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Chapter 1 Optical Resonator

Chapter 1 Optical Resonator.

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Chapter 1 Optical Resonator

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  1. Chapter 1Optical Resonator Fundamentals of Photonics

  2. 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) What is an optical resonator? Fundamentals of Photonics

  3. 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 Contents: Fundamentals of Photonics

  4. 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

  5. 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

  6. 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

  7. A Set of Parallel Transparent Plates Matrices of Cascaded Optical Components Fundamentals of Photonics

  8. 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

  9. From these equation, we have And then: linear differential equations where Fundamentals of Photonics

  10. If we assumed: So that, wehave 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

  11. 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

  12. 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

  13. 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

  14. 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. Home works (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

  15. Resonator Modes Resonator Modes as Standing Waves 1.2 Planar Mirror Resonators This simple one-dimensional resonator is known as a Fabry-Perot etalon. 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 15 Fundamentals of Photonics 2014/9/2

  16. d the modes of a resonator must be the solution of Helmholtz equation with the boundary conditions: So that the general solution is standing wave: With boundary condition, we have Resonance frequencies 16 Fundamentals of Photonics 2014/9/2

  17. 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 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,… 17 Fundamentals of Photonics 2014/9/2

  18. 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. 18 Fundamentals of Photonics 2014/9/2

  19. Losses and Resonance Spectral Width Mirror 2 Mirror 1 U3 U2 U1 U0 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: So that, the sum of the sequential reflective light with field of finally, we have Finesse of the resonator 19 Fundamentals of Photonics 2014/9/2

  20. The resonance spectral peak has a full width of half maximum (FWHM): We have Due to where 20 Fundamentals of Photonics 2014/9/2

  21. 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 21 Fundamentals of Photonics 2014/9/2

  22. 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 22 Fundamentals of Photonics 2014/9/2

  23. 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 23 Fundamentals of Photonics 2014/9/2

  24. Photon Lifetime of Resonator The relationship between the resonance linewidth and the resonator loss may beviewed 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 24 Fundamentals of Photonics 2014/9/2

  25. 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 25 Fundamentals of Photonics 2014/9/2

  26. 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 26 Fundamentals of Photonics 2014/9/2

  27. B. The Resonator as a Spectrum Analyzer r2 r1 t1 t2 U2 U1 U0 Mirror 1 Mirror 2 Transmission of a plane wave across a planar-mirror resonator (Fabry-Perot etalon) Where: The change of the length of the cavity will change the resonance frequency 27 Fundamentals of Photonics 2014/9/2

  28. 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. 28 Fundamentals of Photonics 2014/9/2

  29. Three-Dimensional Resonators Wave vector space Physical space resonator 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 29 Fundamentals of Photonics 2014/9/2

  30. 1.2 Optical resonators and stable condition A. Ray Confinement of spherical resonators z d The rule of the sign: concave mirror (R < 0), convex (R > 0). The planar-mirror resonator is R1 = R2=∞ 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 30 Fundamentals of Photonics 2014/9/2

  31. C. Stable condition of the resonator R2 R1 y1 -q 1 z q 2 y2 q 0 y0 d 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 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 31 Fundamentals of Photonics 2014/9/2

  32. It the way is harmonic, we need f =cos-1b must be real, that is for g1=1+d/R1; g2=1+d/R2 32 Fundamentals of Photonics 2014/9/2

  33. 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 33 Fundamentals of Photonics 2014/9/2

  34. Stable and unstable resonators e d a 1 b -1 0 1 c Symmetrical resonators • Planar • (R1= R2=∞) Non stable b. Symmetrical confocal (R1= R2=-d) c. Symmetrical concentric (R1= R2=-d/2) stable 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 34 Fundamentals of Photonics 2014/9/2

  35. 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 35 Fundamentals of Photonics 2014/9/2

  36. d d Unstable resonators Unstable cavity corresponds to the high loss a. Biconvex resonator 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 e. Some cases in biconcave resonator 36 Fundamentals of Photonics 2014/9/2

  37. 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

  38. 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 z0isRayleigh range q parameter Fundamentals of Photonics

  39. 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

  40. 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

  41. Beam radius z B. The characteristics of Gaussian beam 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

  42. Intensity of Gaussian beam 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

  43. 1 0.5 0 On the beam axis (r = 0) the intensity The normalized beam intensity I/I0 at points on the beam axis (p=0) as a function of z Fundamentals of Photonics

  44. 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

  45. The beam radius W(z) has its minimum value W0at the waist (z=0) reaches at z=±z0 and increases linearly with z for large z. Beam Radius W(z) Beam waist 2W0 W0 q0 z -z0 z0 Beam Divergence Fundamentals of Photonics

  46. 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

  47. 20.5wo wo z 0 2zo 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 20.5of 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 The depth of focus of a Gaussian beam. Fundamentals of Photonics

  48. 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

  49. Wavefront Confocal field and its equal phase front Fundamentals of Photonics

  50. 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 W0or 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

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