Resonator optics - ULisboa · Exercise - Resonator Modes and Spectral Width! Calculate:! • frequency spacing ν F! • spectral width δν! of the modes of a Fabry-Perot resonator

Fabry-Perot: lossless Fabry-Perot: lossy Spherical mirror

1

Resonator optics!Fabry-Perot resonator!

•  no losses / with losses!•  resonator modes!•  spectral width and finesse!•  resonator lifetime and quality factor!

!Spherical mirror resonators!

•  ray confinement!•  Gaussian modes!

Fundamentals of Photonics, Ch. 10!


2

An optical resonator confines and stores light!

… but not any kind of light. The configuration of the resonator determines its resonance frequencies.!


3

Applications of optical resonators!

Can be used as a frequency analizers or optical filters!One of the main uses is in laser resonators.!

Fabry-Perot interferometer HeNe laser cavity


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The simplest type of resonator is the Fabry-Perot!

First we are going to study the case where there are no losses.!

It is composed of two parallel, highly reflective, flat mirrors separated by a distance d.!

Charles Fabry (1867-1945)

Alfred Perot (1863-1925)


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A resonator only allows certain modes!

Consider a monochromatic wave U(r) that satisfies the Helmholtz eq.:!!!!The resonator modes (allowed U(r)) are the solutions under the boundary conditions defined by the geometry of the resonator.!Lossless mirrors U(r)=0 at the mirrors!�

u(r,t ) = Re U(r)exp(i2πνt ){ }

�

U(r) = 0

�

U(r) = A sin(kz ), kd = qπ (q = 1,2,…)

kq = q πd

Uq (r) = Aq sin(kqz )

The modes are standing waves and q=1,2… is the mode number!


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Resonance frequencies and frequency spacing!

Just as the wavenumber k is restricted to discrete values kq, so is the frequency:!

�

ν = c λ = kc2π

kq = q πd

⇒ ν ≡ νq = q c2d

resonance frequencies

frequency spacing = free spectral range (FSR)

c = c0/n speed of light in the medium!!


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Resonance: examples!Consider a 30 cm long resonator filled with air (n=1):!

d = 30 cm νF = c2d

= 500 MHz

λq =2dq

= 60 cm, 30 cm, 20 cm, 15 cm…

For a much smaller resonator (30 µm long) we have!

�

d = 3 µm, νF = c2d

= 50 THz

λq = 2dq

= 6 µm, 3 µm, 2 µm, 1.5 µm…


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Resonator modes must repeat themselves at the same place!

If a given wave is a mode, it must repeat itself after a roundtrip.!In terms of wave optics:!

�

ϕ = 2dk (= q2π)

⇒ k ≡ kq = q πd

This can be understood as a feedback mechanism: only similar waves will add up and build power (i.e. resonate) in the resonator.!


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Real resonator have losses!1. Imperfect reflection at the end mirrors!

•  reflection <100% (e.g. partially reflecting mirror)!•  finite size of the mirror aperture!

2. Absorption and scattering in the medium in-between!!One case when we want to introduce losses is the laser:!


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Calculating the effect of losses: amplitude!

Losses may affect the amplitude and the phase: they are accounted for by a complex roundtrip attenuation factor:!!We have for the consecutive amplitudes U1, U2…!

�

h = r e−iϕ

�

Un = hUn−1 = hnU0

U =U0 +U1 +U2 +

=U0(1+h +h2 +) =U0 /(1− h)

The phase shift φ corresponding to a full roundtrip is:!

�

ϕ = k(2d ) = (2πν /c)(2d ) = 4πνd /c


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Resonator with losses: intensity and finesse!

Let’s now calculate the corresponding intensity:!

�

I = U 2 =U0

2

1− r e−iϕ 2 = I01+ r 2 − 2r cosϕ

We can write this result in a more interesting way:!

�

I = Imax1+ 2F / π( )2 sin2(ϕ / 2)

Imax = I01− r( )2

F =π r1− r

�

F =π r1− r

This is the definition of the finesse. We will see that this parameter is fundamental in defining the characteristics of a resonator.!

sin2(ϕ / 2) = 121− cosϕ( )


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Intensity vs. phase / Finesse!

0 5 10 15 20

0.2

0.4

0.6

0.8

1.0

F=2

F=10 F=50

2π 4π 6π φ

Imin =Imax

1+ 2F / π( )2


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Intensity vs. frequency of a monochromatic wave!

�

I(ν) = Imax1+ 2F / π( )2 sin2(πν / νF )

, νF = c / 2d

�

νq = qνF

frequency spacing

�

δν ≈ νFF

spectral width

(lossless)

(lossy)

ϕ2= 2πνd

c= πννF


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Finding a simpler expression for the finesse!

We can write the losses in the medium as a distributed loss proportional to d!The losses in the mirrors are considered located losses at the mirrors 1 and 2!

�

exp(−2αsd )

R1 = r12, R2 = r2

2

The total losses are then:!

�

r 2 = R1R2 exp(−2αsd )= exp(−2αrd )

→αr = αs + 12d

ln 1R1R2

⎛ ⎝ ⎜

⎞ ⎠ ⎟

If we replace |r |2 in the expression for the finesse we obtain (approx. valid for αrd<<1):!

�

F = πexp(−αrd / 2)1− exp(−αrd )

≈ παrd

αr = loss coefficient!


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Finesse vs. r !


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Exercise - Resonator Modes and Spectral Width!

Calculate:!•  frequency spacing νF!•  spectral width δν!of the modes of a Fabry-Perot resonator whose mirrors have reflectances R1=0.98 and R1=0.99 and are separated by a distance d = 100 cm.!Assume that the medium has refractive index n = 1 and negligible losses.!Is the derived approximation appropriate in this case?!


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Why do resonator losses cause spectral line broadening?!

Consider the expression for the spectral width:!!Note that cαr has dimensions of (time)-1. We define the characteristic decay time as:!!The relation between time width and spectral width has the form of an uncertainty product:!

�

δν ≈ νFF

≈ (c / 2d )(π /αrd )

= cαr

2π

�

τp = 1cαr

resonator or photon lifetime

�

δν ⋅ τp = 12π

0 1 2 3 4 5

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

0.2

0.4

0.6

0.8

1.0

�

∝ exp −t / 2τp( )

�

∝ 11+ (4πντp )

2τp

�

δν

Lorentzian lineshape

FT


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The quality summarizes the resonator characteristics!

�

Q = 2π stored energyenergy lost per cycle

The energy decays exponentially with τp (check expression for E-field, prev. slide):!The energy lost in a cycle (=period T) is:!

�

E(t )∝E0 exp(−t / τp )ΔE = (dE /dt ) ⋅T

So we obtain:!Q = (storage time τp ) / (optical period T)!Q = 2π

(1/τp ) ⋅T= 2πτpν0 =

ν0δν

Finally, using the previous relations, we can arrive to an expression relating!•  quality factor!•  frequency spacing!•  finesse!

�

Q = ν0δν

= ν0νF

F


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Fabry-Perot resonators: summary!

•  Lossless resonator: only selected frequencies are allowed!

•  Spacing between modes (FSR)!

�

νq = q c2d

�

νF = c / 2d

�

δν ≈ νF /F

�

αr (cm−1)τp = 1/cαr (s)

�

F = π /αrdQ = (ν0 / νF )F

•  Lossy ressonator: the resonance line is broadened – the spectral width is!

•  The losses may be described by two parameters!

•  Resonator quality is characterized by two dimensionless parameters!

LOSS

LESS!

LOSS

Y!


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Extra: Fabry-Perot interferometer!In this case a plane wave comes from outside and is transmitted through a mirror with amplitude transmittance t and amplitude reflectance r :!

U0 U0t2e−iϕ

U0t2r 2e−i 3ϕ

U0t2r 4e−i 5ϕ


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By making calculations similar to the FP resonator, we obtain a similar expression for the transmitted intensity!!This depends on the!separation d betweenthe mirrors!

It =I0

1+ 2F / π( )2 sin2(ϕ / 2)

0 5 10 15 20

0.2

0.4

0.6

0.8

1.0


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In particular, this means that the reflectance coefficient of the mirrors is not absolute!!

Detector:!1%!Laser!

Detector:!100%!Laser!

r = 0.99!

r = 0.99! r = 0.99!


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Spherical mirror resonators!

�

R1 < 0, R2 < 0Pay attention! In this case we have:


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Conditions for ray confinement!In ray matrix theory we studied the conditions for a bounded solution in a periodic system.!

For a spherical mirror we have the confinement condition!

�

0 ≤ g1g2 ≤ 1, g1,2 = 1+ dR1,2

⎛

⎝ ⎜ ⎞

⎠ ⎟

�

0 ≤ g1g2 ≤ 1

g1g2< 0> 1

⎧ ⎨ ⎩

g1g2= 0= 1

⎧ ⎨ ⎩

stable resonator!

unstable resonator!

conditionally stable resonator!

The stability depends on the product of the g-parameters!


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Diagram of resonator stability!

�

g1,2 = 1+ dR1,2

⎛

⎝ ⎜ ⎞

⎠ ⎟

R g ∞ 1 -d 0

-d/2 -1

The dotted red line marks the position of symmetric resonators for which g1 = g2


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Symmetric resonators: confinement condition!

In this case R1=R2=R and g1=g2=g :!

�

0 ≤ g2 ≤ 1 ⇔ 0 ≤ d(−R)

≤ 2

A stable symmetric resonator must use mirrors with a radius of curvature greater than (length/2).!Example: symmetric confocal ( R = -d )!


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Gaussian beams: quick reminder!

�

z0 = πW02

λ

W (z ) =W0 1+ zz0

⎛ ⎝ ⎜

⎞ ⎠ ⎟ 2

R(z ) = z 1+ z0z

⎛ ⎝ ⎜

⎞ ⎠ ⎟ 2⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

ζ(z ) = tan−1 zz0

W0 = λz0π

�

I (ρ,z ) = I0W0

W (z )exp − ρ2

W 2(z )⎡

⎣ ⎢

⎤

⎦ ⎥

×exp −i kz − ζ(z )( )[ ]×exp −ik ρ2

2R(z )⎡

⎣ ⎢

⎤

⎦ ⎥


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The Gaussian beam is a mode of the spherical mirror resonator!

A Gausian beam will be reflected at a spherical mirror and retrace its way back exactly if:!

Rwavefront = Rmirror!


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Parameters of the Gaussian beam that obeys the boundary conditions:!

Conditions for Gaussian beam confinement!

We have the following three conditions linking the positions z and radius R1, R2 of the concave mirrors, and the R(z) of the beam:!!!!!!

�

z2 = z1 + dR1 = z1 + z0

2 / z1(−R2 ) = z2 + z0

2 / z2

�

z1 = −d R2 +d( )R2 +R1+2d

z02 = −d R1+d( ) R2 +d( ) R1+R2 +d( )

R2 +R1+2d( )2

W0 = λz0 / π

Wi =W0 1+ ziz0

⎛⎝⎜

⎞⎠⎟

2

, i = 1,2

R2<0, R(z)>0 R1<0, R(z)<0

This gives us the position of the beam center and Rayleigh length:


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Exercise – plano-concave resonator!

When mirror 1 is planar (R1 = ∞), determine as a function of d / |R2|:!•  the confinement condition!•  the depth of focus (= 2z0)!•  beam width at the waist and at each of the mirrors!

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Resonator optics - ULisboa · Exercise - Resonator Modes and Spectral Width! Calculate:! • frequency spacing ν F! • spectral width δν! of the modes of a Fabry-Perot resonator