Optical Resonator Modes EM Review ECE 455 Optical Electronics Gaussian Beams

ECE 455 Lecture 2

ABCD Matrix

Cavity Stability Optical Resonator Modes EM Review ECE 455 Optical Electronics Gaussian Beams

Beams and Cavities Gary Eden Fabry-Perot Cavities Tom Galvin

General Cavities ECE Illinois

Summary

Appendix A Introduction

ECE 455 Lecture 2

ABCD Matrix In this section, we will learn how to do the following things: Cavity Stability Follow the path of paraxial light rays as they propagate EM Review through an optical system Gaussian Beams Determine the stability of optical resonators Beams and Use gaussian beams to describe the ﬁeld inside an optical Cavities resonator Fabry-Perot Cavities Describe the optical containment properties of cavities General Cavities with the quality factor, the ﬁnesse, and the free spectral Summary range Appendix A ABCD Matrix I

ECE 455 Lecture 2 The ABCD matrix is a ray

ABCD Matrix optics formalism that relates

Cavity the distance r from the optical Stability axis and its slope r 0 of a ray as EM Review is propagates through optical Gaussian Beams elements. sin θ≈θ

Beams and Cavities Assumptions θ Fabry-Perot 1 All optical elements are Cavities ”thin” General Cavities 2 the angle of propagation is Summary suﬃciently small that Appendix A sin(θ) ≈ θ ABCD Matrix II

ECE 455 Lecture 2 Position above optic axis and slope are represented by a vector

ABCD Matrix r (1) Cavity r 0 Stability EM Review Cumulative eﬀect of optical elements are matrices Gaussian Beams AB Beams and (2) Cavities CD Fabry-Perot Cavities To ﬁnd the position and slope of a ray after propagating General Cavities through an optical system use Summary Appendix A r2 AB r1 0 = 0 (3) r2 CD r1 ABCD Matrix III

ECE 455 Lecture 2

ABCD Matrix

Cavity Stability

EM Review Beware! Gaussian Different books (notably Hecht and Siegman) use different Beams conventions for ABCD matrices than defined in the previous Beams and 0 Cavities page. Some switch the position of r and r . Another possible Fabry-Perot convention is to multiply the slope by the refractive index. Cavities

General Cavities

Summary

Appendix A Uniform Dielectric distance d Diagram

ECE 455 Lecture 2

ABCD Matrix

Cavity Stability

EM Review

Gaussian Beams

Beams and Cavities

Fabry-Perot Cavities

General Cavities

Summary Axis d Appendix A Uniform Dielectric distance d

ECE 455 Lecture 2 It should be clear that

ABCD Matrix The distance from the optic axis changes as:

Cavity Stability 0 r2 = r1 + r1 · d (4) EM Review Gaussian The slope can be expected to remain constant Beams

Beams and independent of the position above or below the optic axis Cavities 0 0 Fabry-Perot r = r (5) Cavities 2 1

General Cavities Therefore the ABCD matrix is

Summary AB 1 d Appendix A = (6) CD 0 1 Thin Lens Diagram

ECE 455 Lecture 2

ABCD Matrix

Cavity Stability

EM Review

Gaussian Beams

Beams and Cavities

Fabry-Perot Cavities

General Cavities

Summary

Appendix A Thin Lens

ECE 455 Lecture 2 The ABCD matrix of the thin lens can be derived as follows: Because the lens is thin, the distance from the axis has no ABCD Matrix chance to change while light propagates through the lens Cavity Stability r2 ≈ r1 (7) EM Review Gaussian 0 Beams A ray traveling parallel to the optic axis (r1 = 0) is Beams and focused to the origin at f . Therefore C = −1/f Cavities 0 0 T Fabry-Perot A ray originating at the focus of the lens r = (f · r2, r2) Cavities will be turned parallel to the axis by the lens. General Cavities Therefore: D = 1 Summary The ABCD matrix is: Appendix A AB 1 0 = 1 (8) CD − f 1 Spherical Mirror Diagram

ECE 455 Lecture 2

ABCD Matrix 1 Cavity Stability 2 EM Review r Gaussian Beams

Beams and Cavities

Fabry-Perot Cavities

General Cavities

Summary

Appendix A f R = 2f Spherical Mirror

ECE 455 Lecture 2 Changes in the direction of propagation are ignored in ABCD matrix theory ABCD Matrix Only the slope and position of the ray with respect to the Cavity Stability axis is of interest EM Review The spherical mirror is identical to the case of a thin lens. Gaussian Beams Recall that the focus of a spherical mirror with radius R is Beams and Cavities R Fabry-Perot f = (9) Cavities 2

General Cavities The result of the thin lens can then be used with the

Summary appropriate value substituted in place of f

Appendix A AB 1 0 = 2 (10) CD − R 1 Flat Mirror

ECE 455 Lecture 2

ABCD Matrix Consider the ABCD matrix of a spherical mirror.

Cavity Stability 1 0 EM Review 2 (11) − R 1 Gaussian Beams As R → ∞, the mirror becomes ﬂat. The matrix becomes Beams and Cavities 1 0 Fabry-Perot (12) Cavities 0 1 General Cavities The above matrix is simply the identity. The lesson: ﬂat Summary

Appendix A mirrors do not aﬀect the path of rays. A Several Element ABCD Matrix

ECE 455 Lecture 2 To model a system consisting of several elements, the ABCD

ABCD Matrix matrix for each element can be applied in series Cavity Stability rn+1 AB rn EM Review 0 = 0 (13) rn+1 CD rn Gaussian Beams Alternatively the power of matrix multiplication can be used to Beams and Cavities ﬁnd the net ABCD matrix Fabry-Perot Cavities AB An Bn An−1 Bn−1 A1 B1 General = ... Cavities CD Cn Dn Cn−1 Dn−1 C1 D1

Summary (14)

Appendix A ←− progession through optical system Example: Three Elements

ECE 455 Problem: Light source X is placed a distance d away from a Lecture 2 1 lens with focal length f , as shown below. Find the point Y ,

ABCD Matrix where the light source is imaged.

Cavity Stability

EM Review

Gaussian Beams Beams and ? Cavities Fabry-Perot Cavities

General Cavities

Summary

Appendix A

A Three Element Example Continued

ECE 455 Lecture 2 Solution: Point A is imaged to point B when all rays originating from point A, regardless of r 0 travel to point B. ABCD Matrix 0 Cavity Begin by ﬁnding the ABCD matrix. Rays travel distance d Stability 1

EM Review along the optical axis to the lens, are refracted by the lens, and

Gaussian ﬁnally travel an unknown distance d2. Beams Beams and AB 1 d2 1 0 1 d1 Cavities = CD 0 1 − 1 1 0 1 Fabry-Perot f Cavities d2 1 − f d2 1 d1 General = 1 Cavities − f 1 0 1 Summary d2 d1d2 1 − d1 + d2 − Appendix A = f f (15) 1 d1 − f 1 − f A Three Element Example Continued

ECE 455 If element B of an ABCD matrix is zero, the initial slope Lecture 2 doesn’t matter. Setting B = 0 and solving for d2 yields: ABCD Matrix 1 Cavity d2 = 1 1 (16) Stability − f d1 EM Review Gaussian This is of course the thin lens imaging condition. Beams Beams and Next ﬁnd the distance from the axis of the image point: Cavities Fabry-Perot 0 Cavities r2 = Ar1 + Br1 General d2 Cavities = 1 − r f 1 Summary

Appendix A d2 = − r1 (17) d1

where 1 = 1 + 1 , the imaging condition, has been used. f d1 d2 Example: A Several Element ABCD Matrix

ECE 455 Problem: Find the output ray of the system shown below when Lecture 2 the input ray is characterized by r = 0.1 cm and r 0 = 0.1 cm−1

ABCD Matrix R Cavity Stability f EM Review

Gaussian Beams 1 2 3

Beams and 4 Cavities

Fabry-Perot Cavities 5

General Cavities 7 6 Summary

Appendix A

Where f = 2m, R = 4 m, d1 =5 cm and d2 = 5 cm Example: A Several Element ABCD Matrix Continued

ECE 455 Solution: The ﬁrst step in solving this problem is to ‘unwrap’ Lecture 2 the optical system. That leads to the following optical system

ABCD Matrix below

Cavity Stability

EM Review R f f Gaussian 2 Beams

Beams and Cavities

Fabry-Perot Cavities

General Cavities

Summary

Appendix A

d1 d2 d2 d1 Example: A Several Element ABCD Matrix Continued

ECE 455 The output ray can then be found with ABCD matrices. The Lecture 2 focal lengths have been converted to centimeters. ABCD Matrix r 1 d 1 0 1 d 1 0 2 = 1 2 Cavity r 0 0 1 − 1 1 0 1 − 2 1 Stability 2 f R EM Review 1 d2 1 0 1 d1 r1 × 1 0 (18) Gaussian 0 1 − 1 0 1 r Beams f 1 Beams and 1 7 1 0 1 5 1 0 = Cavities 0 1 −0.005 1 0 1 −0.005 1 Fabry-Perot Cavities 1 5 1 0 1 7 0.1 General × (19) Cavities 0 1 −0.005 1 0 1 −0.1 Summary 0.965 7 0.975 5 = Appendix A −0.005 1 −0.005 1 0.975 5 −0.6 × (20) −0.005 1 −0.1 Example: A Several Element ABCD Matrix Continued

ECE 455 Lecture 2

ABCD Matrix r2 0.9059 11.825 −1.085 Cavity 0 = (21) Stability r2 −0.0099 0.975 −.097 EM Review −2.1299 Gaussian = (22) Beams −0.0839

Beams and Cavities The matrix multiplication is shown in full detail above to Fabry-Perot illustrate how to more rapidly multiply matrices. Once values Cavities

General have been placed in the matrices with consistent units, every Cavities pair of matrices is multiplied until there is only one left. Summary

Appendix A You need not only multiply the last two matrices every step! Stable Cavity

ECE 455 Lecture 2 1 2 ABCD Matrix 8 1 3 Cavity 2 Stability 4 EM Review 4 7 Gaussian 5 3 6 Beams

Beams and Cavities

Fabry-Perot Cavities General A stable optical cavity consists of two or more optical Cavities

Summary elements (usually mirrors) in which a ray will eventually

Appendix A replicate itself Have low diﬀraction loss Stable cavities have smaller mode volume Useful for low-gain, low-volume lasers Unstable Cavity

ECE 455 Lecture 2

ABCD Matrix 5

Cavity Stability 4 1 2 EM Review 3 Gaussian Beams

Beams and Cavities

Fabry-Perot Cavities

General Cavities

Summary In an unstable cavity, rays do not replicate themselves Appendix A Each trip through the cavity will take the ray further from the optic axis, resulting in high diﬀraction losses Useful for high-gain, high-volume lasers Mode-Gain Overlap

ECE 455 Lecture 2 Pumped Choice of cavity geometry Area ABCD Matrix aﬀects the mode size Mode Cavity Stability Power is not extracted Area

EM Review from areas outside the Gaussian mode Beams

Beams and Power may go to waste Cavities Larger (possibly undesired) Fabry-Perot Cavities parasitic modes may begin Pumped Area General to lase Cavities

Summary The lesson: the cavity

Appendix A should be designed so the

mode overlaps the pumped Cavity region of the gain medium Mode ABCD Matrices and Stability

ECE 455 Lecture 2 The stability of a cavity may be cast as an eigenvalue problem.

AB r r ABCD Matrix = λ (23) CD r 0 r 0 Cavity round trip Stability EM Review Here λ is not the wavelength, but the eigenvalue of the ABCD Gaussian Beams matrix. The eigenvalue equation is

Beams and Cavities (A − λ)(D − λ) − BC = 0 (24) Fabry-Perot Cavities This equation may be simpliﬁed by noticing for a round trip in General Cavities any cavity Summary AD − BC = 1 (25) Appendix A Why? Recall that if X and Y are matrices, then Det(XY ) = Det(X )Det(Y ). Consider the determinants of all of the ABCD matrices described in these notes. ABCD Matrices and Stability

ECE 455 Placing Equation 25 into Equation 24 yields Lecture 2 λ2 − (A + D)λ + 1 = 0 (26) ABCD Matrix

Cavity Stability The solution is p 2 EM Review λ = m ± m − 1 (27) Gaussian A+D Beams where for convenience m ≡ 2 . Consider the case where Beams and |m| ≤ 1. Then m may be written as m = cos(θ), which means Cavities Fabry-Perot ±ıθ Cavities λ± = cos(θ) ± ı sin(θ) = e (28)

General Cavities The position and slope of a ray after making n round trips Summary through the cavity is then Appendix A ınθ −ınθ rn = c+r+e + c−r−e (29)

where r+ and r− are the eigenvectors of the ABCD matrix ABCD Matrices and Stability

ECE 455 Equation 25 can be rewritten Lecture 2

rn = r0cos(nθ) + s0sin(nθ) (30) ABCD Matrix Note how the maximum displacement is bounded by r0 + s0. In Cavity Stability addition, because it is the sum of sines and cosines, the EM Review position will periodically repeat itself. Gaussian Beams

Beams and Cavities Our assumption that |m| ≤ 1 implies that the cavity is stable. Fabry-Perot Cavities This can also be expressed as General A + D Cavities −1 ≤ ≤ 1 (31) Summary 2

Appendix A Equation 31 can be cast into a diﬀerent form by adding 1 to both sides and dividing the result by 2, which gives A + D + 2 0 ≤ ≤ 1 (32) 4 ABCD Matrices and Instability

ECE 455 Lecture 2 It has been shown that if |m| ≤ 1, the cavity is stable. It will now be shown that the opposite assumption, |m| > 1, implies ABCD Matrix the cavity is unstable. Cavity Stability

EM Review

Gaussian Beams For |m| ≥ 1, the eigenvalues are

Beams and Cavities p 2 λ± = m ± m − 1 (33) Fabry-Perot Cavities Both of these eigenvalues are real and positive. Therefore, after General Cavities n passes through the cavity, the vector Summary nλ+ nλ− Appendix A rn = c+r+e + c−r−e (34)

Both terms will grow without bound. Stability of a Two-Mirror Cavity

ECE 455 Lecture 2 The ABCD matrix for a round trip of a cavity comprising two mirrors with radii R1 and R2 separated by a distance d is ABCD Matrix Cavity Stability AB 1 0 1 d 1 0 1 d = 2 2 EM Review CD − 1 0 1 − 1 0 1 R1 R2 Gaussian " 2d 2d2 # Beams 1 − 2d − R2 R2 = 2 (35) Beams and 4d − 2 − 2 1 + 4d − 4d − 2d Cavities R1R2 R1 R2 R1R2 R1 R2 Fabry-Perot Cavities If d is the mirror separation and the mirror’s radii of curvature General are R and R , then the cavity will be stable if and only if Cavities 1 2

Summary 2d 4d2 4d 2d Appendix A 1 − R + 1 + R R − R − R + 2 0 ≤ 2 1 2 1 2 ≤ 1 (36) 4 Stability of a Two-Mirror Cavity

ECE 455 Lecture 2 When simpliﬁed, this expression becomes ABCD Matrix Cavity d d Stability 0 ≤ 1 − 1 − ≤ 1 (37) R1 R2 EM Review

Gaussian Beams Often, the two terms in the product are deﬁned as

Beams and Cavities d d g1 ≡ 1 − and g2 ≡ 1 − (38) Fabry-Perot R R Cavities 1 2

General Cavities These two quantities will appear later in formulas for the

Summary eigenfrequencies of gaussian modes. The two mirror cavity

Appendix A stability criterion may be plotted on a diagram, shown on the next slide. Stability Diagram

ECE 455 Lecture 2

ABCD Matrix

Cavity Stability EM Review Unstable +1 Gaussian Beams

Beams and Cavities +1

Fabry-Perot -1 Cavities

General Cavities -1 Unstable Summary

Appendix A

Example: The Z-Cavity

ECE 455 Lecture 2 Problem: Determine the minimum radius of curvature of the two mirrors to ensure the following cavity is stable:

ABCD Matrix

Cavity Stability

EM Review

Gaussian Beams

Beams and Cavities

Fabry-Perot Cavities

General Cavities

Summary

Appendix A Solution: The ﬁrst step is to unwrap the cavity. Due to the symmetry of this cavity, it is only necessary to go from M1 to M4 before an equivalent position is reached. Example: The Z-Cavity

ECE 455 Lecture 2

R = ∞ R = ∞ ABCD Matrix

Cavity Stability

EM Review

Gaussian Beams

Beams and Cavities

Fabry-Perot Cavities

General Cavities The ABCD matrix for this half traversal of the cavity is Summary AB 1 d1 1 0 1 d2 Appendix A = CD 0 1 2 0 1 1/2 − R 1 1 0 1 d1 × 2 (39) − R 1 0 1 Example: The Z-Cavity

ECE 455 Lecture 2

AB 1 − 4d1+2d2 + 4d1d2 ABCD Matrix = R R2 (40) CD 4d1+2d2 4d1d2 Cavity 1/2 1 − R + R2 Stability EM Review Here B and C have been omitted because they do not appear Gaussian Beams in the stability equation.

Beams and Using Equation 31, write Cavities 2 Fabry-Perot R − (4d1 + 2d2)R + 4d1d2 Cavities 2 ≤ 1 (41) General R Cavities

Summary The ﬁnal result is is that the cavity is only stable if

Appendix A 2d d R ≥ 1 2 (42) 2d1 + d2 Maxwell’s Equations

ECE 455 Maxwell’s four equations are: Lecture 2 ∂B~ ABCD Matrix ∇ × E~ = − (43) Cavity ∂t Stability ∂D~ EM Review ∇ × H~ = + J~ (44) ∂t Gaussian Beams ∇ · B~ = 0 (45) Beams and Cavities ∇ · D~ = ρ (46)

Fabry-Perot Cavities In free space or a uniform dielectric medium, J~ = 0 and ρ = 0. General Cavities The parameters and µ, which are material parameters, relate

Summary the ﬁelds as show below:

Appendix A D~ = E~ = r 0E~ (47)

B~ = µH~ = µr µ0H~ (48) The Wave Equation

ECE 455 If we take the curl of Equation 43 then we may substitute Lecture 2 Equation 44 into Equation 43 to get:

ABCD Matrix ∂2D~ Cavity ∇ × ∇ × E~ = −µ (49) Stability ∂t2 EM Review ~ ~ 2 ~ Gaussian The vector identity ∇ × ∇ × E = ∇ ∇ · E − ∇ E and Beams ~ ~ Beams and D = E can then be substituted Cavities ∂2E Fabry-Perot ∇ ∇ · E~ − ∇2E~ = −µ (50) Cavities ∂t2 General Cavities In a uniform dielectric medium, the ﬁrst term on the left is √ Summary zero. Then if we deﬁne c ≡ √ 1 and n ≡ µ , we get: µ00 r r Appendix A n2 ∂2E~ ∇2E~ = (51) c2 ∂t2 Plane Waves

ECE 455 Lecture 2 Any ﬁeld of the form: ABCD Matrix Cavity E~ (~r, t) = E~ cos(ωt − ~k · ~r + φ) (52) Stability q q q

EM Review ~ ~ ωn Gaussian where kq is the direction of propagation, |kq| = c , and Beams E~q ⊥ ~kq is a solution to the wave equation. Beams and Cavities ~ ~ Fabry-Perot Because Maxwell’s equations are linear, if Eq and Eq0 are Cavities solutions to the wave equation, then General Cavities ~t ~ ~ Summary E (~r, t) = Eq(~r, t) + Eq0 (~r, t) (53)

Appendix A is also a solution. Phasors

ECE 455 Lecture 2 Combining ﬁeld solutions in the time domain necessitates the use of trigonometric identities

ABCD Matrix In monochromatic fields, the amplitude and phase of the Cavity field can be encoded in complex numbers Stability A field containing arbitrary frequency components can be EM Review created by summing monochromatic fields Gaussian Beams Consider a linearly-polarized plane wave propagating in the Beams and Cavities zˆ direction Fabry-Perot E(z, t) = E cos(kz + φ − ωt) (54) Cavities 0 General = Re{E0exp(ı(kz − φ))exp(−ıωt)} (55) Cavities

Summary The phasor of a plane wave is:

Appendix A ı(kz+φ) E(z, t) = E0e (56) Because all ﬁelds have e−ıωt time dependence, time derivatives may be replaced by −ıω Maxwell’s Equations in Phasor Form

ECE 455 Lecture 2 Maxwell’s Equations in phasor form are then:

ABCD Matrix ∇ × E = ıωB (57) Cavity Stability ∇ × H = −ıωD + J (58) EM Review ~ Gaussian ∇ · B = 0 (59) Beams ∇ · D~ = ρ (60) Beams and Cavities

Fabry-Perot The wave equation in phasor form is: Cavities General ω2n2 Cavities 2 ∇ E + 2 E = 0 (61) Summary c Appendix A To go from phasors to the time domain, multiply the phasor by e−ıωt and take the real part. Optical Beams

ECE 455 Consider an optical beam in a uniform dielectric medium Lecture 2 propagating in thez ˆ direction. Poisson’s equation is

ABCD Matrix ∇ · E~ = 0 (62) Cavity Stability This can be broken up into a transverse and a longitudinal EM Review component Gaussian ∂Ez Beams ∇ E~ + = 0 (63) t t ∂z Beams and Cavities If D is the approximate beam diameter, and |Et | is the peak Fabry-Perot ﬁeld, the transverse derivative can be estimated as: Cavities |E | General ∇ E~ ∼ t (64) Cavities t t D Summary Because the beam is primarily propogating the inz ˆ direction, Appendix A the longitudinal derivative is approximately ∂E 2πn z ∼ ı E (65) ∂z λ z Optical Beams

ECE 455 Lecture 2 The ratio of these two ﬁelds is ABCD Matrix

Cavity |Ez | λ Stability ≈ (66) |Et | 2πnD EM Review Gaussian For any beam with ﬁnite D, the ﬁeld must have a Beams

Beams and longitudinal component. Cavities For a typical HeNe laser, λ = 632.8 nm and D = 1 mm. Fabry-Perot Cavities λ General ≈ (10−4) (67) Cavities 2πD Summary

Appendix A If D λ, the longitudinal component of the ﬁeld is negligible. Paraxial Wave Equation I

ECE 455 Lecture 2 A plane wave has the same amplitude throughout all space ABCD Matrix A laser beam has ﬁnite spatial extent Cavity Stability If the beam diameter is much greater than the wavelength, EM Review it propagates mostly as a plane wave and is polarized Gaussian Beams perpendicular to the direction of propagation

Beams and Cavities The ﬁeld of a wide beam propagating in thez ˆ direction

Fabry-Perot may be written as: Cavities −ıkz General E(x, y, z) = E ψ(x, y, z)e (68) Cavities 0

Summary e−ıkz is the plane wave part of the ﬁeld Appendix A ψ represents the small deviation from a plane wave Paraxial Wave Equation II

ECE 455 Lecture 2 If 68 is substituted into the phasor wave equation ABCD Matrix (Equation 61) and simpliﬁed, we obtain: Cavity Stability 2 2 ∂ψ ∂ ψ EM Review ∇ ψ − 2ık + = 0 (69) t ∂z ∂z2 Gaussian Beams The last term is small compared to the other two and can Beams and Cavities therefore be neglected, leaving Fabry-Perot Cavities 2 ∂ψ General ∇t ψ − 2ık = 0 (70) Cavities ∂z Summary At this point it is convienient to convert to cylindrical Appendix A coordinates in anticipation of a rotationally symmetric solution. Derivation of Gaussian Beam I

ECE 455 In cylindrical coordinates, the wave equation looks like Lecture 2 1 ∂ ∂ψ ∂ψ r − ı2k = 0 (71) ABCD Matrix r ∂r ∂r ∂z Cavity Stability An ansatz (guess) for the proper form of the solution which will EM Review ﬁt the boundary conditions of a laser cavity is now introduced Gaussian Beams kr 2 Beams and ψ0(r, z) = exp −ı P(z) + (72) Cavities 2q(z)

Fabry-Perot Cavities If this ansatz is placed in the cylindrical form the paraxial wave

General equation, we obtain: Cavities 2 Summary k ∂q 2 ∂P ı 2 − 1 r − 2k + ψ0 = 0 (73) Appendix A q (z) ∂z ∂z q(z) ∂q In order for the above equation to be satisﬁed, ∂z − 1 = 0 and ∂P ı ∂z + q(z) = 0 Derivation of Gaussian Beam II

ECE 455 Lecture 2 The solution to the q(z) diﬀerential equation is: q(z) = z + C (74) ABCD Matrix

Cavity The beam has ﬁnite transverse extent, therefore ψ0 must Stability decay away from the optical axis. EM Review For this to happen, q(z) must be complex. Because z is Gaussian Beams real, the constant from integration must be complex

Beams and Cavities q(z) = z + ız0 (75)

Fabry-Perot Cavities At z = 0 2 General kr Cavities ψ0(r, z = 0) = exp (−ıP(z = 0)) exp − (76) 2z0 Summary 2! Appendix A r = exp (−ıP(z = 0)) exp − w0

2 λz0 where w0 ≡ nπ has been deﬁned for convenience Derivation of Gaussian Beam III

ECE 455 Lecture 2 Expanding yields: 1 1 = ABCD Matrix q(z) z + ız0 Cavity z ız0 Stability = − 2 2 2 2 EM Review z + z0 z + z0 Gaussian 1 λ Beams = − ı (77) R(z) πnw 2(z) Beams and Cavities where Fabry-Perot z0 2 Cavities R(z) ≡ z 1 + (78) General z Cavities and Summary " #1/2 z 2 Appendix A w(z) ≡ w0 1 + (79) z0 have been deﬁned because they have a physical interpretation. Derivation of Gaussian Beam IV

ECE 455 Now solve the diﬀerntial equation for dP : Lecture 2 dz Z z dz0 ABCD Matrix ıP(z) = 0 Cavity 0 z + ız0 Stability = ln [1 + ı(z/z0)] EM Review h 21/2 i Gaussian = ln 1 + (z/z0) exp (−ıtan(z/z0)) Beams 1/2 Beams and 2 Cavities = ln 1 + (z/z0) − ıtan(z/z0) (80)

Fabry-Perot Cavities The actual quantity we are interested in is: General Cavities 1 −1 Summary e−ıP(z) = eıtan (z/z0) 2 1/2 Appendix A (1 + (z/z0) ) w0 −1 = eıtan (z/z0) (81) w(z) Gaussian Beams I

ECE 455 Lecture 2 Combining all these results1 and placing them in our ansatz ABCD Matrix yields: Cavity √ √ Stability " # " # 2 2x 2y w0 r EM Review E(x, y, z) = E0Hm Hp · exp − 2 Gaussian w(z) w(z) w(z) w (z) Beams z Beams and × exp −ı kz − (1 + m + p) tan−1 Cavities z0 Fabry-Perot 2 Cavities kr × exp −ı (82) General 2R(z) Cavities Summary This is a very complicated expression. We’ll interpret it piece Appendix A by piece.

1 The derivation of Hm and Hn was left out for simplicity Gaussian Beams II

ECE 455 Lecture 2

ABCD Matrix First, a few deﬁnitions from the above expression Cavity Stability z0 2 EM Review R(z) = z 1 + (83) Gaussian z Beams " 2#1/2 Beams and z Cavities w(z) = w0 1 + (84) z0 Fabry-Perot Cavities 2 πnw0 General z0 = (85) Cavities λ0 Summary

Appendix A The Spot Size w(z) - Interpretation I

ECE 455 Lecture 2 Normalized intensity

Consider the following 1 ABCD Matrix term in Equation 82 Cavity h 2 i Stability w0 r E0 w(z) exp − w 2(z) EM Review

Gaussian w(z) is known as the spot r Beams size 0 Beams and Cavities Describes how rapidly the Fabry-Perot ﬁeld decays away from the Cavities optical axis General Cavities Characteristic radius of the Summary beam 퐸(푟) Appendix A w0 The w(z) factor is

necessary for energy 퐼(푟) conservation. r The Spot Size w(z) - Interpretation II

ECE 455 Lecture 2

ABCD Matrix w(z) Cavity w0 Stability

EM Review

Gaussian Beams z Beams and z = 0 Cavities

Fabry-Perot Cavities

General Cavities

Summary 1/2 Appendix A 2 The beam spot size grows as w(z) = w 1 + z 0 z0

w(z) = w0 is known as the beam waist. It is the narrowest and most intense part of the beam Spot Size w(z) - Divergence Angle I

ECE 455 Lecture 2 Consider the behavior of Equation 50 as z → ∞

ABCD Matrix s z 2 Cavity w(z) = w 1 + (86) Stability 0 z0 EM Review r 2 Gaussian z z0 Beams = w0 1 + (87) z0 z Beams and w 1 z 2 Cavities ≈ 0 z 1 + 0 (88) Fabry-Perot z0 2 z Cavities w0 General ≈ z (89) Cavities z0 Summary λ0 Appendix A = z (90) πnw0 Equation 88 was derived from Equation 87 by Taylor expansion. Spot Size w(z) - Divergence Angle II

ECE 455 From the last slide we have Lecture 2

λ0 ABCD Matrix w(z) ∼ z (91) πnw0 Cavity Stability when z is large. Recall that m = tan(θ), where m is the slope EM Review of a line and θ is the angle that the line makes with the axis. Gaussian Beams The divergence angle is deﬁned as the total angle between the Beams and 1/e2 points (twice the angle made with the axis). Hence: Cavities Fabry-Perot 2λ Cavities θ = tan−1 0 (92) General πnw0 Cavities Summary For small slopes, θ ≈ tan(θ) and Appendix A 2λ θ = 0 (93) πnw0 The Radial Phase Factor R(z)

ECE 455 Lecture 2 Consider the following term in Equation 82 ABCD Matrix h 2 i exp −ı kr Cavity 2R(z) Stability Constant Surfaces of constant phase Phase EM Review are not ﬂat, they are Surface Gaussian r Beams parabolic Beams and Cavities The radius of curvature of z Fabry-Perot the surfaces is R(z) Cavities z = 0 General Surface of constant phase Cavities at beam waist is ﬂat Summary (R(0) = ∞) Appendix A Radius of curvature increases away from the beam waist The Longitudinal Phase Factor

ECE 455 Lecture 2

ABCD Matrix Consider the following term in Equation 82 Cavity h i Stability exp −ı kz − (1 + m + p) tan−1 z z0 EM Review −ıkz Gaussian e is plane wave propagation in thez ˆ direction Beams ı(1+m+p) tan−1 z Beams and e z0 is the deviation from the plane wave Cavities velocity in thez ˆ direction because the wave must also Fabry-Perot Cavities propagate in the transverse direction General Cavities Phase velocity of a Gaussian is lower than that of a plane Summary wave Appendix A Transverse Modes I

ECE 455 Lecture 2 Consider the following term in Equation 82 ABCD Matrix √ √ Cavity h 2x i h 2y i Stability Hm w(z) Hp w(z) EM Review These terms introduce structure onto Gaussian m Hm(x) Beams the beam (see next two slides) 0 1 Beams and Cavities Hermite-Gaussian polynomials. They 1 2x Fabry-Perot can be generated with the following 2 4x2 − 2 Cavities function 3 General 3 8x − 12x Cavities m u2 dm −u2 Hm(u) = (−1) e dum e Summary

Appendix A Higher order modes occupy more volume, diverge more rapidly, and cannot be focused as tightly Transverse Modes II: Electric Field Proﬁle

ECE 455 Lecture 2

ABCD Matrix

Cavity Stability

EM Review

Gaussian Beams

Beams and Cavities

Fabry-Perot Cavities

General Cavities

Summary

Appendix A Figure: Electric ﬁeld of TEM modes Transverse Modes III: Intensity Proﬁle

ECE 455 Lecture 2

ABCD Matrix

Cavity Stability

EM Review

Gaussian Beams

Beams and Cavities

Fabry-Perot Cavities

General Cavities

Summary

Appendix A Figure: Various TEMmp modes with the same w0, normalized to have the same total optical power. Transverse Modes IV: Gauss-Laguerre Modes

ECE 455 Lecture 2

ABCD Matrix

Cavity Stability

EM Review

Gaussian Beams

Beams and Cavities

Fabry-Perot Cavities

General Cavities

Summary

Appendix A Figure: Various Gauss-Laguerre modes with the same w0, normalized to have the same total optical power. Gauss-Laguerre Modes

ECE 455 Lecture 2 Solutions with Hermite polynomials result from solving Equation 70 in Cartesian coordinate ABCD Matrix Equation 70 may also be solved in radial coordinates, Cavity resulting in Gauss-Laguerre modes Stability 2 EM Review In general, any ﬁeld may be written as a linear

Gaussian combination of Hermite-Gaussian or Gauss-Laguerre Beams modes: Beams and Cavities X Earb = TEMmp(x, y) (94) Fabry-Perot Cavities mp

General X Cavities = LG`p(r, θ) (95)

Summary `p Appendix A Inﬁnite number of other solutions may be found in other coordinates (elliptic, hyperbolic) 2Provided that the transverse proﬁle doesn’t contain high enough spatial frequencies to violate the paraxial approximation Beam Quality: M2

ECE 455 Lecture 2 M2 approximately measures how much faster a given beam will diffract as compared to the TEM00 mode ABCD Matrix 2 Cavity A perfect TEM00 mode has M = 1 Stability Gas lasers are prized for their beam quality M2 ∼ 1 EM Review 2 Gaussian M is unchanged by ABCD law elements Beams Astigmatic beams have different values for M2 along Beams and Cavities different axes: In particular for higher-order Gaussian Fabry-Perot modes Cavities

General 2 Cavities Mx = (2m + 1) (96) Summary 2 My = (2p + 1) (97) Appendix A True M2 measurements require a fairly complex measurement device Forcing TEM00 Operation

ECE 455 Lecture 2 Aperture

Gain Medium ABCD Matrix

Cavity Stability

EM Review

Gaussian Beams

Beams and Cavities Multi-mode beams don’t focus as tightly as the TEM00 Fabry-Perot does alone Cavities Higher order modes take power away from the TEM00 General Cavities mode Summary An aperture may be inserted which introduces loss for Appendix A higher order modes It is better to insert the aperture in the cavity rather than ﬁlter the output beam because the change receives positive feedback Gaussian Beams and Cavities

ECE 455 In order to be reﬂected unchanged, the radius of curvature of Lecture 2 the guassian mode must match the curvature of the end

ABCD Matrix mirrors, as illustrated below.

Cavity Stability EM Review Radially Symmetric field Gaussian Plane - like wave Beams

Beams and Cavities

Fabry-Perot Cavities

General Cavities

Summary

Appendix A

Constant phase front; it’s radius of curvature matches that of the mirror but only at the mirror. Gaussian Cavity Stretch

ECE 455 Lecture 2

ABCD Matrix

Cavity Stability

EM Review

Gaussian Beams

Beams and Cavities

Fabry-Perot Cavities

General Cavities

Summary

Appendix A Gaussian Cavity Stretch

ECE 455 Lecture 2

ABCD Matrix

Cavity Stability

EM Review

Gaussian Beams

Beams and Cavities

Fabry-Perot Cavities

General Cavities

Summary

Appendix A Gaussian Beams and Cavities

ECE 455 Lecture 2 Let z1 and z2 represent the z coordinates of the left and right mirror (Note this is not Verdeyen’s convention), with the origin ABCD Matrix being deﬁned as the beam waist. We can write the following Cavity Stability three equations: EM Review z2 − z1 = d (98) Gaussian Beams " 2# z0 Beams and R(z1) = z1 1 + = −R1 (99) Cavities z1 Fabry-Perot Cavities " 2# z0 General R(z2) = z2 1 + = R2 (100) Cavities z2 Summary

Appendix A Note the strange convention in Equation 99. It results from the fact that concave surfaces have a mathematically negative radius of curvature to the left of the beam waist. Gaussian Beams and Cavities

ECE 455 Lecture 2

ABCD Matrix With three equations and three unknowns, it is possible to Cavity solve the system Stability

EM Review 2 d(R1 − d)(R2 − d)(R1 + R2 − d) Gaussian z0 = 2 (101) Beams (R1 + R2 − 2d) Beams and Cavities −d(R2 − d) Fabry-Perot z1 = (102) Cavities R1 + R2 − 2d General Cavities d(R1 − d) z2 = (103) Summary R1 + R2 − 2d Appendix A Use Your Head!

ECE 455 Lecture 2 Rather than memorizing the sign convention used here (it may be diﬀerent in other texts), use your common sense in applying ABCD Matrix Equation 83. Cavity Stability Focusing mirrors are concave and are quoted with positive EM Review radii of curvature Gaussian Beams Diverging mirrors are convex and are quoted with negative Beams and Cavities radii of curvature Fabry-Perot The center of curvature of a gaussian beam is always Cavities towards the beam waist General Cavities The curvature is positive on the right side of the beam Summary waist Appendix A The curvature is negative on the left side of the beam waist Example: Finding the Mode Inside a Cavity

ECE 455 Problem: Locate the beam waist in the cavity shown below. Lecture 2 What are the spot sizes at the beam waist and at both of the

ABCD Matrix mirrors?

Cavity Stability

EM Review

Gaussian 50 cm Beams

Beams and Cavities Fabry-Perot z Cavities

General Cavities

Summary

Appendix A 푅1 = 4 m 푅2 = 3 m Z = 0 λ = 500 nm Example: Finding the Mode Inside a Cavity

ECE 455 Lecture 2 Solution: The ﬁrst step is to ﬁnd the spot size using Equation ABCD Matrix 101. In keeping with the convention that focusing mirrors have Cavity Stability a positive radius of curvature, the values to use are: d = 50 EM Review cm, R1 = +400 cm and R2 = +300 cm. The solutions for z0 is Gaussian Beams z0 = 88.88 cm (104) Beams and Cavities

Fabry-Perot which can be converted to the spot size with Cavities

General r Cavities λz0 w0 = = 376 µm (105) Summary nπ Appendix A Which is the spot size at the beam waist. Example: Finding the Mode Inside a Cavity

ECE 455 According to Equations 101, z and z are Lecture 2 1 2

z1 = −20.83 cm (106) ABCD Matrix

Cavity Stability z2 = 29.17 cm (107) EM Review The beam waist is therefore 20.83 cm to the right of Mirror 1. Gaussian Beams Note the beam waist is nearer the ﬂatter mirror.

Beams and Cavities The spot sizes on the two mirrors are Fabry-Perot Cavities " 2#1/2 General z1 Cavities w(z ) = w 1 + = 386 µm (108) 1 0 z Summary 0

Appendix A " 2#1/2 z2 w(z2) = w0 1 + = 396 µm (109) z0 Power Flux

ECE 455 Lecture 2

ABCD Matrix Because the transverse component of the ﬁeld is negligible, the

Cavity intensity distribution is simply Stability EM Review E ∗(r, z)E(r, z) Gaussian I (r, z) = (110) Beams 2η

Beams and Cavities To ﬁnd the ﬂux of power out to a certain radius at a distance z Fabry-Perot away from the beam waist, use Cavities

General Z r0 Cavities P = I (r, z)2πrdr (111) Summary 0 Appendix A ABCD Law for Gaussian Modes

ECE 455 Lecture 2 The complex beam parameter is deﬁned as

1 1 λ0 ABCD Matrix = − j (112) q(z) R(z) πnw 2(z) Cavity Stability

EM Review Then the complex beam parameter at point q2 is related to the Gaussian complex beam parameter at point q1 by: Beams Beams and Aq1 + B Cavities q = (113) 2 Cq + D Fabry-Perot 1 Cavities or (114) General 1 C + D(1/q ) Cavities = 1 Summary q2 A + B(1/q1) Appendix A where A, B, C and D are the coeﬃcients of the ABCD matrix that connects the two points. See Siegman Chapter 20 for a detailed proof. Example: Focus of a Laser Beam

ECE 455 Problem: A lens with focal length f is inserted at the beam Lecture 2 waist of a Gaussian mode as shown below. Find the spot size

ABCD Matrix at the new beam waist and the distance between the lens and

Cavity the focus. Stability

EM Review Gaussian Before Beams

Beams and Cavities

Fabry-Perot Cavities General After Cavities

Summary

Appendix A

= L Example: Focus of a Laser Beam

ECE 455 Solution: The ABCD matrix for a lens focus f followed by a Lecture 2 distance d is AB 1 − d d ABCD Matrix = f (115) CD − 1 1 Cavity f Stability Because the radius of curvature is inﬁnite at the beam waist, EM Review the complex beam parameter at the lens is Gaussian Beams 1 λ = −ı (116) Beams and 2 Cavities q1 πnw0 Fabry-Perot λ Cavities Deﬁne W1 = − 2 for convenience. πnw0 General 1 C + ıDW Cavities = 1 (117) Summary q2 A + ıBW1 Appendix A C + ıDW A − ıBW = 1 1 (118) A + ıBW1 A − ıBW1 AC + BDW 2 + ı (AD − BC) W = 1 1 (119) 2 2 2 A + B W1 Example: Focus of a Laser Beam

ECE 455 Lecture 2 Using the knowledge that R = ∞ at a beam waist, we can set the real part of the expression above to zero to ﬁnd at which d

ABCD Matrix the beam waist occurs.

Cavity Stability 2 AC + BDW1 = 0 (120) EM Review Gaussian After substitution, this becomes Beams Beams and d 1 dλ2 Cavities 1 − − + = 0 (121) 2 2 4 Fabry-Perot f f π n w Cavities 0

General Cavities The ﬁnal answer is Summary f f d = = (122) Appendix A 2 2 2 1 + W1 f f λ 1 + 2 πnw0 Which is slightly less than the focal length Example: Focus of a Laser Beam

ECE 455 To ﬁnd the spot size at the new focus, substitute this value of Lecture 2 d into Equation 119

ABCD Matrix 2 1 1 λ πnw0 Cavity = W1 + 2 = 2 + 2 (123) Stability q2 f W1 πnw0 λf EM Review s Gaussian w 2λ2f 2 Beams w = 0 (124) 02 2 2 2 2 4 Beams and λ f + π n w0 Cavities

Fabry-Perot To make the focus as small as possible we need: Cavities Small λ General Cavities Small f Summary Large w0 Appendix A Large n Lenses with large f /d (known as f #) ratios are diﬃcult (and therefore expensive) to make. ABCD Matrices for Resonator Modes

ECE 455 Lecture 2 After exactly one round trip through the cavity, the mode must repeat itself. Therefore, for one round-trip of the cavity, the ABCD Matrix

Cavity beam parameter must obey: Stability

EM Review 1 C + D(1/q) = (125) Gaussian q A + B(1/q) Beams

Beams and Cavities When solved for the beam parameter, we find: Fabry-Perot p Cavities 1 −(A − D) ± (A − D)2 + 4BC General = (126) Cavities q 2B Summary Note that because the round-trip ABCD matrix is different at Appendix A different locations inside the cavity, the beam parameter will be different depending on where the round trip was started. Fabry-Perot Cavity Electric Field Diagram

ECE 455 Lecture 2

Z = 0 Z = L ABCD Matrix z

Cavity Stability t

EM Review

Gaussian Beams Beams and Cavities Fabry-Perot Cavities

General Cavities Summary

Appendix A . . . . .

Reflectivity Reflectivity Fabry-Perot Electric Field

ECE 455 Lecture 2 Fabry-Perot is the simplest optical cavity – two mirrors with a uniform medium in between ABCD Matrix Field may be represented as infinite sum of reflected fields Cavity Stability Define φ = nkL EM Review r1 = r2 = r has been assumed for simplicity Gaussian r and t are field reflectivities, so conservation of energy Beams 2 2 Beams and implies |r| + |t| = 1 Cavities |t| = p1 − |r|2 Fabry-Perot Cavities The transmitted field is then: General Cavities 2 jφ 2 2 j3φ 2 4 j5φ Et = E0t e + E0t r e + E0t r e + ··· Summary 2 jφ 2 ıφ 4 ı2φ Appendix A = E0t e 1 + r e + r e + ··· E t2ejφ = 0 (127) 1 − r 2ei2φ Fabry-Perot Transmitted Intensity

Fabry-Perot Cavities Expanding the denominator yields General Cavities T 2 I Summary I = 0 (130) t 1 − R2 4R 2 Appendix A 1 + (1−R)2 sin (φ) Note the intensity in the cavity is larger than the transmitted 1 ﬁeld by a factor of 1−R Fabry-Perot Resonances

ECE 455 Lecture 2 When φ is a multiple of π, the transmitted intensity will have a maximum. Solving φ = nkL = 2πn L = qπ, the wavelengths and ABCD Matrix λq th Cavity frequencies of the q modes are determined to be: Stability EM Review 2nL λ = (131) Gaussian q Beams q Beams and and Cavities qc Fabry-Perot νq = (132) Cavities 2nL General Notice how the frequency spacing between adjacent maxima is Cavities constant. This quantity is deﬁned as the free spectral range Summary and is denoted Appendix A c ∆ν ≡ (133) 2nL Fabry-Perot Transmission Proﬁle

ECE 455 Lecture 2

ABCD Matrix Cavity Reflectivity vs. Frequency Cavity Stability 0 EM Review +1 +2 … Gaussian 1 Beams Δ = Beams and 2 Cavities

Fabry-Perot Cavities

General Cavities

Summary

Appendix A

0 Frequency Example: Fabry-Perot Free Spectral Range

ECE 455 Lecture 2 Problem: Find the free spectral range of a GaAs (n = 3.6) diode laser with end facets separated by 100 µm. ABCD Matrix

Cavity Solution: Stability c EM Review ∆ν = Gaussian 2nL Beams 3 × 108 m/s Beams and = −4 Cavities 2 · (3.6) · (1 × 10 m) Fabry-Perot ≈ 417 GHz (134) Cavities

General Cavities Problem: Find the free spectral range of an Argon Ion laser, Summary with mirrors separated by 1.5 m and n = 1. Appendix A Solution: ∆ν = 100 MHz (135) Finesse

ECE 455 It can be show with a mess of algebra, that if the mirror Lecture 2 reﬂectivies are unequal, the transmission through the cavity is

ABCD Matrix (1 − R1)(1 − R2) I0 Cavity I+ = √ √ (136) Stability 2 4 R1R2 2 1 − R1R2 1 + √ 2 sin (φ) EM Review (1− R1R2)

Gaussian Beams A quantity called the ﬁnesse is deﬁned as the free spectral Beams and range divided by the full width at half maximum of the peak. Cavities Fabry-Perot FSR Cavities F ≡ (137) General FWHM Cavities c/(2nL) Summary = (138) ∆ν1/2 Appendix A π(R R )1/4 = 1 2 (139) 1/2 1 − (R1R2) A Graphical View of Finesse

ECE 455 Lecture 2

ABCD Matrix 퐼푡

Cavity 퐼 Stability 0 FSR EM Review 1 Gaussian Beams 휈1 Δ 2 Beams and Cavities （FWHM） Fabry-Perot Cavities

General Cavities

Summary

Appendix A

0 휈푚 휈푚+1 Frequency 휈 The Quality Factor

ECE 455 Lecture 2 Another quantity used to describe cavities is the center frequency divided by the full width at half maximum of the ABCD Matrix transmission peak. Cavity Stability Center Frequency EM Review Q ≡ (140) FWHM Gaussian ν Beams = 0 (141) Beams and ∆ν1/2 Cavities λ Fabry-Perot 0 Cavities = (142) ∆λ1/2 General Cavities 2πnL (R R )1/4 = 1 2 (143) Summary 1/2 λ0 1 − (R1R2) Appendix A In crystal oscillator RF circuits, Q can be 105 − 106. In laser cavities, Q can be in excess of 108!! Further Interpretations of the Quality Factor

ECE 455 Lecture 2 The quality factor can be related directly to cavity ABCD Matrix parameters, rather than the cavity’s spectral Cavity characteristics: Stability EM Review Max Stored Energy Q = 2π (144) Gaussian Beams Energy Lost Per Cycle Max Stored Energy Beams and = 2πν (145) Cavities 0 Average Power Loss Fabry-Perot Cavities Loss may come from scattering or absorption in the cavity General Cavities or light coupling out of the cavity Summary Q may also be interpreted as the number of oscillations Appendix A observed before the amplitude of the oscillation decays below e−2π A Graphical View of the Quality Factor

ECE 455 Lecture 2 휈1 Δ 2 （FWHM） ABCD Matrix 퐼푡 Cavity Stability 퐼0 EM Review Gaussian 1 Beams

Beams and Cavities

Fabry-Perot Cavities

General Cavities

Summary

Appendix A

휈0 Frequency 휈 Example: FSR, Q, F

ECE 455 Problem: Find the FSR, Q, and F of the cavity shown below at Lecture 2 a wavelength of 1 µm.

ABCD Matrix

Cavity Stability

EM Review

Gaussian Beams

Beams and Cavities

Fabry-Perot Cavities

General Cavities

Summary

Appendix A Solution: The free spectral range is c FSR = = 150 GHz (146) 2nL Example: FSR, Q, F Continued

ECE 455 Lecture 2

ABCD Matrix Cavity The quality factor of the cavity is Stability

EM Review 1/2 2 · (0.001 m)π(0.995) 6 Gaussian Q = ≈ 1.25 × 10 (147) Beams (1 × 10−6 m)(1 − 0.995) Beams and Cavities Its ﬁnesse is Fabry-Perot π(0.995)1/2 Cavities F = = 627 (148) General 1 − 0.995 Cavities

Summary

Appendix A Fabry-Perot Tuning: Changing Mirror Separation

ECE 455 Lecture 2

ABCD Matrix

Cavity Stability

EM Review

Gaussian Beams

Beams and Cavities

Fabry-Perot Cavities

General Cavities Fabry-Perot cavities can be used to examine ﬁne spectral

Summary features. Appendix A A length change of λ/4 will shift a cavity from resonance to anti-resonance ∆L L 1 for the FSR to remain constant over tuning range Fabry-Perot Tuning: Changing Mirror Separation

ECE 455 Recall that if φ is the phase shift from a one way pass through Lecture 2 the cavity, the intensity of light transmitted through a FP

ABCD Matrix cavity is: Cavity 1 Stability It ∝ 4R EM Review 1 + (1−R)2 sin(φ) Gaussian Beams 1 = 4R Beams and 1 + 2 sin(nkL) Cavities (1−R)

Fabry-Perot 1 Cavities = 4R 2π (149) 1 + 2 sin( νnL) General (1−R) c Cavities

Summary The last equation tells us that a plot of transmitted intensity

Appendix A will have the same shape regardless of whether we sweep the index of refraction, the frequency of the incident light, or the length of the cavity while holding the other two constant. Therefore the ﬁnesse may be esimated from any of these plots. Example: Fine Spectral Features

ECE 455 Problem: A CW HeNe laser is shone upon a FP cavity with a Lecture 2 FSR of 4.25 GHz. One mirror of the FP is rapidly moved with a

ABCD Matrix piezo-electric actuator. The transmitted intensity as a function

Cavity of time is shown below. What is the Finesse of the Fabry-Perot Stability cavity? How far apart are the end mirrors of the HeNe laser? EM Review

Gaussian Beams 퐼 Beams and 푡 Cavities

Fabry-Perot Cavities 1

General Cavities

Summary

Appendix A

t Example: Fine Spectral Features

ECE 455 Lecture 2 Solution: The ﬁrst step is to interpret the data:

ABCD Matrix The periodic pattern is from the same frequencies going in

Cavity and out of resonance as the FP cavity is tuned Stability The distance from the one of the high peaks to the next is EM Review

Gaussian the free spectral range of the FP cavity Beams Assume the taller and shorter peaks are adjacent Beams and Cavities longitudinal modes of the HeNe. The distance between Fabry-Perot them is then one FSR of the HeNe Cavities

General Not quite resolved next to the larger peak is a higher-order Cavities transverse mode. Summary The finesse can be estimated by simply finding the ratio of the Appendix A FSR to the FWHM found by measuring the figure. A good guess is F = 45. Example: Fine Spectral Features

ECE 455 Lecture 2

ABCD Matrix

Cavity Stability

EM Review In the same way, the frequency separation between the two

Gaussian longitudinal modes is estimated to be 500 MHz. Beams c c Beams and L = = = 30 cm (150) Cavities 2 · FSR 2 · 500MHz Fabry-Perot Cavities

General Cavities

Summary

Appendix A Fabry-Perot Tuning: Tilting

ECE 455 Lecture 2

ABCD Matrix

Cavity Stability

EM Review θ

Gaussian Beams

Beams and Cavities

Fabry-Perot Cavities

General If the cavity is tilted with respect to the incident beam, the Cavities phase shift per round trip is no longer 2kL. Instead the phase Summary shift is: Appendix A 2kL φ0 = (151) sin(θ) Fine tuning of a laser’s wavelength frequently accomplished by inserting a high-Q Fabry-Perot etalon into a larger cavity. Photon Lifetime

ECE 455 Consider a cavity such as the one shown below Lecture 2

ABCD Matrix 1

Cavity Stability N EM Review p Gaussian R N Beams 2 p R1R2Np Beams and Cavities

Fabry-Perot L Cavities

General Cavities

Summary After each trip through the cavity, only a fraction S Appendix A (0 ≤ S ≤ 1) of photons remain. Hence if there are Np photons in the cavity initially, the change in the number of photons in the cavity is ∆Np = −(1 − S) · Np (152) Photon Lifetime

ECE 455 The time required for light to make one round trip in the cavity Lecture 2 is 2nL ABCD Matrix t = (153) RT c Cavity Stability The change in the number of photons per unit time is EM Review dNp ∆Np Gaussian ≈ Beams dt ∆t Beams and (1 − S)Np Cavities = − (154) t Fabry-Perot RT Cavities The familiar solution to this equation is General Cavities − t N (t) = N e τp (155) Summary p p0

Appendix A Where τp is called the photon lifetime and is deﬁned as t τ = RT (156) p 1 − S Survival

ECE 455 Lecture 2 Most generally, the photon lifetime is deﬁned as ABCD Matrix

Cavity round trip time Stability τ = (157) p fractional loss per round trip EM Review

Gaussian Beams

Beams and Cavities The loss may come from several sources Fabry-Perot Cavities Finite mirror reﬂectivity General Cavities Unsaturated intracavity absorption Summary Scattering loss Appendix A Diﬀraction loss A Two-Mirror Cavity

ECE 455 Lecture 2

ABCD Matrix

Cavity Consider the following cavity with end mirrors separated by L Stability and reflectivities R1 R2. The loss of the cavity is dominated by EM Review the finite reflectivity of the mirrors. In this case the survival Gaussian 2nL Beams rate is S = R1R2. The round trip time is tRT = c , which Beams and means the photon lifetime is Cavities Fabry-Perot 2nL Cavities τc = (158) General c · (1 − R1R2) Cavities

Summary

Appendix A Example: Cavity with an Absorber

ECE 455 Problem: Determine the photon lifetime of the cavity below. Lecture 2

ABCD Matrix 푅1=0.95 푅2=0.95

Cavity Stability

EM Review

Gaussian Beams Beams and Loss of α = 0.02푐푚−1 Cavities n=1 Fabry-Perot Cavities

General Cavities 20 cm

Summary

Appendix A

100 cm Example: Cavity with an Absorber

ECE 455 Lecture 2 Solution: The ﬁrst step should be to determine the survival fraction of photons during one round trip ABCD Matrix

Cavity Stability S = R1exp(−αL)R2exp(−αL) = 0.4055 (159) EM Review

Gaussian Note that during the round trip, light must make two trips Beams through the absorber. Next the round trip time is Beams and Cavities 2nL Fabry-Perot tRT = = 6.67 ns (160) Cavities c General Cavities Going back to Equation 157, the photon survival time is Summary 6.667 ns Appendix A τ = = 11.2 ns (161) p 1 − 0.4055 A Useful Relation

ECE 455 Lecture 2

ABCD Matrix

Cavity Stability Cavity Q is related to the cavity lifetime, τc , and angular EM Review frequency of light, ω, by: Gaussian Beams Q = ωτ (162) Beams and c Cavities Fabry-Perot This useful relation allows measurement of the cavity lifetime Cavities without performing a temporal measurement. General Cavities

Summary

Appendix A General Cavity Resonances

ECE 455 Lecture 2 The condition for resonance for any mode is: Z ABCD Matrix kz dz = q2π (163) Cavity RT Stability EM Review which states that the phase shift during a round trip must be Gaussian an integer multiple of 2π. It is a consequence of the fact that Beams

Beams and the ﬁeld must be single-valued at every point. Cavities

Fabry-Perot Cavities Consider a plane wave in a cavity with a non-uniform index of General 2πνqn(z) Cavities refraction. We can use kz = c in Equation 163 to get Summary qc Appendix A νq = R (164) RT n(z)dz as the resonance frequencies of the cavity. Resonant Wavelengths of Gaussian Modes

ECE 455 Lecture 2

ABCD Matrix

Cavity Applying Equation 163 to Equation 82, we ﬁnd the Stability eigenfrequencies the gaussian modes within a resonator are: EM Review Gaussian c 1 + m + p −1√ Beams νm,p,q = q + cos g1g2 (165) Beams and 2nd π Cavities

Fabry-Perot where m and p are the transverse quantum number, q is the Cavities d longitudinal quantum number, and gi = 1 − is the General Ri Cavities stability parameter. Summary

Appendix A Example: Cavity with an Absorber

ECE 455 Lecture 2 Problem: Find the FSR, F, τp, and Q for the cavity shown below.

ABCD Matrix

Cavity = 0.99 = 0.98 Stability

EM Review Gaussian : Crystal Beams = 9 cm Beams and L Cavities = 1064 nm =

Fabry-Perot = 1.958 cm 17 Cavities = 0.001 cm-1 General Cavities

Summary

Appendix A

L = 26 cm = 0.80 Example: Cavity with an Absorber

ECE 455 Lecture 2 Solution: First, we must ﬁnd the optical path length

ABCD Matrix Lopt = 17 cm + (26 − 9) cm + 1.958 · 9 cm Cavity Stability = 51.62 cm (166) EM Review

Gaussian Beams Equation 164 tells that the FSR is: Beams and c Cavities FSR = = 290.4 MHz (167) Fabry-Perot 2Lopt Cavities General To ﬁnd the ﬁnesse, we can modify Equation 139 Cavities Summary 2 −2αL 1/4 π(R1R2 R3e ) Appendix A F = = 21.53 (168) 2 −2αL 1/2 1 − (R1R2 R3e ) Example: Cavity with an Absorber

ECE 455 Lecture 2

ABCD Matrix The photon lifetime is: Cavity Stability 2L EM Review τ = opt = 13.6 ns (169) p 2 −2αL Gaussian c(1 − R1R2 R3e ) Beams Beams and Finally, the cavity Q is: Cavities

Fabry-Perot 2 −2αL 1/4 Cavities 2L π(R R R e ) Q = opt 1 2 3 = 2.09 × 107 (170) General λ 1 − (R R2R e−2αL)1/2 Cavities 0 1 2 3

Summary

Appendix A Non-Traditional Cavities

ECE 455 Lecture 2

ABCD Matrix

Cavity Stability

EM Review

Gaussian Beams

Beams and Cavities

Fabry-Perot Cavities Figure: General Source: http://www.ireap.umd.edu/NanoPhotonics/PCdevices.html Cavities

Summary Appendix A Not all optical cavities consist of mirrors Any structure which conﬁnes light may behave as a laser cavity Fiber Cavity

ECE 455 Lecture 2 Light may be confined in a ABCD Matrix waveguide with ncore > nclad Cavity Stability The electric field for such a EM Review waveguide is: nclad Gaussian Beams ımφ ıkz z Ecore (r, φ, z) = E0J(kr r)e e ncore Beams and Cavities q Fabry-Perot where k = n2 k2 − k2 Cavities r core 0 z General Mirrors on end of fiber form a Cavities cavity Summary Reflection from index change Appendix A at end of fiber can also be considered a mirror Summary

ECE 455 Lecture 2

ABCD Matrix

Cavity The paths of paraxial light rays through optical systems Stability can be described with ABCD matrices EM Review

Gaussian Determine the stability of optical resonators Beams The electric ﬁeld inside an optical cavity is described with Beams and Cavities gaussian beams Fabry-Perot Cavities Describe the optical containment properties of cavities

General with the quality factor, the ﬁnesse, and the free spectral Cavities range Summary

Appendix A Useful Matrix Identities

ECE 455 Lecture 2

ABCD Matrix

Cavity Stability Let X , Y , Z, and W be matrices. Then the following EM Review properties hold Gaussian Beams (XY )Z = X (YZ) Associative Property Beams and det(XY ) = det(X )det(Y ) Cavities XYZW = (XY )(ZW ) Fabry-Perot Cavities XY 6= YX Non-commutativity General Cavities

Summary

Appendix A