LAS TROPAS BELGAS DEBEN ABANDONAR EL CONGO

To derive an expression for the intra-cavity field we can start from the input field, with amplitude↵_in, and follow its propagation within the cavity after entering from the first mirror. Let r₁, r₂ and t₁, t₂ be respectively the Fresnel reflection and transmission coefficients of the two end mirrors, which we identify by the subscript i 2 {1,2}. We then define the reflectivities _Ri = _|r_i|2 and the transmissivities _Ti = _|t_i|2, and consider each mirror to have other scattering or absorption losses described by _Li so that the relationship Ri +Ti +Li = 1 always holds true [41]. Also, let µ be

the attenuation coefficient within the cavity. The distance between the two reflective surfaces determines the length of the cavity, L₀. A cavity resonates only when its length is an integer multiple of the half-wavelength /2, or else boundary conditions would not allow the fully resonant build-up of a stationary wave.

A diagram for the following discussion is provided in Fig. 2.1. Initially, the light inside the cavity is produced by the transmission of the input field through the input mirror (leading to ↵₀ =t₁↵_in). The light then propagates for the length of the cavity (gathering a phase shift equal toeikL0 and an attenuation of e µL0), is reflected at the

second mirror, propagates back to the first mirror, and is reflected once more (resulting into↵₁). The total round-trip time of these steps is ⌧ ..= 2L₀/c. Multiple passes (↵_n) keep repeating the same process until the field leaks outside of the resonator due to the losses and residual transmissivity of the mirrors. This is translated into the equations

𝛼in 𝛼0𝛼1 𝛼n

𝑟1, 𝑡1 𝑟2, 𝑡2

ℓ1

ℓc ℓ2

Figure 2.1: Schematic of a Fabry–P´erot cavity displaying resonance via the build-up of inter- ference through subsequent passes.

for the field at the nth pass as

↵_{in !} ↵₀ =t₁↵_in ! ↵₁ =r₁e( µ+ik)L0_r 2e( µ+ik)L0t1↵in . . . ! ↵_n=rn₁en( µ+ik)L0_rn 2en( µ+ik)L0t1↵in. (2.36)

The phase accumulated after a single round trip is ₀ ..= k·2L₀, where k ..= 2⇡/

is the wave number. Analogously, the losses due to attenuation within the cavity are described by`<sub>c</sub>..=µ·2L<sub>0</sub>. The total cavity field, as a consequence of the superposition principle, is given by the sum of each single pass contribution. The transmitted and reflected fields can also be obtained by using Fresnel conditions at the first and second mirror, respectively. ↵<sub>cav</sub>= 1 X n=0 ↵<sub>n</sub>= t1 1 r<sub>1</sub>r<sub>2</sub>e `c+i 0↵in, (2.37) ↵_tra=t₂↵_cav = t1t2 1 r₁r₂e `c+i 0↵in, (2.38) ↵<sub>ref</sub> = r⇤<sub>1</sub>↵<sub>in</sub>+t<sub>1</sub>⇤r<sub>2</sub>e( µ+ik)2L0<sub>↵</sub> cav = r⇤1+ r<sub>2</sub>|t1|2e `c+i 0 1 r₁r₂e `c+i 0 ! ↵_in. (2.39)

To infer the power, which we know is proportional to the absolute value of the corre- sponding field thanks to Eq. 2.18, we first introduce thecoefficient of finesse

f ..= 4

p

R1R2e `c

1 p_R1R2e `c 2

and then express everything in terms of the input power P_in: P_cav= T1 1 p_R1R2e `c 2 1 +fsin2( <sub>0</sub>/2) P<sub>in</sub>, (2.41) P<sub>tra</sub>= T1T2 1 p<sub>R1R2</sub>e `c 2 1 +fsin2( ₀/2) P_in, (2.42) P_ref= R1+ (1 L1) 2 R2e `c<sub>+ 2T1</sub>p<sub>R1R2e</sub> `c_cos( 0) 1 p_R1R2e `c 2 1 +fsin2( ₀/2) P_in. (2.43)

We can use the quantities introduced during the derivation to identify certain dis- tinctive properties of the cavity field. The frequency-domain equivalent of the phase 0 accumulated at each pass, commonly referred to as free spectral range, can be defined from the inverse of the round-trip time⌧,

!_FSR..= 2⇡

⌧ =

⇡c

L₀. (2.44)

The free spectral range corresponds to the spacing in frequency between di↵erent reso- nances of the cavity, a direct consequence of the periodicity of Eq. 2.41. The coefficient of finesse gives a measure of the quality of the resonance, as large values of f corre- spond to a higher build-up of constructive interference into a narrower portion of the free spectral range. To have a measure of the spectral width of the resonance is, we can consider the phase for which the cavity power is half of its maximum resonant value, i.e. the that satisfies P_cav|

0= ⌘

1

2 Pcav| ₀=0. The equivalent of this phase in the frequency domain is the cavity half-linewidth,

!

2 ..=!FSR

arcsin(1/pf)

⇡ . (2.45)

Together, the free spectral range and the cavity linewidth can be used to define an optical analogue of a quality factor to indicate the number of reflections that light undergoes before escaping from the resonator. This is known as the finesse:

F ..= !FSR_! = ⇡

2 arcsin(1/pf). (2.46)

The definition of Eq. 2.40 suggests that f, and consequently _F, can be specified by a unique parameter pR1R2e `c_{. Having the loss factors corresponding to each mirror}

indicated by`<sub>i</sub>, chosen such thate `i ₌p_R

0.0 0.5 1.0 1.5 2.0 100 50 10 5 2 1 20 ℓ ℱ

Figure 2.2: Finesse_F as a function of total cavity losses`. The approximated expression for

F (dashed line) significantly diverges from the original definition only for `&1.

the total losses of the cavity. For`⌧1, the finesse can then be approximated to

F ' ⇡ p

e `

1 e `. (2.47)

In Fig. 2.2, where Eq. 2.46 and 2.47 are compared, we see that the approximation does not require extreme system purities to hold. As few as five reflections before the field leaks out of the cavity are enough to make the two results indistinguishable. In this approximation, the intra-cavity power can be expressed as

P_cav' T1

1 +4_⇡F22 sin2( 0/2) F2

⇡2Pin. (2.48)

The reduction induced by the necessarily low transmissivity of the input mirror is compensated by the square of the finesse, and as a rule of thumb the intra-cavity power scales as F times the input power.

The cavity dynamics obtained so far are general enough for regular purposes, but we may sometimes be interested in scanning through the cavity length (or equivalently the optical frequency) and the speed of the scan might be high enough that it might a↵ect the regular build-up of resonance. To make provisions for the ring-down inter- ference e↵ects arising, we can look back at Eq. 2.36 and consider a length which is now dependent on the number of reflections of the light within the cavity. Using the initial cavity length L₀ as reference, we consider a scan actuated through motion of the input mirror at speedv. When light is reflected at the second mirror after travers- ing the cavity once, the length is altered to L₀(1 v/c). When light has travelled again to return to the input mirror, the correction due to the back-and-forth reflec- tion gives a revised length L₁ = L₀(1 v/c)/(1 +v/c), which can be linearized to

L_{1 '}L₀(1 2v/c) for a mirror travelling much slower than light. Successive iterations give a lengthLn'L0(1 2nv/c) at thenth pass. The field amplitude is revised to

↵_{in !} ↵₀ =t₁↵_in ! ↵₁ =r₁e( µ+ik)L0(1 2v/c)_r 2e( µ+ik)L0t1↵in . . . ! ↵_n=r₁e( µ+ik)L0(1 2nv/c)_r 2e( µ+ik)L0[1 2(n 1)v/c]↵n 1 =t₁(r₁r₂)ne(n n2v/c)( `c+i 0)_↵ in. (2.49)

The full series remains unresolved due to the quadratic exponent and the intra-cavity field cannot be expressed analytically. An accurate comparison with Eq. 2.37 can still be performed if we consider that after a number of reflections comparable to the finesse most of the light has escaped and the series can be truncated to the first leading terms. However, this method could prove computationally demanding and unappealing, especially in light of an alternative, more efficient method to describe the intra-cavity field presented in the next section.