longitudinal Mode Fundamentals of Photonics 2018/9/11
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Laser output Aperture Active medium Output mirror High reflectance mirror Prism Unwanted line Figure A paticular atomic line may be selected by the use of a prism placed inside the resonator. A transverse mode may be selected by means of a spatial aperture of carefully chosen shaped and size. Fundamentals of Photonics 2018/9/11
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High reflectance mirror
Output mirror Polarized laser output qB Brewster window Active midum Figure The use of Brewster windows in a gas laser provides a linearly polarized laser beam. Light polarized in the plane of incidence (the TM wave) is transmitted without reflection loss through a window placed at the Brewster angle. The orthogonally polarized (TE) mode suffers reflection loss and therefore does not oscillate. Fundamentals of Photonics 2018/9/11
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Selection of Longitudinal Mode
Etalon Active midum High reflectance mirror Output mirror d1 d Resonator loss c/2d Resonator mdoes Etalon mdoes c/2d1 Laser output Figure Longitudianl mode selection by the use of an intracavity etalon. Oscillation occurs at frequencies where a mode of the resonator coincides with an etalon mode; both must, of course, lie within the spectral window where the gain of the medium exceeds the loss. Fundamentals of Photonics 2018/9/11
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Multiple Mirror Resonators (a) (b) (c) Figure Longitudinal mode selection by use of (a) two coupled resonators (one passive and one active); (b) two coupled active resonators; (c) a coupled resonator-interferometer. Fundamentals of Photonics 2018/9/11
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Characteristics of Common Lasers
Solid State Lasers: Ruby, Nd3+:YAG, Nd3+:Silica, Er3+:Fiber, Yb3+:Fiber Gas Lasers: He-Ne, Ar+; CO2, CO, KF; Liquid Lasers: Dye Plasma X-Ray Lasers Free Electron Lasers Fundamentals of Photonics 2018/9/11
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Pulsed Lasers Method of pulsing lasers External Modulator or Internal Modulator? t Average power CW power Modulator (a) (b) Figure Comparison of pulsed laser outputs achievable with (a) an external modulator, and (b) an internal modulator Gain switching Q-Switching 3. Cavity Dumping 4.Mode Locking Fundamentals of Photonics 2018/9/11
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Gain Switching Fundamentals of Photonics 2018/9/11
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Q- Switching t Modulator Loss Gain Laser output Figure Q-switching. Fundamentals of Photonics 2018/9/11
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Cavity Dumping Figure Cavity dumping. Fundamentals of Photonics 2018/9/11
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Mode locking Laser modes coupling together
Lock their phases to each other
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Rate equation for the photon-number density
photon lifetime Probability density for induced absorption/emission From We have Photon-Number Rate Equation Fundamentals of Photonics 2018/9/11
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Rate equation for the Population Difference
For a three level system Note Then Where the small signal population difference Substituting Population-difference rate equation (Three-level system) We have Fundamentals of Photonics 2018/9/11
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Situation for the gain switching
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Situation for the Q-switching
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Dynamics of a Q-Switching process Dynamics of the Q-switching process Note the time relationship between the photon density and the population inversion variations! Fundamentals of Photonics 2018/9/11 174174
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Determination of the peak power, energy, width and shape of the optical pulse Dividing Fundamentals of Photonics 2018/9/11
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Power Peak pulse power When Ni>>Nt It is clear So The larger the initial population inversion, the higher the Q-switched pulse peak power. Fundamentals of Photonics 2018/9/11
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c. Pulse energy: The final population difference Nf Fundamentals of Photonics 2018/9/11
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d. Pulse width: A rough estimation of the pulse width is the ratio of the pulse energy to the peak pulse power. When Ni>>Nth and Ni>>Nf The shorter the photon life time, the shorter the Q-switched pulses. Fundamentals of Photonics 2018/9/11
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Techniques for Q-switching
1. Mechanical rotating mirror method: Q-switching principle: rotating the cavity mirror results in the cavity losses high and low, so the Q-switching is obtained. Advantages: simple, inexpensive. Disadvantages: very slow, mechanical vibrations. Fundamentals of Photonics 2018/9/11
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2. Electro-optic Q-switching
Disadvantages: complicate and expensive Advantages: very fast and stable. Pockels effect: applying electrical field in a uniaxial crystal results in additional birefringence, which changes the polarization of light when passing through it. Electro-optic effects: optical properties change with the applied electrical field strength. In the case of Pockels cell, the polarization of the light changes with the applied electrical field strength. Combined with a polarization beam splitter, one can use them to realize the Q-switching. Q-switching principle: placing an electro-optic crystal between crossed polarizers comprises a Pockels switch. Turning on and off the electrical field results in high and low cavity losses. Fundamentals of Photonics 2018/9/11
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Electro-optic Q-switch operated at (a) quarter-wave and (b) half-wave retardation voltage Fundamentals of Photonics 2018/9/11
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3. Acousto-optic Q-switching
1. Part of the light intensity is deflected away by the acoustic wave. The stronger the acoustic wave, the stronger the deflection. Fundamentals of Photonics 2018/9/11
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Diffracted light Incident light q q L Sound Transmitted light Piezoelectric transducer RF 1. Part of the light intensity is deflected away by the acoustic wave. The stronger the acoustic wave, the stronger the deflection. Bragg scattering: due to existence of the acoustic wave, light changes its propagation direction. Q-switching principle: through switching on and off of the acoustic wave the cavity losses is modulated. Advantages: works even for long wavelength lasers. Disadvantages: low modulation depth and slow. Fundamentals of Photonics 2018/9/11
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4. Saturable absorber Q-switching
What’s a saturable absorber? Absorption coefficient of the material is reversely proportional to the light intensity. Is: saturation intensity. Saturable absorber Q-switching: Insertion a saturable absorber in the laser cavity, the Q-switching will be automatically obtained. Fundamentals of Photonics 2018/9/11
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General characteristics of laser Q-switching
Pulsed laser output: Pulse duration – related to the photon lifetime. Pulse energy - related to the upper level lifetime. Laser operation mode: Single or multi-longitudinal modes. Active verses passive Q-switching methods: Passive: simple, economic, pulse jitter and intensity fluctuations. Active: stable pulse energy and repetition, expensive. Comparison with chopped laser beams: Energy concentration in time axis. Function of gain medium Energy storage Fundamentals of Photonics 2018/9/11
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Laser mode-locking Aims: Familiarize with the principle of laser mode-locking. Familiarize with different techniques of achieving laser Mode-locking. Outlines: Principle of laser mode-locking. Methods of laser mode-locking. Active mode-locking. Passive mode-locking. Transform-limited pulses. Fundamentals of Photonics 2018/9/11
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Principle of laser mode-locking
1. Lasing in inhomogeneously broadened lasers: i) Laser gain and spectral hole-burning. ii) Cavity longitudinal mode frequencies. iii) Multi-longitudinal mode operation. Fundamentals of Photonics 2018/9/11
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2. Laser multimode operation: Single mode lasers: Multimode lasers: Mode-frequency separations: Phase relation between modes: Random and independent! Total laser intensity fluctuates with time ! The mean intensity of a multimode laser remains constant, however, its instant intensity varies with time. Fundamentals of Photonics 2018/9/11
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3. Effect of mode-locking: (i) Supposing that the phases of all modes are locked together: (ii) Supposing that all modes have the same amplitude: purely for the convenience of the mathematical analysis (iii) Under the above two conditions, the total electric field of the multimode laser is: For the sack of simplicity, we assume that all laser modes have the same amplitude. As there exist a lot of modes, we select the mode at the center of the mode frequency distribution as the reference frequency, and denote all the other mode in relation to this reference mode. The reason for doing this is for the easy of mathematical treatment. where Fundamentals of Photonics 2018/9/11
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0 is the frequency of the central mode, N is the number of modes in the laser, c is the mode frequency separation. i is the frequency of the i-th mode. Calculating the summation yields: Note this is the optical field of the total laser Emission ! The optical filed can be thought to consist of a carrier wave of frequency 0 that amplitude modulated by the function Fundamentals of Photonics 2018/9/11
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4. Characteristics of the mode-locked lasers: The intensity of the laser field is: The output of a mode-locked laser consists of a series of pulses. The time separation between two pulses is determined by RT and the pulse width of each pulse is tp. Fundamentals of Photonics 2018/9/11
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5. Properties of mode-locked pulses: i) The pulse separation RT: The round-trip time of the cavity! ii) The peak power: Considering when is small, N times of the average power. N: number of modes. The more the modes the higher the peak power of the Mode-locked pulses. Fundamentals of Photonics 2018/9/11
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iii)The individual pulse width: a: bandwidth of the gain profile. Narrower as N increases. The mode locked pulse width is reversely proportional to the gain band width, so the broader the gain profile, the shorter are the mode locked pulses. Fundamentals of Photonics 2018/9/11
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Techniques of laser mode-locking
Active mode-locking: Actively modulating the gain or loss of a laser cavity in a periodic way, usually at the cavity repetition frequency c/2nL to achieve mode-locking. Amplitude modulation: A modulator with a transmission function of is inserted in the laser cavity to modulate the light. Where is the modulation strength and < 0.5. Under the influence of the modulation phases of the lasing modes become synchronized and as a consequence become mode-locked. Operation mechanism of the technique: Fundamentals of Photonics 2018/9/11
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Time domain analysis: Consider the extreme case where a shutter is inside the cavity, and the opens only for a short time every second. Is the cavity round trip time. In this case only a pulse with pulse width narrower than the opening time can survive in the cavity, all the CW type of operation will be blocked by the shutter. To have a pulse moving in the cavity the phase of all lasing modes must be synchronized. The laser modes will arrange themselves to realize such a state. It is a natural competition and the fittest will survive. Shutter losses Fundamentals of Photonics 2018/9/11
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Frequency domain analysis: Amplitude modulation Sidebands generation Electrical filed of each mode Amplitude of each mode is modulated Sidebands are generated by the modulation m m+ m- Without modulation After amplitude modulation Fundamentals of Photonics 2018/9/11
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In the case of a multimode laser As all modes are modulated by the same frequency, the sidebands of one mode will drive its adjacent modes, and as a consequence, all modes will oscillate with locked phase. From both the time domain and the frequency domain analysis it is easy to understand why the modulation frequency must be exactly the cavity longitudinal mode separation frequency. Fundamentals of Photonics 2018/9/11
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Passive mode-locking: Inserting an appropriately selected saturable absorber inside the laser cavity. Through the mutual interaction between light, saturable absorber and gain medium to automatically achieve mode locking. A typical passive mode locking laser configuration: laser medium satur able absorber Mechanism of the mode-locking: i) Interaction between saturable absorber and laser gain: Survival takes all! ii)Balance between the pulse shortening and pulse broadening: Final pulse width. Fundamentals of Photonics 2018/9/11
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Transform limited pulses
Gaussian pulses: In our analysis we have assumed that En=E0 The real gain line has a Gaussian profile,which results in that the lasing mode amplitudes have a Gaussian distribution. A Gaussian gain line shape function a Gaussian mode-locked pulse intensity variation, namely a Gaussian pulse. Fundamentals of Photonics 2018/9/11
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Fundamentals of Photonics
Intensity of the pulse: Gaussian intensity profile! Transform limited pulses: If the product of pulse width and spectral bandwidth of a Gaussian pulse equals 0.441, then the Gaussian pulse is called a transform limited pulse as in this case the pulse width is purely determined by the Fourier transformation of the pulse spectral distribution. For transform limited Gaussian pulses! tp: pulse width, L: spectral bandwidth Fundamentals of Photonics 2018/9/11
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