Gaussian Beam Optics, Ray Tracing, and Cavities

Gaussian Beam Optics, Ray Tracing, and Cavities Revised: /4/14 1:01 PM /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 1

I. Gaussian Beams (Text Chapter 3) /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities

Gaussian Beams Real optical beams are not plane waves Real optical beams are not rays Real optical beams are of finite transverse extent Laser beams tend to be Gaussian in cross-section why? /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 3

Gaussian Beams Observation : Most real optical beams are almost pure TEM Certainly for free space: and i E = 0 i H = 0 /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 4

Gaussian Beams Decompose the divergence into transverse and longitudinal components: i E = 0 i ( E t + ẑe ) z = 0 = t + ẑ z i Et + ẑe z ( ) = t + ẑ z i E t + t + ẑ z i ẑe z = t i E t + z ẑ i E t = t i E t + z E z = 0 + i ẑe + ẑ i ẑe t z z z /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 5

Gaussian Beams The beam propagates as the speed of light and hence must at least approximately contain a factor of the form: Hence e jkz, k = nk o = n π λ z E z jn π λ E z At optical frequencies the wavelength is small hence the factor multiplying E z is large. Also note i E = t i E t + z E z = 0 z E z = t i E t /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 6

Gaussian Beams Approximate as: Where D is the transverse extent of the beam. Thus from Since in general t i E t t i E t z E = i E z t t n π λ E E t z D E z λ D 1 E t D λ π nd E t at optical wavelengths D ~ 1 cm, but be careful!) E z E t /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 7

Gaussian Beams It is thus reasonable (even intuitive) to examine an electric field of the form: E( x, y,z) = E o ψ ( x, y,z) e jkz ( ) slowly varying The factor ψ x, y,z captures how the beam differs from a uniform plane wave. This form must satisfy the wave equation: E + k E = 0 t E + z E + k E = 0 E( x, y,z) = E o ψ ( x, y,z)e jkz plane wave-type /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 8

Gaussian Beams t E + z E + k E = 0 Substitute: E( x, y, z) = E o ψ ( x, y, z)e jkz z E = E o z E = E o ψ ( x, y, z) z ψ x, y, z ( ) z e jkz jke o ψ ( x, y, z)e jkz ( ) t E = E o e jkz t ψ x, y, z jk ψ ( x, y, z) z ( ) k ψ x, y, z e jkz /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 9

Gaussian Beams Substitute: t E + z E + k E = 0 E o ( ) + ψ ( x, y,z) t ψ x, y,z z jk ψ ( x, y,z) k ψ ( x, y,z) + k ψ ( x, y,z) z e jkz = 0 ψ ( x, y,z) ( ) + 1 t ψ x, y,z z neglect, k>>1 ψ ( x, y,z) j k z = 0 t ψ ( x, y,z) jk ψ ( x, y,z) z = 0 Paraxial Wave Equation /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 10

Gaussian Beams: TEM 0,0 Mode Express the transverse gradient in cylindrical coordinates. The simplest beam will have cylindrical symmetry (d/dφ = 0) t ψ ( r,φ,z) = 1 r r r ψ r + 1 ψ r φ 0 t ψ ( x, y, z) jk 1 r r r ψ r ψ ( x, y, z) z jk ψ z = 0 = 0 /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 11

Gaussian Beams: TEM 0,0 Mode Try a Gaussian function for a possible solution ψ 0 = e j P( z)+ kr q z ( ) /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 1

Gaussian Beams: TEM 0,0 Mode Derivatives 1 r r r ψ 0 r jk ψ 0 z = 0, j ψ = e P z 0 jk ψ 0 z = k P ( z ) + j k r q ( z) ψ q ( z) 0 ψ 0 1 r r = j kr q z r r ψ 0 r = 1 r ( ) ψ, ψ 0 0 r = j ψ 0 r + ψ 0 r k q z = j ( )+ kr q z ( ) ( ) ψ k r 0 q ( z) ψ 0 k ( ) ψ k r 0 q ( z) ψ 0 q z /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 13

Gaussian Beams: TEM 0,0 Mode 1 r r r ψ 0 r j k q( z) ψ 0 k r q ( z) ψ 0 Group powers of r: jk ψ 0 z ( )+ j k r q ( z) q ( z) k P z ( ) ψ 0 = 0 j k q( z) + P ( z ) ψ 0 k r ( q ( z) 1 q ( z )) ψ 0 = 0 = 0 P ( z) = j = 0 q ( z) = 1 q z q( z) = q 0 + z Where is z = 0? /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 14

Gaussian Beams: TEM 0,0 Mode Thus, ψ 0 = e j P( z)+ kr q z j ( ) jp z = e ( ) e kr q z ( ) If q(z) were purely real, then for a fixed value of z the phase would continue to increase more and more rapidly with increasing radial distance with a constant amplitude, and this is impossible. kr ψ 0 = e jp ( z ) e j q z ( ) = 1 Consider then a complex q, q( z) = q 0 + z = z + jz 0 This gives (at z = 0) /4/14 ψ ( 0 z = 0) = e jp ( 0 ) kr z e o 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 15

Gaussian Beams: TEM 0,0 Mode Examine ψ ( 0 z = 0) = e jp ( 0 ) kr e z o ( ) = e 1 kr 1 ψ 0 z = 0 z o = π nw o λ o = 1 r z 1 = w o o = z o k = z λ o o π n w o is the Beam Waist or Spot Size (radius), Really the minimum spot size /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 16

Gaussian Beams: TEM 0,0 Mode z = 0 plane /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 17

Gaussian Beams: TEM 0,0 Mode For z 0 We really want q -1 : 1 ( ) = 1 q z ψ 0 = e jp ( z j ) e z jz 0 = z jz 0 z + jz 0 z jz 0 z + z = 0 ψ 0 = e jp ( z j ) kr e kr q z 1 q z ( ) = e jp z ( ) q( z) = z + jz 0 P ( z) = j q z z o = π nw o λ o z z + z j z 0 0 z + z 0 j kr ( ) e z z +z 0 j z 0 z +z 0 ( ) /4/14 j = e jp ( z ) e 1 kzr z +z 0 Rapid phase variation with r e 1 kz 0 r z +z 0 Vanishing amplitude with r 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 18

Gaussian Beams: TEM 0,0 Mode w ψ 0 Spot size: 1 = e kz 0 r z +z 0 = e ( z) = z + z 0 = kz 0 r w z ( ), z o = π nw o λ o z + π n w o 4 λ o π n w o λ o = z + π n w o λ o π n w o λ o w o = z π n w o λ o + w λ o = w o 1+ o z π nw o Minimum spot size is at z = 0 /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 19

Gaussian Beams: TEM 0,0 Mode What about P(z)? Recall: Or, P ( z) = j q z ( ) = j z + jz 0, z o = π nw o λ o ( ) = P z z 0 = j ln z + jz 0 z P ( j ζ )dζ = j dζ = j ln ζ + jz ζ + jz 0 0 ( ) ln( jz ) 0 0 ( ) ζ =0 = j ln z + jz 0 jz 0 = j ln 1 j z z 0 z /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 0

Gaussian Beams: TEM 0,0 Mode Thus, ( ) = j ln 1 j z z 0 P z 1 j z z 0 = 1 j z z 0 e z jarg1 j z 0 z = 1+ z 0 e j tan 1 z z 0 We need e jp ( z ) /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 1

Gaussian Beams: TEM 0,0 Mode e jp z j j ln1 j ( ) z = e z 0 = e ln 1 j z z 0 = e ln 1+ z z 0 e j tan 1 z z 0 = 1 e ln 1+ z z 0 e j tan 1 z z 0 = 1+ z z 0 1 e j tan 1 z z 0 = 1 1+ z z 0 e j tan 1 z z 0 /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities

Gaussian Beams: TEM 0,0 Mode 1 1+ z z 0 = z o = π nw o λ o 1+ 1 zλ o π nw o = w o w( z), w ( z ) = w o 1+ λ o z π nw o /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 3

Gaussian Beams: TEM 0,0 Mode Putting it all together, E( x, y,z) = E o ψ ( x, y,z)e jkz, ψ ( x, y,z) = e jp z = w o w( z) e j tan where w ( z) = w o Define R z 1 z z 0 kr j R z e ( ) e 1 λ 1+ o z π nw o ( ) = z + z 0 z kz 0 r z +z 0 = w o w( z) e j tan 1 z z 0 ( ) e j kr j R z e, w = z λ o o o π n, z = π nw o, o λ o = z 1+ z 0 z kr q z ( ) r ( ) w e ( z) /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 4

Gaussian Beams: TEM 0,0 Mode Rearranging, E( x, y,z) = E ψ ( x, y,z)e jkz o w = E o o w( z) e Amplitude Factor r ( ) w z jkz tan e z 1 z 0 j e Longitudinal Phase Factor kr ( ) R z Radial Phase Factor /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 5

Gaussian Beams: TEM 0,0 Mode Field Amplitude: ( ) = E o w o E x, y,z ( ) e w z r ( ) w z w ( z) = w o λ 1+ o z π nw o For increasing (or decreasing) z, The field amplitude decreases The beam waist increases The narrowest beam waist is w o occurring at z = 0 /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 6

Gaussian Beams: TEM 0,0 Mode w o e 1 e 1 w o e 1 e -1 points of the field θ w o e 1 e 1 w o e 1 z = 0 z = z 0 w z w( z) = w o z=zo ( ) = w o 1+ z z o /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 7

Gaussian Beams: TEM 0,0 Mode For large z the beam waist increases linearly w( z) = w o 1+ and spreads with angle λ o z π nw o z λ o z π nw o ( ) θ = dw z dz = w o λ o πnw o z λ 1+ o z πnw o z λ o πnw o /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 8

Gaussian Beams: TEM 0,0 Mode Longitudinal Phase: ( ) = kz tan 1 z φ z z 0 λ = kz o tan 1 π nw z o The phase velocity of a Gaussian beam is close to, but slightly greater than, the velocity of light in the equivalent uniform medium. v p ( z) ω z φ z ( ) = ck o z λ nk o z tan 1 πnw z o = c n 1 λ λ πnz tan 1 πnw z o /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 9

Gaussian Beams: TEM 0,0 Mode Radial Phase: j e kr ( ) R z For z = constant, the phase in not a constant (the equiphase surface is not a plane) but varies with radius r, hence we do not have a plane wave. The phase front is curved, not flat as with a plane wave. /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 30

Gaussian Beams: TEM 0,0 Mode To better understand the radial phase factor, consider a point source which emits a spherical wave. The electric field can be expressed as E 1 R e jkr Point Source R r z R = r + z = z 1+ r z 1+ 1 r z z r z via the binomial theorm zr z + 1 r R /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 31

Gaussian Beams: TEM 0,0 Mode Radial Phase Factor close to the z-axis: R = r z + 1 R R z for phase terms for amplitude terms E 1 R e jkr 1 jk z+ R e 1 r R /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 3

Gaussian Beams: TEM 0,0 Mode But for a Gaussian beam the apparent center for the curved wavefront changes. For a Gaussian beam recall, ( ) z ( ) = z 1+ z 0 R z When z z 0, R z and the wave appears to originate from the origin z = 0. z As we move closer to the origin however, center of curvature is at infinity and the wavefront is planar. R( z) z 0 z 0 z the /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 33

Gaussian Beams: TEM 0,0 Mode Where is z = 0? Where the spot size is minimum and the wavefront is planar. /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 34

Gaussian Beams: Higher Order Modes What if the assumption that φ is relaxed? Any type of imperfection. Intentional or otherwise, even dust in an optical system, can cause this to occur. Now the wave equation becomes: t ψ ( x, y, z) jk 1 r ψ ( x, y, z) z added term = 0 r r ψ r + 1 ψ jk ψ r φ z = 0 /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 35

Gaussian Beams: Higher Order Modes The solution is significantly more involved. It is simply stated here as: ( ) x = E m,p H m w( z) E x, y,z j y kz 1+ m+ p w H p w( z) o w( z) e w ( z) tan 1 z 0 j e R( z) e new terms The H m (u) are Hermite Polynomials r new term original terms z kr /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 36

Gaussian Beams: (Hermite-Gaussian) Higher Order Modes Hermite Polynomials: H m ( u) = ( 1) m d m e u e u du m H 0 H 1 H H 3 H 4 H 5 ( u) = 1 ( u) = u ( ) ( ) ( ) ( u) = u 1 ( u) = 4 u 3 3u ( u) = 4 4u 4 1u + 3 ( u) = Homework /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 37

Gaussian Beams: Higher Order Modes /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 38

Gaussian Beams: (Hermite-Gaussian) Higher Order Modes The idea of spot size can be a bit vague here. The spot size definition for w(z) is the same for all the modes illustrated, but the field occupies a bigger area as the mode number gets larger. /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 39

Gaussian Beams Lasers produce Gaussian beams The laser beam is generally produced by a cavity We need to understand what a cavity is along with methods of analyzing them. /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 40

II. Ray Tracing (Text Chapter ) /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 41

To trace a ray in an optical system two (very simple) things must be known: 1. Where is the ray at a given point?. In what direction is it going? /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 4

Ray Tracing 1 Ray θ ( = θ ) 1 θ r 1 r 1 Length d of free space Clearly, if we know where the ray is at plane 1 and we know its slope w.r.t. the optical axis, then we know where the wave is when it exits at plane. We assume a paraxial approximation, namely d Optical Axis tanθ = sinθ = θ ( ray slope) r = r r 1 d = tanθ = θ /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 43

Ray Tracing 1 Ray θ θ r 1 r 1 d Optical Axis r = r 1 + r 1 d r = r 1 ( y = mx + b) r r = 1 d 0 1 r 1 r 1 r out r out = A B C D ABCD Matrix r in r in /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 44

Ray Tracing 1 1 r 1 r 1 r d 1 d Optical Axis r 1 r 1 = 1 d 1 0 1 r 1 r 1 r r = 1 d 0 1 r 1 r 1 /4/14 r r = 1 d 0 1 1 d 1 0 1 Note the reverse order r 1 r 1 = d = 1 d 1 + d 0 1 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 45 r 1 r 1

Ray Tracing A Thin Lens A thin lens means this distance is negligible, hence r 1 = r regardless of angle of incidence. f f 1 r r = A B C D r 1 r 1 = 1 0 C D r 1 r 1 /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 46

Ray Tracing A Thin Lens f f For the blue ray: 1 r 1 = 0, r = r 1 f r r = A B C D r 1 r 1 1 0 = 1 D f r 1 r 1 /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 47

Ray Tracing A Thin Lens f f For the red ray: 1 r = 0, r 1 = + r 1 f r = 0 = r 1 f + D r 1 f D = 1 r r = A B C D r 1 r 1 1 0 = 1 D f r 1 r 1 1 0 T = 1 1 f /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 48

Ray Tracing Free Space and a Thin Lens d 1 Note the reverse order 3 T = 1 0 1 f 1 1 d 0 1 = 1 d 1 f 1 d f /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 49

Ray Tracing A Spherical Mirror Tangent Plane θ R R f = R r 1 = 0, r = r 1 R r r = A B C D r 1 r 1 1 0 = R D r 1 r 1 /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 50

Ray Tracing A Spherical Mirror θ R R r = 0, r 1 = r 1 R r = 0 = R r 1 + D R r 1 D = 1 r r = A B C D r 1 r 1 1 0 = R D r 1 r 1 1 0 = R 1 r 1 r 1 /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 51

Ray Tracing A Spherical Mirror T = 1 0 R 1 R Note the similarity to the lens. /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 5

Ray Tracing Summary r out r out = A B C D r in r in, AD BC = 1 /4/14 Length d of free space T d = T f = 1 d 0 1 1 0 1 f 1 Thin lens with focal length f Length d of free space followed by a thin lens with focal length f 1 d T df = 1 1 d f f Spherical mirror, radius R T R = 1 0 R 1 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 53, f = R

III. Ray Tracing in an Optical Cavity /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 54

Ray Tracing An Optical Cavity The most important part of a laser is the feedback system. A ray inside the cavity bounces back and forth between the two mirrors. 1. If the rays stays close to the optical axis even after many bounces it is called a stable cavity.. If the ray walks off one of the mirrors it is called unstable. 3. If the mirrors have to be perfectly aligned to keep the ray near the axis is is called conditionally stable. /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 55

Ray Tracing Optical Cavity Stability Equivalent Lens System: d R R 1 M 1 M d M d M 1 d M d M d M d M 1 1 Unit Cell /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 56

Ray Tracing Optical Cavity Stability Equivalent Lens System: d M d M 1 d M d M d M d M 1 1 Unit Cell Unit Cell f 1 f 1 f 1 f 1 f f f d d d d /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 57

Ray Tracing Optical Cavity Stability Equivalent Lens System: T 1 T d M d M 1 d M d M d M d M 1 1 T 1 = T = 1 0 1 f 1 1 0 1 f 1 1 T = T T 1 = 1 d 0 1 1 d 0 1 1 d 1 f 1 1 d f 1 = = Unit Cell 1 d 1 f 1 d f 1 d 1 f 1 1 d f 1 1 d 1 f 1 d f = 1 d d + d 1 d f f 1 1 1 d f 1 f f 1 1 d f 1 d f d f 1 1 /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 58

Ray Tracing Optical Cavity Stability Equivalent Lens System: 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 59 T n = 1 d f d + d 1 d f 1 f 1 1 f 1 d f 1 1 d f 1 1 d f d f 1 = A B C D r n+1 r n+1 = T n r n r n /4/14

Ray Tracing Optical Cavity Stability r n+1 r n+1 = T n r n r n r n+1 = Ar n + Br n r n = 1 ( B r Ar ) r n+1 n n+1 = 1 ( B r Ar ) n+ n+1 r n+1 = Cr n + Dr n r n+1 = 1 ( B r Ar ) = Cr n+ n+1 n + Dr n = Cr n + D 1 ( B r Ar ) n+1 n 1 ( B r Ar ) = Cr n+ n+1 n + D 1 ( B r Ar ) n+1 n /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 60

Ray Tracing Optical Cavity Stability 1 ( B r Ar ) = Cr n+ n+1 n + D ( B r Ar ) n+1 n 1 B r n+ A + D B r n+1 r n+ ( A + D)r n+1 + r n = 0 A second order difference equation. =1 AD BC + B r n = 0 /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 61

Ray Tracing Optical Cavity Stability Assume a solution of the form: Substitute: r o x n+ ( A + D)r o x n+1 + r o x n = 0 r o x ( A + D)x +1 xn = 0 r n+ ( A + D)r n+1 + r n = 0 x = A + D x = A + D ± 1 r n = r o x n ( A + D) 4 = A + D +1 A + D cos( ϕ) = A + D, sin( ϕ) = 1 A + D ± j 1 A + D = 1 x = e ± jϕ = cos( ϕ) ± jsin( ϕ) /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 6

Ray Tracing Optical Cavity Stability 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 63 x = e ± jϕ = cos ϕ ( ) ± jsin ϕ ( ) cos ϕ ( ) = A + D ϕ = cos 1 A + D x n = e ± jnϕ = cos nϕ ( ) ± jsin nϕ ( ) = cos ncos 1 A + D ± jsin ncos 1 A + D = exp ± j tan 1 tan ncos 1 A + D = exp ± jncos 1 A + D /4/14

Ray Tracing Optical Cavity Stability But x must be real. The only option is A = B 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 64 x n = exp ± jncos 1 A + D x n = Aexp jncos 1 A + D + Bexp jncos 1 A + D = cos ncos 1 A + D x n = cos ncos 1 A + D /4/14

Ray Tracing Optical Cavity Stability x n = cos ncos 1 A + D r n = r o x n A + D > 1 A + D = 1 A + D < 1 /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 65

Ray Tracing Optical Cavity Stability Clearly for a bounded solution: A + D 0 A + D The condition for a stable cavity is: An unstable cavity: 1 1 A + D 1 +1 0 A + D + 4 A + D > 1 1 0 A + D + 4 1 Unstable cavities are sometimes used in high-power lasers. /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 66

Ray Tracing Optical Cavity Stability For the cavity being studied: A = 1 d, D = 1 d f f 1 1 d f d f 1 1 d + 1 d A + D + f = f 1 1 d f d + f 1 4 4 1 d +1 d d + d d d + f = f f 1 f 1 f f 1 = 1 1 d 1 4 f = 1 1 d f 1 1 1 d f d f 1 + 1 4 d f 1 f /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 67

Ray Tracing Optical Cavity Stability The condition for a stable cavity is: Or 0 1 1 d f 1 1 1 d f 1 0 A + D + 4 1 Since f 1 = R 1, f = R The stability condition reduces to: 0 1 d R 1 1 d R 1 /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 68

Ray Tracing Optical Cavity Stability Stability condition: 0 1 d R 1 1 d R 1 d R 1 Unstable Stable Stable 1 d R 1 Unstable /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 69

Ray Tracing Confocal Geometry Borderline stability R 1 = R = R, d = R 0 1 R R 1 R R 1 d R =0 =0 1 Unstable R R Stable Stable 1 d R 1 Unstable /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 70

Ray Tracing Example Flat Mirror R 1 = R d = 3 4 R Tangent Planes r 0 f = R The entering horizontal ray will pass through the Focal point of M. M 1 M This is an example of a repetitive ray path. /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 71

Ray Tracing Summary r out r out = A B C D r in r in, AD BC = 1 /4/14 Length d of free space T d = T f = 1 d 0 1 1 0 1 f 1 Thin lens with focal length f Length d of free space followed by a thin lens with focal length f 1 d T df = 1 1 d f f Spherical mirror, radius R T R = 1 0 R 1 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 7, f = R

Ray Tracing Example Recall: 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 73 1 d R 1 1 d R = 1 3 4 = 1 4 stable T = 1 d f d + d 1 d f 1 f 1 1 f 1 d f 1 1 d f 1 1 d f d f 1 = 1 d f d + d 1 d f 1 f 1 d f = 1 4 5 4 d 3 4 1 4 /4/14

IV. ABCD Law for Gaussian Beams /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 74

ABCD Law for Gaussian Beams Recall our earlier result: ψ 0 = e j P( z)+ kr q z j ( ) jp z = e ( ) e The ABCD law relates the complex beam parameter q of a Gaussian beam at plane to the value q 1 at plane 1 using the elements of the ABCD matrix. q z = Aq 1 + B Cq 1 + D The proof of this result is tedious, but it is easy to to convince ourselves on its validity. kr q z ( ) /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 75

ABCD Law for Gaussian Beams Recall that: q = 1 q( z) = q 1 + z Also recall that for free space of length z that: T = The same result. 1 z 0 1 q z = Aq + B 1 Cq 1 + D = q + z 1 /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 76

ABCD Law for Gaussian Beams q z We were more interested in 1 ( ) = z z 0 = π nw o λ o 1 q z z + z j z 0 0 z + z, 0 ( ) = 1 w o = z λ 0 o π n, w z 1+ z 0 z j z 0 ( z) = w o z + z 0 = 1 R z 1 ( ) q z 1+ z z 0 ( ) j w o, R z 1 z 0 w z ( ) = z 1+ z 0 z ( ) = 1 R( z) j λ o π n w 1 ( z) Also: q z = Aq 1 + B Cq 1 + D 1 q z = C + D 1 q 1 A + B 1 q 1 /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 77

ABCD Law for Gaussian Beams If we assume a beam with a minimum spot size w o and a planar wavefront at z = 0 and utilize the ABCD parameters for free space we can confirm our previous results that: 1 ( ) = 1 q 1 = j 1 z 0 = j q 0 1 q z = 1 z ( ) = C + D 1 q 1 A + B 1 q 1 = λ o π nw, w ( z) = w o o 0 +1 1 q 1 1+ z 1 q 1 = j 1 jz 1 1+ z z + j z z 0 0 1+ z z = 1 0 R z λ o π nw o λ o π nw o ( ) + j λ o 1+ jz 1+ jz 1+ z z 0, R z λ o π nw o λ o π nw o ( ) = z 1+ z 0 Please read the discussion in Verdeyen (pp. 77-79) on gaussian beam transformation by a thin lens. π n w z ( z) = z z 0 + j z z 0 1+ z z 0 z /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 78

V. Gaussian Beams in Cavities /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 79

Gaussian Beams in Simple Stable Resonators Cavities How are the parameters of a Gaussian beam determined by a real cavity? The objective is to choose a set of mirrors and adjust their positions and curvature so that their surfaces exactly match the surfaces of the constant phase-front of the beam. Then, since the rays associated with the Gaussian beam impinge perpendicular to the mirror surface, they will be reflected back on themselves and return to the other, yielding a self-consistent description of a normal mode of the cavity. /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 80

Gaussian Beams in Simple Stable Resonators Cavities Three equations are needed: z 0 = π nw o λ o w R z ( z) = w o ( ) = z 1+ z 0 w o = z λ 0 o π n 1+ z z z 0 /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 81

Gaussian Beams in Cavities TEM 0,0 If we knew w o, we could predict everything about the beam. w o d w ( z ) curvature R( z) R 1 = z = 0 flat phase front here (infinite radius) Note how we ve picked mirrors that exactly match the phase fronts. R /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 8

Gaussian Beams in Cavities We choose the value of w o such that the equiphase surfaces coincide with the choice of mirrors. On the previous slide the flat mirror matches the phase surface at z = 0. We then force the phase surface to match the mirror at z = d. ( ) = R = d 1+ z 0 R d d z 0 = πnw o = R λ d 1 d o R z = d R 0 d 1 = R d 1 d R for n = 1. Note that z 0 and w o are real so long as Once z 0 is known so is w o. 0 d R 1 /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 83

Application of ABCD Laws to Stable Cavities This self-consistent description defines a cavity mode. Definition: A cavity mode is a field distribution that reproduces itself in relative shape and in relative phase after a round trip through the system. Finding these modes rigorously is complicated. 1. Here we assume that the Hermite-Gaussian beams are the characteristic modes of the cavity.. For this to be true we require the complex beam parameter q(z) to repeat itself after a round trip. /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 84

Application of ABCD Laws to Stable Cavities Require the complex beam parameter q(z) to repeat itself after a round trip: q ( z ) = q ( 1 z 1 + Ζ) = q ( 1 z ) 1 q ( 1 z ) 1 = Aq ( z ) + B 1 1 ( ) + D Cq 1 z 1 Cq ( 1 z ) 1 + ( D A)q ( 1 z ) 1 B = 0 B 1 q ( 1 z ) ( D A ) 1 1 ( ) C = 0 q 1 z 1 /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 85

Application of ABCD Laws to Stable Cavities Require the complex beam parameter q(z) to repeat itself after a round trip: B 1 q ( D A ) 1 C = 0 1 q 1 1 q 1 = D A B ± 1 B D A + BC Recall AD BC = 1 1 = D A q 1 B ± 1 B A + D AD + 4BC 4 = D A B ± 1 B ( A + D) 4 4 /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 86

Application of ABCD Laws to Stable Cavities Recall that 1 ( ) = 1 R( z) + j λ o q z π n w z ( z) Compare: 1 ( ) = A D q 1 z 1 B ± j 1 B 1 A + D Radius of curvature: Spot Size: nπ w ( z ) 1 = λ 0 R( z ) 1 = B A D 1 B A + D /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 87

Application of ABCD Laws to Stable Cavities These parameters are found at the plane z 1 where the unit cell starts and stops. T = 1 d + z 1 0 1 1 d z 1 1 f 1 d z 1 f Unit Cell f = R Flat Mirror Flat Mirror Flat Mirror d z 1 d z 1 d z 1 d z 1 d z 1 d z 1 /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 88

Application of ABCD Laws to Stable Cavities General procedure: 1. Assume that Hermite-Gaussian modes are the normal modes of the cavity.. Formulate an equivalent transmission system for the cavity showing at least one round trip. 3. Identify a unit cell. Is the cavity stable? a. The starting point is arbitrary. However the beam parameters ( ) = B R z 1 A D and nπ w ( z 1 ) = B 1 A + D λ 0 correspond to R and w at the corresponding planes in the cavity. Considerable arithmetic can be avoided by an intelligent choice of the unit cell. /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 89

Application of ABCD Laws to Stable Cavities General procedure continued: 4. Force the complex beam parameter to transform into itself after a round trip by the ABCD law. 5. Evaluate R and w via: R( z ) 1 = B A D nπ w ( z ) 1 = λ 0 1 B A + D /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 90

Application of ABCD Laws to Stable Cavities This procedure applies for stable cavities only, i.e., A + D 1 This procedure does not apply to unstable cavities. /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 91

Some Numbers Spot sizes are small at optical wavelengths. π nw o λ o = R d 1 d R, π nw ( d) = λ o R d For: d = 1 meter ( ) ( flat mirror) R = 0 meter reasonably flat R 1 = λ = 63.8 nm 1 d R w ( o 0) = 9.37 10 4 meter ( flat mirror) w( d) = 9.614 10 4 meter ( spherical mirror) /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 9

Some Numbers T = 1 d + z 1 0 1 1 d z 1 1 f 1 d z 1 f = 1 d z 1 f 1 f ( ) ( d z 1) d z 1 1 d z 1 f f A + D = 1 d z 1 f +1 d z 1 f = 1 d z 1 f = 1 d z 1 = 1 d 1 z d 1 R 10 Stable if 0 < 1 d 1 z 1 d 10 < 1 /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 93

Mode Volume in Stable Resonators Another important parameter for a cavity is the mode volume. We ll see that the active atoms in a laser interact with the square of the electric field, hence we would like to know the effective mode volume of a Gaussian beam. Knowing this volume, we can estimate the number of atoms that must be present and radiating to generate a given optical power. What volume does a cavity mode occupy? /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 94

Mode Volume in Stable Resonators Define the mode volume as: E o V = d 0 E( x, y,z) E * ( x, y,z)dx dy dz E o is the peak electric field (occurs on the beam waist and on the optical axis) and will cancel out when defined in this way. Recall that the electric field for mode (m, p) is given by ( ) x = E m,p H m w( z) E x, y, z H p y ( ) w z w o w z ( ) e r w z jkz 1+m+ p ( ) e ( )tan 1 z kr z 0 j R z e ( ) /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 95

Mode Volume in Stable Resonators E o is the peak electric field (occurs on the beam waist and on the optical axis) For a given mode (m, p) E m,p V m,p = d 0 = E m,p E( x, y,z) E * ( x, y,z)dx dy dz d w o w 0 ( z) H m x w( z) H p y ( ) w z e x + y w z ( ) dx dy dz /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 96

Details of the integration: Let E m,p u = x ( ) dx = w ( z ) w z V m,p = E m,p d w o 0 w w = E o m,p ( z) d 0 H m H m du, v = y w z ( ) dy = w ( z ) ( ) w z ( u)e u du ( u)e u du H p dv H p The inner integral can be looked up in most tables: ( u)e u du dz ( ) w z ( v)e v dv dz H m ( u)e u du = m m! π /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 97

Mode Volume in Stable Resonators E m,p w V m,p = E o m,p w = E o m,p d 0 d 0 V m,p = w o H m ( u)e u du ( m m! π )( p p! π ) dz d 0 π m+ p m!p!dz H p ( u)e u du dz = = 1 wo π d area volume length m+ p m!p! HOM Factor /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 98

Mode Volume in Stable Resonators For the previous example where: d = 1 meter R = 0 meter λ = 63.8 nm w o 0 ( ) = 9.37 10 4 V 0,0 = 1 w πd m+ p m! p! = w o o πd0 0!0! ( 9.37 ) 10 4 = π 1 = 1.379 10 6 m 3 = 1.379 10 6 m 3 = 1.379 cm 3 So what? /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 99

Mode Volume in Stable Resonators So what? Suppose we had a neon filled tube with a pressure of 0.1 torr. And that each atom is excited on average of ten times per second via gas discharge which produces a photon at 63.8 nm. The maximum power that we could expect this laser to produce is: Power = Energy Average Excitation Average Emission Number of atoms Photon Atom Atom = hν = hc λ 10 sec Average Excitation Average Emission = 1.96 ev ( 0.1 3.54 10 17 V 0,0 ) Atom Atom = 15.3 mw Number of neon atoms /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 100

Example: Consider a 441.6 nm He-Cd laser operating in the TEM 00 mode in a cavity (flat/curved mirrors) with mirror spacing of d = 0.6 m and a radius of curvature of R = m. Note that the cavity is stable (R > d). Determine the minimum beam waist w o. w o = λ o nπ R d 1 d R TEM 00 mode volume: V m,p = 1 wo π d area volume length m+ p m!p! HOM Factor Not a very large volume, but this is typical! /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 101

VI. Resonance /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 10

Resonance We ve discussed cavities a great deal (we could go on!). We ve noted that cavities are the basic feedback mechanism for a laser. The cavity also ultimately determines the laser frequency via its resonant properties. To understand cavity resonance consider the simplest possible cavity The Fabry-Perot Cavity. /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 103

Resonance The Fabry-Perot Cavity Partially Reflecting Mirror: ρ 1, σ 1, σ 1 = 1 + ρ 1 Partially Reflecting Mirror: ρ, σ, σ = 1 + ρ Incident Plane Wave H i E i ki ρ 1 n 1 n n 3 /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 104

Resonance The Fabry-Perot Cavity ρ Ei 1 E i σ 1 Ei /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 105

Resonance The Fabry-Perot Cavity ρ Ei 1 E i σ Ei σ 1 E 1 i e jk /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 106

Resonance The Fabry-Perot Cavity ρ Ei 1 E i σ 1 E i e jk σ Ei 1 ρ σ Ei 1 e jk σ 1 σ Ei e jk /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 107

Resonance The Fabry-Perot Cavity E i ρ 1 Ei σ 1 Ei σ 1 E i e jk σ 1 σ Ei e jk ρ σ 1 Ei e jk ρ σ 1 Ei e jk /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 108

Resonance The Fabry-Perot Cavity ρ Ei 1 E i n ρ n σ 1 E i e jk 1 σ 1 E i e jk σ Ei 1 ρ 1 ρ σ 1 Ei e jk ρ σ 1 Ei e jk σ 1 σ Ei e jk /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 109

Resonance The Fabry-Perot Cavity ρ Ei 1 E i n ρ n σ 1 E i e jk 1 σ 1 E i e jk σ Ei 1 ρ 1 ρ σ 1 Ei e jk ρ σ 1 Ei e jk ρ 1 ρ σ 1 Ei e j3k σ 1 σ Ei e jk ρ 1 ρ σ 1 σ Ei e j3k ρ 1 ρ σ 1 Ei e j3k /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 110

Resonance The Fabry-Perot Cavity ρ Ei 1 E i n n 1 σ 1 ρ σ 1 Ei e jk n ρ n 1 ρ σ 1 E i e j4k 1 σ 1 E i e jk σ Ei 1 ρ 1 ρ σ 1 Ei e jk ρ 1 ρ σ 1 Ei e j4k ρ σ 1 Ei e jk ρ 1 ρ σ 1 Ei e j3k ρ 1 ρ σ 1 Ei e j3k σ 1 σ Ei e jk ρ 1 ρ σ 1 σ Ei e j3k ρ 1 ρ σ 1 Ei e j4k /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 111

Resonance The Fabry-Perot Cavity ρ Ei 1 E i n n 1 σ 1 ρ σ 1 Ei e jk n ρ n 1 ρ σ 1 E i e j4k 1 σ 1 E i e jk σ Ei 1 ρ 1 ρ σ 1 Ei e jk ρ 1 ρ σ 1 Ei e j4k ρ σ 1 Ei e jk ρ 1 ρ σ 1 Ei e j3k ρ 1 ρ σ 1 Ei e j3k ρ 1 ρ σ 1 Ei e j5k σ 1 σ Ei e jk ρ 1 ρ σ 1 σ Ei e j3k ρ 1 ρ σ 1 σ Ei e j5k ρ 1 ρ σ 1 Ei e j4k ρ 1 ρ 3 σ 1 Ei e j5k /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 11

Resonance The Fabry-Perot Cavity ρ Ei 1 E i n n 1 σ 1 ρ σ 1 Ei e jk n ρ n 1 ρ σ 1 E i e j4k 1 σ 1 E i e jk σ Ei 1 ρ 1 ρ σ 1 Ei e jk ρ 1 ρ σ 1 Ei e j4k ρ σ 1 Ei e jk ρ 1 ρ σ 1 Ei e j3k ρ 1 ρ σ 1 Ei e j3k ρ 1 ρ σ 1 Ei e j5k σ 1 σ Ei e jk ρ 1 ρ σ 1 σ Ei e j3k ρ 1 ρ σ 1 σ Ei e j5k n ρ n 1 ρ 3 σ 1 E i e j6k 1 ρ 1 ρ σ 1 Ei e j4k ρ 1 ρ 3 σ 1 Ei e j6k ρ 1 ρ 3 σ 1 Ei e j5k ρ 1 3 ρ 3 σ 1 Ei e j6k /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 113

Resonance The Fabry-Perot Cavity ρ Ei 1 E i n ρ n σ 1 E i e jk 1 n ρ n 1 ρ σ 1 E i e j4k 1 σ 1 E i e jk σ Ei 1 ρ 1 ρ σ 1 Ei e jk ρ 1 ρ σ 1 Ei e j4k ρ σ 1 Ei e jk ρ 1 ρ σ 1 Ei e j3k ρ 1 ρ σ 1 Ei e j3k ρ 1 ρ σ 1 Ei e j5k σ 1 σ Ei e jk ρ 1 ρ σ 1 σ Ei e j3k ρ 1 ρ σ 1 σ Ei e j5k n ρ n 1 ρ 3 σ 1 E i e j6k 1 ρ 1 ρ σ 1 Ei e j4k ρ 1 ρ 3 σ 1 Ei e j6k ρ 1 ρ 3 σ 1 Ei e j5k ρ 1 3 ρ 3 σ 1 Ei e j7k ρ 1 3 ρ 3 σ 1 σ Ei e j7k /4/14 ρ 1 3 ρ 3 σ 1 Ei e j6k 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 114

Resonance The Fabry-Perot Cavity Transmitted wave (field): N z n = 1 z N +1 n=0 1 z E T = σ 1 σ Ei e jk + ρ 1 ρ σ 1 σ Ei e j3k + ρ 1 ρ σ 1 σ Ei e j5k + ρ 1 3 ρ 3 σ 1 σ Ei e j7k + = σ 1 σ Ei e jk 1+ ρ 1 ρ e jk + ρ 1 ρ e j4k + ρ 3 1 ρ 3 e j5k + ( ) n = σ 1 σ Ei e jk ρ 1 ρ e jk N n=0 = σ 1 σ Ei e 1 ( ρ ρ jk 1 e jk ) N +1 1 ρ 1 ρ e jk N E i e jk σ 1 σ 1 ρ 1 ρ e jk /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 115

Resonance The Fabry-Perot Cavity Transmitted wave (intensity): Note: I T = 1 E η T E * T N 1 σ E 1 σ η i 1 ρ 1 ρ e jk = I i σ 1 σ 1+ ρ 1 ρ ρ 1 ρ cos k ( ) ( )( 1 ρ 1 ρ e + jk ) The field reflection coefficient is ρ The power reflection coefficient is R what we usually use, i.e., R 1, = ρ 1,. R is known as the reflectivity Similarly, T 1, = σ 1,, and for a lossless mirror. T 1, = 1 R 1, /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 116

Resonance The Fabry-Perot Cavity Transmitted wave (intensity): I T = 1 η Let R 1 R = R and consider the case when Also let E T E 1 * 1 R T E 1 N η i 1 R 1 R e jk ( )( 1 R ) ( ) 1 R = I 1 i 1+ R 1 R R 1 R cos k R 1 = R = R, then ( )( 1 R ) ( ) = I i 1 R I T = I 1 i 1+ R Rcos qπ ( )( 1 R ) ( )( 1 R 1 R e + jk ) ( 1 R) 1+ R R = I i k = qπ ( 1 R) ( 1 R) = I i /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 117

Resonance The Fabry-Perot Cavity I T I i = ( 1 R) 1+ R Rcos( k ) /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 118

Resonance The Fabry-Perot Cavity I T I i = ( 1 R) 1+ R Rcos( k ) /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 119

Resonance The Fabry-Perot Cavity I T I i = ( 1 R) 1+ R Rcos( k ) /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 10

Resonance The Fabry-Perot Cavity E T E i = 1 ( ρ ρ 1 e jk) N +1 1 ρ 1 ρ e jk /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 11

Resonance The Fabry-Perot Cavity The distance between peaks is known as the Free Spectral Range (FSR) in Hertz k = n π f o c = qπ ( ) π f ( k + Δk) = n o + FSR c π f n o + FSR c FSR = c n ( ) = ( q +1)π n π f o c = ( q +1)π qπ /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 1

Resonance The Fabry-Perot Cavity /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 13

Resonance The Fabry-Perot Cavity Linewidth (or 3 db Bandwidth, Sharpness): I T I i = = I T I i = ( 1 R) 1+ R Rcos( k ) = cos( x)=1 sin x ( 1 R) ( ) 1+ R R 1 sin k ( 1 R) ( 1+ R R + 4Rsin ( k ) = 1 R) ( 1 R) + 4Rsin ( k ) ( 1 R) ( 1 R) + 4Rsin k 3dB ( ) = 1 ( 1 R) sin( δ k 3dB ) = 1 R R δ k 1 R 3dB R δ f c 3dB 4R = sin k 3dB 1 R 4πn R ( ( )) ( ) = sin k o + δ k 3dB = sin qπ ( δ k 3dB ) δ f 3dB = 1 1 R π R c n Δf = δ f = 1 R 3dB 3dB π R c n /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 14

Resonance The Fabry-Perot Cavity Linewidth (or 3 db Bandwidth, Sharpness): /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 15

Resonance The Fabry-Perot Cavity Linewidth (or 3 db Bandwidth, Sharpness): HPBW = Δf 3dB = 1 R π R c n Free Spectral Range: Finesse (or cavity Q): FSR = F c n FSR HPBW = Δf Δf 3dB = c n 1 R c π R n = π R 1 R /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 16

Photon Lifetime Closely related to the finesse. Represents a time constant describing the build up or decay of energy in the cavity; i.e., the time dynamics of a cavity. Recall: /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 17

Photon Lifetime Closely related to the finesse. Consider a cavity with a packet of N p photons, frequency f o in it at t = 0. The total energy is N p hf o. N p R 1 R N p R N p Index n d The number of photons lost in one round trip is: ΔN p = N p R 1 R N p /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 18

Photon Lifetime Closely related to the finesse. The number of photons lost in one round trip is: ΔN p = N p R 1 R N p ΔN p Δt ( = 1 R R ) N 1 p τ RT lim Δt 0 dn p dt ΔN p Δt dn p dt = lim τ RT 0 = 1 R 1 R τ RT N p = 1 τ p N p ( 1 R 1 R ) N p τ RT τ p = τ RT 1 R 1 R Photon Lifetime /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 19

Photon Lifetime dn p dt = 1 R 1 R τ RT N p = 1 τ p N p N p ( t) = N p ( 0)e t τ p In words, it takes on the order of 1 R 1 R round trips for the energy in the cavity to fall to e -1 of its initial value. If the cavity is lossy (with loss factor α) the fall off is even quicker: ( 1 R 1 R e αd ) 1 ( ) 1 Also note that index n. τ RT = nd c for cavity length d and refractive /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 130

Summary: The Fabry-Perot Cavity R = R 1 R τ p = τ RT 1 R HPBW = Δf 3dB = 1 R π R FSR = c n c n F FSR HPBW = Δf = π R Δf 3dB 1 R Q = f o = π R Δf 3dB 1 R = cπ n τ p /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 131

Some numbers: /4/14 014, Henry Zmuda Set 1 Gaussian Beams and Optical Cavities 13