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CHAPTER 1---- Optical Resonator Chapter 1 Optical Resonator Fundamentals of Photonics 2017/5/24 1 CHAPTER 1---- Optical Resonator 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. (b) (a) (d) (c) Fundamentals of Photonics 2017/5/24 2 CHAPTER 1---- Optical Resonator 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 2017/5/24 3 CHAPTER 1---- Optical Resonator 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: y2 Ay1 B1 2 Cy1 D1 y2 Ay1 B1 the high : the angle above z axis y2 A B y1 C D 1 2 2 Cy1 D1 Fundamentals of Photonics y: 2017/5/24 4 CHAPTER 1---- Optical Resonator y2 y1 d tg1 2 1 2 1 For the paraxial rays y2 y1 tg y2 1 d y1 0 1 1 2 y2,2 -2 y1,1 1 y2 y1 2 2 y1 1 R y1 -R Along z upward angle is positive, and downward is negative Fundamentals of Photonics 2017/5/24 5 CHAPTER 1---- Optical Resonator Free-Space Propagation Refraction at a Planar Boundary 1 M 0 1 d M 0 1 Refraction at a Spherical Boundary 1 M (n2 - n1 ) n2 R Transmission Through a Thin Lens 0 n1 n2 Reflection from a Planar Mirror 1 M 1 f 0 1 Reflection from a Spherical Mirror 1 0 M 0 1 Fundamentals of Photonics 0 n1 n2 2017/5/24 1 M 2 R 0 1 6 CHAPTER 1---- Optical Resonator A Set of Parallel Transparent Plates 1 M 0 di n i 1 Matrices of Cascaded Optical Components M M1M 2 ....M N Fundamentals of Photonics 2017/5/24 7 CHAPTER 1---- Optical Resonator 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, ym A C m Fundamentals of Photonics m B y0 D 0 2017/5/24 ym1 Aym B m m1 Cym D m 8 CHAPTER 1---- Optical Resonator ym1 Aym B m m1 Cym D m From these equation, we have ym 1 - Aym B y - Aym 1 m 1 m 2 B m And then: ym 2 2bym1 - F 2 ym where linear differential equations A D 2 b , F AD - BC det M 2 Fundamentals of Photonics 2017/5/24 9 CHAPTER 1---- Optical Resonator If we assumed: y m y0 h m So that, we have If we defined h 2 - 2bh F 2 0 h b i F 2 - b2 cos-1 b F We have b F cos , F 2 - b2 F sin then h F (cos i sin ) Fei ym y0 F m e im 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 ym y0 F m sin(m 0 ) ymax F m sin(m 0 ) Fundamentals of Photonics 2017/5/24 10 CHAPTER 1---- Optical Resonator If F=1, then ym ymax sin(m 0 ) Condition for a Harmonic Trajectory: if ym be harmonic, the cos-1b must be real, We have condition b 1 or A D 1 2 The bound b 1 therefore provides a condition of stability (boundedness) of the ray trajectory If, instead, |b| > 1, 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 2017/5/24 11 CHAPTER 1---- Optical Resonator 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: s = 2pq, where q is an integer Fundamentals of Photonics 2017/5/24 12 CHAPTER 1---- Optical Resonator 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 0 (1 1 M 1 f 2 0 1 1 d 1 1 0 1 f1 Fundamentals of Photonics d d )(1 ) 1 2 f1 2 f2 d 1 0 f1 1 d 1 0 1 d 1 1 - f1 f 2 f1 f 2 2017/5/24 d d d - (1 - )(1 - ) f2 f1 f2 d2 2d f1 13 CHAPTER 1---- Optical Resonator 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 xz plane. Fundamentals of Photonics 2017/5/24 14 CHAPTER 1---- Optical Resonator 1.2 Planar Mirror Resonators This simple one-dimensional resonator is known as a Fabry-Perot etalon. A. Resonator Modes Resonator Modes as Standing Waves A monochromatic wave of frequency v has a wavefunction as u (r , t ) Re{U (r ) exp( j 2p vt )} 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 2017/5/24 15 CHAPTER 1---- Optical Resonator the modes of a resonator must be the solution of Helmholtz equation with the boundary conditions: z 0 U (r ) 0 z d So that the general solution is standing wave: U (r ) A sin kz d With boundary condition, we have kd qp F c 2d q q 1 qp kq d c , q 1, 2,..., 2d c F 2d Resonance frequencies Fundamentals of Photonics q q 2017/5/24 16 CHAPTER 1---- Optical Resonator lq The resonance wavelength is: The length of the resonator, d = q lq /2, wavelength Attention: c c0 / n c q 2d q is an integer number of half 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. k 2d Fundamentals of Photonics 4p n l0 d 4p d q 2p c 2017/5/24 q= 1,2,3,… 17 CHAPTER 1---- Optical Resonator 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 M ( ) 4 c For 1D resonator The number of modes in a resonator of length d within the frequency interval v is: 4 d c 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 2017/5/24 18 CHAPTER 1---- Optical Resonator 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: Mirror 1 U1 hU 0 e U 0 e - i -i 4p nd l U 0 e-i 2 kdU 0 U2 So that, the sum of the sequential reflective light with field of U U 0 U1 U 2 U3 ... U 0 (1 h h2 h3 ...) IU 2 U0 2 1- e finally, we have - i 2 I I0 (1 - 2 cos ) 2 I0 U0 U1 U0 (1 - h) (1 - ) 2 4 sin 2 2 I max , 1 (2F / p ) 2 sin 2 ( / 2) p 1/ 2 F 1- Fundamentals of Photonics Mirror 2 U3 2017/5/24 I max I0 (1 - ) 2 Finesse of the resonator 19 CHAPTER 1---- Optical Resonator The resonance spectral peak has a full width of half maximum (FWHM): c 4p d F F I max I max 4p d I I We have Due to 1 (2F / p ) 2 sin 2 (p / F ) min 1 (2F / p ) 2 c where F c 2d q q F , q 1, 2,..., c F 2d Fundamentals of Photonics 2017/5/24 F F 20 CHAPTER 1---- Optical Resonator Spectral response of Fabry-Perot Resonator The intensity I is a periodic function of with period 2p. The dependence of I on , which is the spectral response of the resonator, has a similar periodic behavior since = 4pd/c is proportional to . This resonance profile: I I max 1 (2 F / p ) 2 sin 2 (p / F ) The maximum I = Imax, is achieved at the q q F , q 1, 2,..., resonance frequencies whereas the minimum value I min The FWHM of the resonance peak is Fundamentals of Photonics 2017/5/24 I max 1 (2 F / p ) 2 c 4p d F F 21 CHAPTER 1---- Optical Resonator Sources of Resonator Loss • • Absorption and scattering loss during the round trip: exp (-2asd) Imperfect reflectance of the mirror: R1, R2 R1 R2 exp(-2 s d ) 2 Defineding that 2 exp(-2 r d ) 1 1 r s ln 2d R1 R2 we get: ar is an effective overall distributed-loss coefficient, which is used generally in the system design and analysis Fundamentals of Photonics r s m1 2017/5/24 1 1 ln s m1 m 2 2d R1 R2 1 1 ln 2d R1 m1 1 1 ln 2d R1 22 CHAPTER 1---- Optical Resonator If the reflectance of the mirrors is very high, approach to 1, so that The above formula can approximate as r s R1 R2 R 1 m1 1 1 ln s m1 m 2 2d R1 R2 1 - R1 1 - R2 1- R m2 2d 2d 2d r s 1- R d The finesse F can be expressed as a function of the effective loss coefficient ar, F p exp(- r d / 2) 1 - exp(- r d ) Because ard<<1, so that exp(-ard)=1-ard, we have: F p r d The finesse is inversely proportional to the loss factor rd Fundamentals of Photonics 2017/5/24 23 CHAPTER 1---- Optical Resonator 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 c c / 2d r p / r d 2p 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 2017/5/24 p 1 c r 1 2p p 24 CHAPTER 1---- Optical Resonator 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: Q 2p ( storedenergy) Large Q factors are associated with low-loss resonators energylosspercycle For a resonator of loss at the rate cr (per unit time), which is equivalent to the rate cr /0 (per cycle), so that Q 2p 1 (c r / 0 ) c r 2p Q 0 The quality factor is related to the resonator lifetime (photon lifetime) 1 p 1 c r 2p Q 2p 0 p The quality factor is related to the finesse of the resonator by Q Fundamentals of Photonics 2017/5/24 0 F F 25 CHAPTER 1---- Optical Resonator • In summary, three parameters are convenient for characterizing the losses in an optical resonator: – the finesse F – the loss coefficient r (cm-1), – photon lifetime p = 1/cr, (seconds). • In addition, the quality factor Q can also be used for this purpose Fundamentals of Photonics 2017/5/24 26 CHAPTER 1---- Optical Resonator B. The Resonator as a Spectrum Analyzer Transmission of a plane wave across a planar-mirror resonator (Fabry-Perot etalon) t1 r1 r2 t2 T ( ) It I T ( ) Tmax 1 (2F / p ) 2 sin 2 (p / F ) Where: U2 Tmax U0 Fundamentals of Photonics 2 (1 - ) 2 , t t1t2 , 1 2 p 1/2 F 1- U1 Mirror 1 t Mirror 2 The change of the length of the cavity will change the resonance frequency q d qc q - 2 d 2d d 2017/5/24 27 CHAPTER 1---- Optical Resonator C. Two- and Three-Dimensional Resonators • Two-Dimensional Resonators q yp qp ky , k z z , q y 1, 2,..., qz 1, 2,..., d d k 2 k y2 k z2 ( 2p 2 ) c • Mode density Determine an approximate expression for the number of modes in a two-dimensional resonator with frequencies lying between 0 and , assuming that 2p/c >> p/d, i.e. d >>l/2, and allowing for two orthogonal polarizations per mode number. Fundamentals of Photonics 2017/5/24 M ( ) 4p c2 28 CHAPTER 1---- Optical Resonator Three-Dimensional Resonators Physical space resonator Wave vector space q yp qxp qp kx , ky , k z z , qx , q y , qz 1, 2,..., d d d k 2 k x2 k y2 k z2 ( 2p 2 ) c Mode density M ( ) 8p c3 2 Fundamentals of Photonics 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 2017/5/24 29 CHAPTER 1---- Optical Resonator 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=∞ z d 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 2017/5/24 30 CHAPTER 1---- Optical Resonator C. Stable condition of the resonator For paraxial rays, where all angles are small, the relation between (ym+1, m+1) and (ym, m) is linear and can be written in the matrix form y1 - 1 A B 1 z C D 2 R1 2 y2 ym1 A B ym C D m m1 R2 R1 0 reflection from a mirror of radius R1 y0 d Attention here: we just take general case spherical so doesn’t take the sign Fundamentals of Photonics 0 1 d 1 1 0 1 R22 2017/5/24 0 1 d 1 0 1 reflection from a mirror of radius R2 propagation a distance d through free space 31 CHAPTER 1---- Optical Resonator A 1 2d R2 B 2d (1 d C 2 R1 D 2d 2 R1 R2 R2 ( 2d det M Ad - BC 1 F 2 ) 4d R1 ym ymax sin(m 0 ) d d b ( A D) / 2 2 1 1 - 1 R1 R2 R1 R2 1)(2d R1 1) It the way is harmonic, we need cos-1b must be real, that is b 1 b ( A D ) / 2 2 1 d d 1 -1 1 R1 R2 for g1=1+d/R1; g2=1+d/R2 d d 0 1 1 1 R1 R2 Fundamentals of Photonics 2017/5/24 0 g1 g 2 1 32 CHAPTER 1---- Optical Resonator resonator is in conditionally stable, there will be: d d 0 1 1 1 R1 R2 0 g1 g 2 1 In summary, the confinement condition for paraxial rays in a sphericalmirror 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 2017/5/24 33 CHAPTER 1---- Optical Resonator Stable and unstable resonators a. Planar g2 (R1= R2=∞) e 1 Non stable d b. Symmetrical confocal (R1= R2=-d) a c. Symmetrical concentric stable b -1 0 1 g1 (R1= R2=-d/2) d. confocal/planar c Non stable (R1= -d,R2=∞) e. concave/convex (R1<0,R2>0) d/(-R) = 0, 1, and 2, corresponding to planar, confocal, and concentric resonators Fundamentals of Photonics 2017/5/24 34 CHAPTER 1---- Optical Resonator The stable properties of optical resonators Crystal state resonators g1 g2 0 or g1 g2 1 a. Planar (R1= R2=∞) b. Symmetrical confocal Stable (R1= R2=-d) c. Symmetrical concentric unstable Fundamentals of Photonics (R1= R2=-d/2) 2017/5/24 35 CHAPTER 1---- Optical Resonator Unstable resonators Unstable cavity corresponds to the high loss g1 g2 0 g1 g2 1 or a. Biconvex resonator d b. plan-convex resonator c. Some cases in plan-concave resonator When R2<d, unstable R1 d. Some cases in concave-convex resonator When R1<d and R1+R2=R1-|R2|>d d e. Some cases in biconcave resonator g1 g 2 (1 d / R1 )(1 d / R2 ) 0 g1 g 2 (1 d / R1 )(1 d / R2 ) 1 Fundamentals of Photonics R1 R2 d 2017/5/24 36 CHAPTER 1---- Optical Resonator 1.2 Gaussian waves and its characteristics A. Gaussian beam Helmholtz equation 2U k 2U 0 Normally, a plan wave (in z direction) will be U U 0 exp{-i(k r)} U 0 exp(-ikz ) When amplitude is not constant the wave is U A( x, y, z ) exp(-ikz ) An axis symmetric wave in the amplitude U A(r ) exp(-ikz ) frequency Fundamentals of Photonics 2p Wave vector 2017/5/24 z k 2p l 37 CHAPTER 1---- Optical Resonator Paraxial Helmholtz equation Substitute the U into the Helmholtz equation we have: A A - i 2k 0 z where 2 T 2 T 2 2 2 2 x y A1 2 One simple solution is spherical wave： A(r ) exp(- jk ) 2 x 2 y 2 z 2z The other solution is Gaussian wave： W0 2 2 U (r ) A0 exp[- 2 ]exp[-ikz - ik i ( z )] W ( z) W ( z) 2 R( z ) where z z 2 1/ 2 ) ] ( z ) tan -1 z0 is Rayleigh range z0 z0 1 1 l l z0 1/2 z0 2 i R( z ) z[1 ( ) ] W0 ( ) W ( z ) z 0 W (0) q( z ) R( z ) pW 2 ( z ) W ( z ) W0 [1 ( z p q parameter Fundamentals of Photonics 2017/5/24 38 CHAPTER 1---- Optical Resonator Electric field of Gaussian wave propagates in z dirextion A0 -( x 2 y 2 ) x2 y 2 E ( x, y, z ) exp[ ] exp[ -ik ( z) i ( z)] 2 W ( z) W ( z) 2 R( z ) Beam width at z W ( z ) W0 [1 ( z 2 1/ 2 ) ] z0 W0 W (0) Waist width pW02 2 z0 2 pW02 Radii of wave front at z R( z ) z[1 ( ) ] z[1 ( ) ], z0 lz z l Phase factor Fundamentals of Photonics ( z ) arctan 2017/5/24 lz -1 z tg p W02 z0 39 CHAPTER 1---- Optical Resonator Gaussian beam at z=0 A0 r2 E ( x, y,0) exp[ - 2 ] where r 2 x 2 y 2 W0 W0 Beam width: W ( z ) W0 [1 ( E A0 W0 z 2 1/ 2 ) ] will be minimum z0 A0 eW0 2 2 p W 0 wave front: lim R ( z ) lim z 1 z 0 z 0 l z -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 2017/5/24 40 CHAPTER 1---- Optical Resonator B. The characteristics of Gaussian beam Beam radius z 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 2017/5/24 41 CHAPTER 1---- Optical Resonator Intensity of Gaussian beam Intensity of Gaussian beam z=0 y W0 2 2 I ( , z) I0[ ] exp[- 2 ] W ( z) W ( z) z=z0 z=2z0 y y x I I0 I I0 1 0 x -1 0 1 W0 I I0 1 0 x -1 0 1 W0 1 0 -1 0 1 W0 The normalized beam intensity as a function of the radial distance at different axial distances Fundamentals of Photonics 2017/5/24 42 CHAPTER 1---- Optical Resonator On the beam axis ( = 0) the intensity I (0, z ) I 0 [ W0 2 I0 ] z W ( z) 1 ( )2 z0 I I0 1 1 0.5 - zo 0 0 zo z The normalized beam intensity I/I0 at points on the beam axis (p=0) as a function of z Fundamentals of Photonics 2017/5/24 43 CHAPTER 1---- Optical Resonator Power of the Gaussian beam The power of Gaussian beam is calculated by the integration of the optical intensity over a transverse plane P 1 I 0p W02 2 So that we can express the intensity of the beam by the power 2P 2 2 I ( , z) exp[- 2 ] 2 pW ( z ) W ( z) The ratio of the power carried within a circle of radius r. in the transverse plane at position z to the total power is 202 1 0 I ( , z )2p d 1 - exp[- 2 ] P 0 W ( z) Fundamentals of Photonics 2017/5/24 44 CHAPTER 1---- Optical Resonator Beam Radius W ( z ) W0 [1 ( W z 2 1/ 2 W ( z ) 0 z 0 z ) ] z0 z0 W(z) Beam waist 2W 0 W0 -z0 0 z z0 The beam radius W(z) has its minimum value W0 at the waist (z=0) reaches 2W0 at z=±z0 and increases linearly with z for large z. Beam Divergence 1 dW ( z ) 2lz p 2W02 2 2 2 2 2 [( ) z ] dz pW0 l Fundamentals of Photonics 2017/5/24 0 l p W0 45 CHAPTER 1---- Optical Resonator The characteristics of divergence angle z 0, 2 0 p W02 z , 2 2l / p W0 l z , 2 2l 2W ( z ) or 2 lim x p W0 z Let’s define f=z0 pW02 as the confocal parameter of Gaussian beam f z0 l The physical means of f ：the half distance between two section of width fl z2 2 (1 2 ) p f 2W ( z ) l 2 lim lim 2 z z z z fp Fundamentals of Photonics 2017/5/24 46 CHAPTER 1---- Optical Resonator 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 20.5o o 0 z 2 z0 2p W02 l 2zo The depth of focus of a Gaussian beam. Fundamentals of Photonics 2017/5/24 47 CHAPTER 1---- Optical Resonator Phase of Gaussian beam The phase of the Gaussian beam is, ( , z ) kz - ( z ) R( z ) z[1 ( k 2 R( z ) 2 z0 2 ) ] z On the beam axis (p = 0) the phase (0, z ) kz - ( z ) kz Phase of plan wave ( z) 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 2017/5/24 48 CHAPTER 1---- Optical Resonator Wavefront pW02 2 f2 R( z ) z[1 ( ) ] z lz z Confocal field and its equal phase front Fundamentals of Photonics 2017/5/24 49 CHAPTER 1---- Optical Resonator 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 2017/5/24 50 CHAPTER 1---- Optical Resonator 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 1 1 l -i q( z ) R( z ) p W 2 ( z ) 1 1 q( z ) z iz0 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 2017/5/24 51 CHAPTER 1---- Optical Resonator C. TRANSMISSION THROUGH OPTICAL COMPONENTS a). Transmission Through a Thin Lens kz k 2 2R - - k 2 2f kz k 2 2R ' - 1 1 1 R' R f 1 1 1 - R R' f 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 2017/5/24 52 CHAPTER 1---- Optical Resonator In the thin lens transform, we have W0 , z1 , f W W ' 1 1 1 R' R f If we know W0, Z1, f, we can get p W 2 2 -1 W W [1 ( ) ] lR ' l R ' 2 -1 - z ' R '[1 ( ) ] 2 pW '2 0 2 The minus sign is due to the waist lies to the right of the lens. Fundamentals of Photonics 2017/5/24 53 CHAPTER 1---- Optical Resonator W0 ' W [1 (p W 2 / l R ')2 ]1/ 2 because R z[1 ( z0 / z ) 2 ] -z ' and R' 1 (p R '/ lW 2 ) 2 W W0 [1 ( z / z0 ) 2 ]1/2 W0 ' MW0 Waist radius Waist location ( z '- f ) M ( z - f ) Depth of focus 2 z0' M 2 (2 z0 ) Divergence angle ， magnification Fundamentals of Photonics 2 2 0 M Mr M (1 r 2 )1/ 2 2 0' 2017/5/24 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. where r z0 f Mr z- f z- f 54 CHAPTER 1---- Optical Resonator 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 z0 r 0 z- f z and M Mr z’ W0 ' MW0 1 1 1 z' z f 2W0’ 2W0 M Mr f z- f 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 2017/5/24 55 CHAPTER 1---- Optical Resonator b). Beam Shaping Beam Focusing If a lens is placed at the waist of a Gaussian beam, so z=0, then W0 ' z' W0 [1 ( z0 / f ) 2 ]1/ 2 f 1 ( f / z0 ) 2 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 W0 ' l p W0 f 0 f z' f 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 2017/5/24 56 CHAPTER 1---- Optical Resonator Focus of Gaussian beam W' 2 0 W02 W02 2 z (1 - ) ( ) f lf 2 For given f, W '0 changes as •when z1 0 •when •when •when z1 f W '02 decreases as z decreases W0 ' reaches minimum, and M<1, for f>0, it is focal effect z1 f , W0 ' increases as z increases z1 f z f the bigger z, smaller f, better focus W '0 reaches maximum, when • p W02 f l , it will be focus Fundamentals of Photonics 2017/5/24 57 CHAPTER 1---- Optical Resonator 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 2W0 ' 4 p l F# F# f D where F# is the F-number of the lens. A microscope objective with small F-number is often used. Fundamentals of Photonics 2017/5/24 58 CHAPTER 1---- Optical Resonator Beam collimate locations of the waists of the incident and transmitted beams, z and z’ are z' z / f -1 -1 f ( z / f - 1) 2 ( z0 / f ) 2 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 2017/5/24 59 CHAPTER 1---- Optical Resonator 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 2017/5/24 60 CHAPTER 1---- Optical Resonator C). Reflection from a Spherical Mirror Reflection of a Gaussian beam of curvature R1 from a mirror of curvature R: W2 W1 1 1 2 f = -R/2. R2 R1 R R > 0 for convex mirrors and R < 0 for concave mirrors, Fundamentals of Photonics 2017/5/24 61 CHAPTER 1---- Optical Resonator R R1 R1 - R 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 2017/5/24 62 CHAPTER 1---- Optical Resonator 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 2017/5/24 63 CHAPTER 1---- Optical Resonator 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 q2 Aq1 B Cq1 D 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 2017/5/24 64 CHAPTER 1---- Optical Resonator 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 W0 2x 2y x2 y 2 U l ,m ( x, y, z ) Al ,m [ ]Gl [ ]Gm [ ] exp[- jkz - jk j (l m 1) ( z )] W ( z) W ( z) W ( z) 2 R( z ) -u 2 ), where Gl (u ) H l (u ) exp( 2 l 0,1, 2,..., 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 2017/5/24 65 CHAPTER 1---- Optical Resonator 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 2017/5/24 66 CHAPTER 1---- Optical Resonator Intensity Distribution The optical intensity of the (I, m) Hermite-Gaussian beam is I l ,m ( x, y, z ) Al ,m Fundamentals of Photonics 2 W0 2 2 2 x 2 2 y [ ] Gl [ ]Gm [ ] W ( z) W ( z) W ( z) 2017/5/24 67 CHAPTER 1---- Optical Resonator 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 Beam radius z 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 2017/5/24 68 CHAPTER 1---- Optical Resonator 2 Gaussian beam intensity: I I W0 e 0 W ( z ) The Rayleigh range z0 2 2 2( x 2 y 2 ) - i k z x y -tg -1 z 2 R z0 W 2 (z) e p W02 z0 l where z0 is the distance called Rayleigh range, at which the beam wavefronts are most curved or we usually called confocal prrameter Beam width W ( z ) W0 [1 ( z 2 1/ 2 ) ] z0 minimum value W0 at the beam waist (z = 0). The radius of curvature z02 R( z ) z z z 2 z z0 z0 2 R R( z ) z 1 z0 z z z0 z0 z Beam waist Fundamentals of Photonics W0 l z0 p 2017/5/24 69 CHAPTER 1---- Optical Resonator Gaussian Mode of a Symmetrical SphericalMirror Resonator z2 z1 d d R1 z02 R1 z1 z1 2 R2 z0 - R2 z2 z2 z1 z1 0 Fundamentals of Photonics -d ( R2 d ) , z2 z1 d R2 R1 2d z2 z z 2 -d ( R1 d )( R2 d )( R2 R1 2d ) 0 2 ( R2 R1 2d ) 2017/5/24 70 CHAPTER 1---- Optical Resonator the beam radii at the mirrors Wi W0 [1 ( zi 2 1/ 2 ) ] , i 1, 2. z0 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 0 (1 Fundamentals of Photonics d d )(1 ) 1 R1 R2 2017/5/24 71 CHAPTER 1---- Optical Resonator Gaussian Mode of a Symmetrical SphericalMirror 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 R d z0 (2 - 1)1/ 2 2 d ld R W (2 - 1)1/ 2 2p d 2 0 W12 W22 ld / p {(d / R )[2 - (d / R )]}1/ 2 The confinement condition becomes 0 Fundamentals of Photonics d 2 R 2017/5/24 72 CHAPTER 1---- Optical Resonator 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 d z0 2 Fundamentals of Photonics W0 ( l d 1/ 2 ) W1 W2 2W0 2p 2017/5/24 73 CHAPTER 1---- Optical Resonator C. Resonance Frequencies The phase of a Gaussian beam, ( x, y, z ) kz - tg ( z -1 k ( x2 y 2 ) ) z0 2 R( z ) At the locations of the mirrors z1 and z2 on the optical axia (x2+y2=0), we have, (0, z2 ) - (0, z1 ) k ( z2 - z1 ) - [ ( z2 ) - ( z1 )] kd - where z z0 ( z ) tg -1 As the traveling wave completes a round trip between the two mirrors, therefore, its phase changes by 2kz - 2 For the resonance, the phase must be in condition 2kz - 2 2qp , q 1, 2,3... If we consider the plane wave resonance frequency k 2p We have q q F F p Fundamentals of Photonics 2017/5/24 c and F c 2d 74 CHAPTER 1---- Optical Resonator Spherical-Mirror Resonator Resonance Frequencies (Gaussian Modes) q q F F p 1. The frequency spacing of adjacent modes is VF = c/2d, which is the same result as that obtained for the planar-mirror resonator. 2. For spherical-mirror resonators, this frequency spacing is independent of the curvatures of the mirrors. 3. The second term in the fomula, which does depend on the mirror curvatures, simply represents a displacement of all resonance frequencies. Fundamentals of Photonics 2017/5/24 75 CHAPTER 1---- Optical Resonator 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 W0 2x 2y x2 y 2 U l ,m ( x, y, z ) Al ,m [ ]Gl [ ]Gm [ ] exp[- jkz - jk j (l m 1) ( z )] W ( z) W ( z) W ( z) 2 R( z ) (0, z ) kz - (l m 1) ( z ) 2kd - 2(l m 1) 2p q, q 0, 1, 2,..., Spherical mirror resonator Resonance Frequencies (Hermite -Gaussian Modes) l ,m,q q F (l m 1) Fundamentals of Photonics 2017/5/24 p F 76 CHAPTER 1---- Optical Resonator 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 l ,m,q q F (l m 1) p F 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 l ,m,q - l ',m ',q [(l m) - (l ' m ')] Fundamentals of Photonics 2017/5/24 p F 77 CHAPTER 1---- Optical Resonator *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 2017/5/24 78 CHAPTER 1---- Optical Resonator Diffraction loss The Fresnel number NF a2 a2 a2 NF d l 2 z l pW 2 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 2017/5/24 79 CHAPTER 1---- Optical Resonator 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 a12 a22 NF 2 2 p W1 p W2 For general stable concave mirror resonator, the Fresnel number for two mirrors are: 1 a12 a12 g1 N F1 [ (1 - g1 g 2 )]2 2 p W1 d l g2 NF 2 1 a22 a22 g 2 [ (1 - g1 g 2 )]2 2 p W2 d l g1 Fundamentals of Photonics 2017/5/24 80 CHAPTER 1---- Optical Resonator 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? W0 ' MW0 M M r W0 ' f z- f W0 [1 ( z0 / f ) 2 ]1/ 2 Fundamentals of Photonics 2017/5/24 81 CHAPTER 1---- Optical Resonator 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 2017/5/24 82 CHAPTER 1---- Optical Resonator b). Parameter q transform for thin lens If put the original point at the waist of beam, the parameter q will be 1 1 l -i q( z ) R( z ) p W 2 ( z ) 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)→∞ q0 q(0) ip W02 l Finally we have： pW02 q( z ) i z q0 z l Fundamentals of Photonics 2017/5/24 83 CHAPTER 1---- Optical Resonator Parameter q in the thin lens transform W1 W2 1 1 1 R' R f 1 1 1 q' q f q01 q(0) ip W02 l q1 q01 z 1 1 1 q2 q1 f q1 q z1 q2 q '- z2 1 1 1 - q1 q2 f 1 1 1 q '- z2 q z1 f Fundamentals of Photonics 2017/5/24 84 CHAPTER 1---- Optical Resonator continue (1 q' Real part z2 zz )q ( z1 z2 - 1 2 ) f f z q -( ) (1 - 1 ) f f f - z1 W012 2 f - z2 W02 z z (1 - 1 )q02 (1 - 2 )q01 f f p W012 q q01 i l p W022 ' q q02 i l Imaginary part q01q02 z1 z2 z1 z2 f f W012 W 2 02 z1 2 pW01 2 ) ( ) f lf z (1 - 1 ) f z2 f [1 ] z1 2 pW01 2 (1 - ) ( ) f lf Fundamentals of Photonics (1 - 2017/5/24 W0 ' p 2W012W022 z1 z z ( 1 ) 1 2 l2 f f W [1 (p W 2 / l R ')2 ]1/ 2 85 CHAPTER 1---- Optical Resonator 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 • p W012 z1 when l f 1 - f 2 Fundamentals of Photonics 1 1 1 , then z1 z2 f 2017/5/24 86 CHAPTER 1---- Optical Resonator ， Beam waist radius W0 ' MW0 Waist location ( z '- f ) M 2 ( z - f ) Depth of focus 2 z0' M 2 (2 z0 ) Divergence Magnification Fundamentals of Photonics 2 0' M 2 0 M z0 Mr r (1 r 2 )1/ 2 z- f 2017/5/24 Mr f z- f 87