Understanding Electromagnetic Resonators in Waveguides
Explore the concepts of electromagnetic resonators and waveguides through the teachings of Uday Khankhoje at IIT Delhi, delving into Maxwell's equations, transverse components, and solving for Ez and Bz. Discover how to cap the ends with metal plates and calculate resonator losses for practical applications.
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Presentation Transcript
Electromagnetic Resonators ELL212: Uday Khankhoje EE, IIT Delhi 1
Waveguides (recap) [1] 1. Maxwell: 2. Expand fields into components: ? ?,?,?,? = ?0?,? ?? ?? ?? ?0= ?? ? + ?? ? + ?? ? 3
Waveguides (recap) [1] 3. Transverse components in terms of Ez, Bz: 4 (Note: replace i by j above)
Waveguides (recap) [1] 4. Solve for Ez, Bz from here: 5
Waveguides (recap) [1] TE Solutions of the form ??(?,?,?,?) = ?0cos ??? ? ??? ? ??(?? ??) cos i.e. a fwd travelling wave Cut off freqs: 2 2 ? ? ? ? ???= ?? + 6
Cap the ends with metal plates [2] ??? ? ??? ? +? ???+ ?0 Fwd and bkwd waves ????cos ???? ??= ?0 cos Boundary conditions at ? = 0,? = ?: ? ? =??? ?= ???? ?? ??? ?? = 0 7
Cap the ends with metal plates [2] += ?0 p?? ? cos ,?? = ?? and : ??? ? cos This implies that ?0 ??= 2??0 ??? ? +sin ???? The frequency of a waveguide mode earlier was ?? = ?2 ??? 2 Now, k is restricted, so 2 2 2 ? ? ? ? ? ? ?0= ?? + + Waveguide frequency 8
Losses in a resonator Perfect metal No losses Finite ? Boundary conditions get slightly modified System supports small range of frequencies around ?0 9
Waveguide resonator [2] ?????? ?????? ????? ???? ? = ?0 10
Waveguide resonator [2] ? ?0 ? Calculating resonator loss allows Q calc. 11
Resonators in the real world Fiber-Bragg mirrors [3] 12
Resonators in the real world Fiber-based resonator [4] 13
Laser Physics in brief [5] Fast Slow The presence of a photon encourages another photon to be emitted: Stimulated emission Only 2 level system, ?2 ?1= exp( ? ??) 15
Laser Physics in brief [5] So, the resonant frequency of the resonator must match the energy transition of the gain/laser medium! 16
Gain medium in the fiber? [6] Introduce a rare Earth ion into the fiber: Called doping Erbium Doped Fiber Energy band diagram of ??3+ 19
Applications [7] As a gain medium in a fiber-laser As an amplifier in fiber optics: backbone of ALL telecom! 20
VCSEL Vertical Cavity Surface Emitting Laser [8] 21
GRACE mission (NASA) [9] Revealed ground water depletion over North India to be 1 ft (2002-08) 25
References [1] Griffiths, Introduction to Electrodynamics, 4th Ed. [2] Ulaby et al., Fundamentals of applied electromagnetics, 6th Ed. [3] http://www.hoestarinsp.com.sg/FBG1-large.jpg [4] http://spie.org/x8609.xml [5] http://www.orc.soton.ac.uk/61.html [6] http://www.rp- photonics.com/erbium_doped_gain_media.html [7] http://www.rp- photonics.com/erbium_doped_fiber_amplifiers.html [8] http://japaneseclass.jp/trends/about/VCSEL [9] http://www.csr.utexas.edu/grace/ 26
