OPTI510R: Photonics. Khanh Kieu College of Optical Sciences, University of Arizona Meinel building R.626

Transcription

1 OPTI510R: Photonics Khanh Kieu College of Optical Sciences, University of Arizona Meinel building R.626

2 Announcements No class Monday, Feb 26 Mid-term exam will be on Feb 28 th (open books/notes)

3 Homework solutions

4 Homework solutions

5 Homework solutions

6 Homework solutions

7 Homework solutions

8 Homework solutions

9 Review Properties of light Wave optics Interference and devices Diffraction and devices Polarization optics Guided-wave optics

10 Properties of light Corpuscular theory of light: light consists of corpuscles or very small particles flying at finite velocity (Isaac Newton). The wave theory of light: At first, the nature of light is wave propagating in a medium called luminiferous aether. But all attempts to detect the aether have failed so far. So the theory based on the aether was more or less abandoned. The electromagnetic theory of light was developed culminating in the Maxwell s equations. The quantum theory of light: In an attempt to explain blackbody radiation, Planck postulated that electromagnetic energy could be emitted only in quantized form, in other words, the energy could only be a multiple of an elementary unit,, where h is Planck's constant.

11 Properties of light Velocity of light Frequency and wavelength Polarization Coherence Other light characteristics

12 Wave optics Maxwell s equations Wave equation Plane wave solution Classical theory of permittivity Sellmeier Equation and Kramers-Kroenig Relations

13 Maxwell s equations Here, E and H are the electric and magnetic field, D the dielectric flux, B the magnetic flux, J f the current density of free charges, ρ f is the free charge density.

14 Maxwell s equations Material equations: P is the polarization, M is the magnetization

15 Uniform optical medium Wave equation

16 Plane-wave solution Simple harmonic plane wave Dispersion relation

17 Plane-wave solution Transverse electromagnetic wave

18 Boundary conditions at the interface Maxwell s equations in integral form The normal component of D and B are continuous across the dielectric interface

19 Boundary conditions at the interface Maxwell s equations in integral form The tangential component of E and H are continuous across the dielectric interface

20 Pointing vector, energy and Intensity

21 Classical theory of permittivity Glass A dipole (Induced polarization)

22 Classical theory of permittivity The electron motion equation: Where: Trial solution: Induced dipole moment:

23 Classical theory of permittivity,for 1 dipole or:,where: (Plasma frequency)

24 Classical theory of permittivity Real part (dashed line) and imaginary part (solid line) of the susceptibility of the classical oscillator model for the dielectric polarizability

25 Sellmeier Equation and Kramers- Kroenig Relations The refractive index and absorption of a medium are not independent

26 Sellmeier Equation and Kramers- Kroenig Relations If the media are used in a frequency range far away from resonances. Then the imaginary part of the susceptibility related to absorption can be approximated by: The Kramers-Kroenig relation results in the Sellmeier Equation for the refractive index: 2

27 Pulse propagation Optical pulses often have relatively small spectral width compared to the center frequency of the pulse ω 0. In such cases it is useful to separate the complex electric field into a carrier frequency ω 0 and an envelope A(t) and represent the absolute frequency as Ω = ω 0 + ω. We can then rewrite: The optical spectrum of a pulse can be calculated through the Fourier transform of the envelop.

28 Pulse propagation Time domain and frequency domain: Spectrum Long pulse time frequency Short pulse time frequency

29 Pulse propagation Pulse shapes, corresponding spectra and time bandwidth products

30 Pulse propagation We can separate an optical pulse into a carrier wave at frequency ω 0 and a complex envelope: By introducing the offset frequency ω, the offset wave-number k(ω) and spectrum of the envelope Ã(ω): We have:

31 Dispersion In general dispersion is a function of frequency:,in the frequency domain,in the time domain

32 Group velocity If we keep only the first term: This is the equation describing a wave-package moving at the speed of v g0

33 Group velocity The pulse travels with the group velocity without changing its shape!

34 Pulse spreading due to second order dispersion This equation does not have an analytical solution, generally. Decomposition of a pulse into wave packets with different center frequencies. In a medium with dispersion the wave-packets move at different relative group velocities.

35 Pulse spreading due to second order dispersion k is the group velocity dispersion (GVD) Fortunately, for a Gaussian pulse, the pulse propagation equation can be solved analytically. The initial pulse is then of the form: (We have this from the Fourier transform)

36 Pulse spreading due to second order dispersion Since: Fourier transform of the Gaussian function is a Gaussian function: Therefore, in the spectral domain the solution at an arbitrary propagation distance z is:

37 Pulse spreading due to second order dispersion Now we go back to the time domain (again using the Fourier transform): After splitting the real and imaginary part we get: Here is the starting point:

38 Pulse spreading due to second order dispersion Initial pulse duration: FWHM pulse duration after propagating distance L:

39 Pulse spreading due to second order dispersion Pulse spreading is proportional to propagation distance Pulse spreading is proportional to the GVD Inversely proportional to the (initial pulse duration) 2

40 Pulse spreading due to second order dispersion Important parameters

41 Dispersion example: Fused silica Sellmeier equation:

42 Dispersion example-fused silica Normal GVD Anomalous GVD

43 Chirped pulses GVD > 0 GVD < 0 In general, we need to use numerical simulation to simulate the propagation of short optical pulses.

44 Other important topics Fresnel reflection Brewster angle Total internal reflection (phase shift, evanescent field) Goos-Haenchen phase shift Frustrated TIR Metamaterials

45 Interference and Devices Interference Interferometers (Michelson, Mach-Zehnder, Fabry- Perot, Sagnac, etc.) Autocorrelator Mach-Zehnder Modulators Sagnac interferometer (Rotation sensor)

46 Interference Two plane waves (solutions of the wave equation) interference: (still a solution of the wave equation) We detect the intensity instead of the amplitude:

47 Interference

48 Interference Since the oscillation is very fast, the time average becomes: Interference pattern generated by two monochromatic plane waves

49 Fabry-Perot Resonator

50 Fabry-Perot Resonator

51 Fabry-Perot Resonator

52 Fabry-Perot Resonator (α = 0, normal incident)

53 Sagnac Interferometer The sagnac sensor has the best sensitivity compared to other type of sensors.

54 Fiber Optics Gyroscope

55 Laser Gyroscope We can easily measure f beat with <1Hz precision. What would be the smallest rotation rate that we can measure using a ring resonator with 1m radius?

56 Diffraction and Devices Diffraction Diffraction gratings Ruled grating Holographic grating Volume grating Applications Tunable laser Spectroscopy Laser stabilization Pulse compression Volume grating

57 Diffraction Diffraction relies on the interference of waves emanating from the same source taking different paths to the same point on a screen Diffraction can be explained by interference Diffraction of a laser beam through a small circular hole (Airy disk) Young's double-slit interferometer (Homework)

58 Diffraction and nature of light Need to be in the near field: Arago spot, Fresnel bright spot, or Poisson spot This experiment confirmed the wave nature of light!

59 Huygens Fresnel principle

60 Diffraction limit How to overcome the diffraction limit?

61 Diffraction Grating A periodic structure that diffracts light into different directions. Grating can be flat, concave, convex and arbitrary shape HeNe laser incident on a diffraction grating showing zero, first and second order beams

62 Basic equations Monochromatic source White light

63 Polarization optics Polarization optics Anisotropic media Index ellipsoid Uniaxial crystal Double refraction Calcite CaCO3 Polarization devices Polarizer Waveplates Isolators Polarization microscope Potassium Niobate

64 Polarization optics The polarization of light is determined by the trajectory of the end of the electric vector in time at a given position.

65 Polarization optics Plane wave propagating in the z direction: Where: (A: complex envelope) describes an ellipse (z = const)

66 Polarization optics Plane wave propagating in the z direction: Where: describes an ellipse (z = const)

67 Polarization optics (Homework)

68 Poincaré sphere

69 Other representations Stokes parameters: Jones vector:

70 Anisotropic media The electric permittivity is a matrix (tensor) D and E may point to different directions By choice of the coordinate system we can simplify the math: (Principle refractive index)

71 Anisotropic media Index ellipsoid (representation of the tensor):

72 Uniaxial crystals Calcite, Rutile TiO2, Yttrium Vanadate

73 Uniaxial crystals The direction of energy flow is not the same as the wave front propagation direction Double refraction

74 Polarization devices Polarizer: Glan-Thompson polarizer Dichroic polarizer (Stretched Polyvinyl Alcohol (PVA) Double refraction polarizer Glan-Thompson polarizer (Polarization extinction ratio: PER)

75 Isolators

76 Polarization microscope

77 Phase matching Second harmonic generation Input IR beam Phase matching concept LBO crystal Angle and Wavelength Dependence of Refractive index of BBO

78 Guided-wave optics Introduction Overview of guided-wave devices Optical fibers Planar waveguides Integrated optical devices Compact lasers Planar mirror waveguides Waveguide modes Dispersion relation

79 Why guided wave optics? Propagate light over long distances without the need for lenses Escape the slavery of diffraction Take advantage of semiconductor manufacturing processes to determine dimensions and designs Enable unique devices impossible to make in other ways (arrayed waveguide grating) Drive the size of photonics down dramatically (compact and silicon photonics)

80 Important trends Integrated optical circuits: combination of multiple optical functions on a single substrate Functions include splitting, filtering, switching, modulating, isolating, coupling (general passive functions), generation (lasers) and detection Monolithic integration a single material is used Hybrid integration multiple materials are used that perform different functions Optoelectronic integrated circuits (OEIC) Includes both integrated optical circuits and conventional electronic circuits on the same substrate Generally limited to semiconductor substrate materials High refractive indices create the potential for ultracompact circuitry Increasing need for optical interconnections between and within computers

81 Planar mirror waveguides / n 0 0 k nk0 c c / n TEM plane wave TE: E polarized in x-direction TM: H polarized in x-direction π phase shift for each reflection (boundary conditions) Amplitude and polarization do not change (perfect mirror). Not practical due to the fact that there is no perfect metal mirror

82 Planar mirror waveguides Self-consistency: The wave reflects twice and reproduces itself Therefore the phase shift in travelling from A to B must be equal to or differ by an integer multiple of 2p from the phase shift from A to C Modes are fields that maintain the same transverse distribution and polarization at all locations along the waveguide axis.

83 Planar mirror waveguides A guided wave consists of the superposition of two plane waves in the y-z plane at angle ± with respect to the z axis. The components of the mode wave vector are k ym = nk 0 sinq m = mp / d b 2 2 m = k zm = k 2 - m 2 p 2 d 2

84 Mode field profile TE modes E x ( y, z) a u ( y)exp( j z) m m or m u m ( y ) 2 d 2 d mpy cos( ), d mpy sin( ), d m 1,3,5... m 2,4,6... Modes are orthogonal and normalized

85 Mode properties TE modes E x ( y, z) a u ( y)exp( j z) m m or m u m ( y ) 2 d 2 d mpy cos( ), d mpy sin( ), d m 1,3,5... m 2,4,6... Modes are orthogonal and normalized Orthogonal condition Normalized condition Any field distribution can be discomposed into a sum of modes

86 Number of modes, Cutoff Number of modes sin m / 2d 1, M 2d / m Reduce to nearest integer Dispersion relation 2 m / c m p / d Cutoff wavelength and frequency c 2d, n c c / 2d For > c or n < n c there is no guided mode

87 Dispersion relation Dispersion relation. 2 m / c m p / d This leads to waveguide dispersion

88 Group velocity Dispersion relation 2 m / c m p / d Cutoff frequency =

89 Group velocity Is this normal or anomalous dispersion?

90 TE versus TM

91 Planar dielectric waveguide y x z Core film sandwiched between two layers of lower refractive index Bottom layer is often a substrate with n = n s Top layer is called the cover layer (n c n s ) Air can also acts as a cover (n c = 1) n c = n s in symmetric waveguides

92 Planar dielectric waveguide Symmetric waveguide Reflection due to TIR (similar to planar mirror waveguide) sinq c = n 2 / n 1 q p 2 -q = p c 2 - æ n ö 2 sin-1 ç è ø = æ n 2 cos-1 ç è n 1 n 1 ö ø Self Consistency 2p 2 d sin 2 r 2 p m 2k d 2 2pm y r

93 Phase shift for TIR TE wave tan j r 2 = sin2 q c sin 2 q -1 TM wave tan j r 2 = -n n 2 sin 2 q c sin 2 q -1 We can now arrive at an equation for the mode angles

94 Transcendental equation for modes 2 pd mp sin c tan sin sin dielectric waveguide mirror waveguide p, or tan( / 2) r r

95 Number of modes,or,where Single mode 2d l 0 NA <1

96 Transcendental equation for modes m nk cos 1 0 m b m = N m eff w /c 0

97 Cut-off frequency Mirror waveguide Dielectric waveguide There is no gap for dielectric waveguide always one guided mode for a symmetric slab (not so for asymmetric)

98 Oscillating field component In the core The electric field in a symmetric dielectric waveguide is harmonic within the slab and exponentially decaying outside the slab. E ( y, z) a u ( y)exp( j z) x m m m ì æ cos 2p l sinq ö ç my, m = 0,2,4,... ï è ø u m (y) µ í æ sinç 2p, - d ï l sinq ö 2 y d 2 my, m =1,3,5,... î ï è ø

99 Evanescent field component u m ( y) exp( m y), y > d / 2 exp( m y), y d / 2 The z dependence must be identical in order to satisfy continuity at ± d/2. Signs are chosen to obtain a decaying field E x ( ( y, z) a 2 n 2 k 2 0 M m 0 ) E x m u m ( y)exp( ( y, z) 0 j z) m Extinction coefficient m m 2 0 m nk nk cos cos m c 1

100 TE field distribution E x M ( y, z) a u ( y)exp( j z) m 0 m m m

101 Dispersion relation Self Consistency 2p 2 d sin 2 r 2 p m 2k d 2 2pm y r TE wave tan j r 2 = sin2 q c sin 2 q -1

102 Dispersion relation Dispersion relation in parametric form: (n: effective refractive index)

103 Dispersion relation n 1, n 2 are constant

104 Dispersion relation Normal or anomalous dispersion?

105 What is the smallest waveguide? Mirror waveguide Dielectric waveguide There is no gap for dielectric waveguide always one guided mode for a symmetric slab (not so for asymmetric) Can we then make an infinitely small dielectric waveguide?

106 Using Maxwell s equations An optical mode is solution of Maxwell's equations satisfying all boundary conditions Its spatial distribution does not change with propagation Modes are obtained by solving the curl equations These six equations are solved in each layer of the waveguide Boundary condition: Tangential component of E and H be continuous across both interfaces Waveguide modes are obtained by imposing the boundary conditions

107 Using Maxwell s equations Assume waveguide is infinitely wide along the x axis E and H are then x-independent For any mode, all components vary with z as exp(i z). Thus,

108 Using Maxwell s equations These equations have two distinct sets of linearly polarized solutions For Transverse-Electric (TE) modes, E z = 0 and E y = 0 TE modes are obtained by solving: Magnetic field components are related to E x as:

109 Using Maxwell s equations G. Agrawal

110 Using Maxwell s equations G. Agrawal

111 Using Maxwell s equations G. Agrawal

112 TE mode for symmetric waveguide G. Agrawal

113 TE mode for symmetric waveguide G. Agrawal

114 Modes of asymmetric waveguide G. Agrawal

115 Modes of asymmetric waveguide G. Agrawal

116 Universal dispersion curve G. Agrawal

117 Rectangular mirror waveguide 2k x d = 2pm x m x =1,2,... 2k y d = 2pm y m y =1,2,... k ym = nk 0 sinq m = mp / d

118 Number of modes Number of modes 2k x d = 2pm x m x =1,2,... 2k y d = 2pm y m y =1,2,... M = Quadrant area Unit cell area = pn 2 k p 2 d 2 æ = 2d ö ç è l ø 2 p 4

119 Rectangular dielectric waveguide k x 2 + k y 2 n 1 2 k 0 2 sin 2 q c æ n q c = cos -1 2 ö ç è ø n 1

120 Rectangular dielectric waveguide Number of TE modes: k 2 x + k 2 y n 2 1 k 2 0 sin 2 q c æ n q c = cos -1 2 ö ç è ø n 1

121 Slab directional coupler phase mismatch per unit length

122 Slab directional coupler ) ( sin (0) ) ( ) ( sin 2 ) ( cos (0) ) ( z C P z P z z P z P C C L p Coupling length 3dB coupler

123 Phase-mismatched vs. phase-matched L 0 P1 P1 P2 Phase mismatched 0 P2 Phase matched = 0

124 Switching with directional coupler B Power transfer ratio Power Transfer Ratio T P2 ( L0 ) P (0) 1 2 p 4 sin( px) sin c( x) px sin c L 1 p Phase Mismatch Phase mismatch can be tuned electrically in directional couplers. In tuning the phase mismatch from 0 to 3p, light is switch from WG 2 to 1. Tuning can be done electro-optically or thermally, for example.

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