Chapter III Optical Resonators Lecture 4 Chapter III Optical Resonators Highlights 1. Quality Factor and Mode Density of a Resonator 2. Fabry-Perot Etalon Spherical Mirror Resonator Free Spectral Range Finesse Resolution Gaussian Beam Mode Stability Criteria Stability Diagram 3. The Self-Consistent Method 4. Resonance Frequency of Optical Resonators 5. Loss in Optical Resonators

§3.0 Quality Factor and Mode Density I. Quality Factor of Resonator 1. Optical Resonator is used to build up large field intensity; Consist in two, more, curved mirrors; Wave oscillation as a standing wave.

§3.0 Quality Factor and Mode Density Quality Factor of a Resonator Field Energy Stored in a Resonator

§3.0 Quality Factor and Mode Density Power Dissipate in a Resonator At steady state, the input power is equal to the dissipate power, i.e., P So On the other hand:

§3.0 Quality Factor and Mode Density I. Mode Density in Optical Resonator What is the optical mode? 在自由空间，具有任意波矢 k 的单色平面波都可以存在。但在一个有边界条件限制的空间 V（例如谐振腔）中，只能存在一系列独立的具有特定波矢 k 的单色平面驻波。这种能够存在于腔内的驻波（以某一波失 k 为标志）称为光波模式。不同模式以不同的 k 区分，同一波失对应着两个具有不同偏振方向的模式。

§3.0 Quality Factor and Mode Density To design a resonator, smaller high Q modes is better, ideally only one, however, this requires resonator dimension to be the order of the wavelength. That seems too hard… Example: one-dimension resonator size For example: We need find other ways

§3.0 Quality Factor and Mode Density To build large resonators, only small fraction of modes possess higher Q , e.g., plane-parallel cavity, etc; Cavity contains an amplifying medium, oscillation occurs at high Q modes; Due to the atomic excitation happens only within a limited frequency region, other modes outside this region decay even if have high Q. So, the strategy of modal discrimination by controlling Q is sensible! Question: How many modes will oscillation in the resonator within a given frequency interval with a given optical resonator? ?

§3.0 Quality Factor and Mode Density x y z a b c

§3.0 Quality Factor and Mode Density Each mode elemental volume in K space Then, total number of modes with k from 0 to k is: k > 0 Recall the relation of k and n :

§3.0 Quality Factor and Mode Density Mode density, that is the number of modes per unit n near n, is thus Two polarized modes are included So, the number of modes that fall within the n to n+dn is Example: calculate the mode number

§3.1 Fabry-Perot Etalon n’ n l Ai A1 A2 A3 B1 B2 B3 B4 … q ’ r : Reflectivity in n’ r’: Reflectivity in n t : Transmission from n’ to n t’: Transmission from n to n’ Reflected wave: …

§3.1 Fabry-Perot Etalon n’ n l q ’ q A B C O 1 2 Transmitted wave: …

§3.1 Fabry-Perot Etalon

§3.1 Fabry-Perot Etalon Let’s consider the transmission characteristics now: It / Ii d 2mp 2(m+1)p R=0.046 R=0.27 R=0.64 R=0.87 1 The line approach unity at some points, while others is close to zero; The bigger the R , the sharper the line is. Transmission maximum at: Free Spectral Range

§3.1 Fabry-Perot Etalon The meaning of FSR Exercise: If the fraction intensity loss per pass is (1-A), i.e., the left fraction intensity is A per pass. Then, the maximum transmission drops to:

§3.2 Fabry-Perot Etalons As Optical Spectrum Analyzers When That is when etalon length change by half a wavelength, the peak transmission frequency can be tuned by Dn . This property is utilized etalon as a scanning interferometer.

§3.2 Fabry-Perot Etalons As Optical Spectrum Analyzers Obviously, the etalon resolution, that is ability to distinguish details in the spectrum, is limited by the finite width of its transmission peaks. If we define the limiting resolution as the frequency width of half maximum of the transmission peak, i.e., or If Which means the width of half maximum peak is smaller than peak separation

§3.2 Fabry-Perot Etalons As Optical Spectrum Analyzers Finesse Remember the above equation, able to calculate F by knowing and Example: Design of a Fabry-Perot Etalon