La movilidad del género con varios “trancones” para el referente masculino

2.2

Multiheterodyne equidistance measurement prin-

ciple

An experimental approach for determination of the frequencies of the four-wave mixing modes in a microresonator is multiheterodyne spectroscopy with an additional frequency comb. In this experiment, several beat notes are generated between the modes from the microresonator and the modes of a stabilized frequency comb (termed reference comb). Since the modes of the reference comb are known to be equidistant, the generated beat notes between adjacent modes from microresonator and reference comb are expected to be equidistant as well in case of an equal spacing of the microresonator modes. Multi- heterodyne spectroscopy has been first developed and used for gas sensing applications by superimposing two frequency combs at slightly different repetition rates and measurement of the radio frequency beat notes [135–137]. Another similar example of this principle is Fourier transform spectroscopy with only one frequency comb, in which a Michelson inter- ferometer with a movable arm is used to increase or decrease the repetition rate of part of the frequency comb, which also enables the generation of low frequency beat notes between the frequency shifted modes and the original modes of the comb [138].

Multiheterodyne spectroscopy typically employs frequency combs with nearly the same repetition rate, such that the frequency difference of the generated beat notes is much smaller than the mode spacing. However, to prove the equidistance of a microresonator based comb with a fiber laser reference comb it has to be taken into account that the mode spacings are different by a factor of around 10,000 (with _≈1 THz mode spacing of the microresonator comb and _≈100 MHz mode spacing of the reference comb). To derive the most general case of beat note generation between two frequency combs, the modes of the two frequency combs fc1 and fc2 can be defined as

fc1 = fceo1+n·frep1 (2.1)

fc2 = fceo2+m·frep2 , (2.2)

with fceo1 and fceo2 being the carrier envelope offset frequencies, frep1 and frep2 the rep-

etition rates and n, m _{∈ N}. Without loss of generality we assume frep1 frep2 and

consequently we can derive the beat note fB0 of the nth mode of comb 1 with the nearest

mode of comb 2 as

fB0 =|fceo2+m0 ·frep2−fceo1−n·frep1| (2.3)

with m0 =

fceo1+n·frep1−fceo2

frep2

, (2.4)

where _dx_cdenotes the rounded value of x (to the closest integer number).

The general principle of multiheterodyne beat note generation is depicted in figure 2.3. To determine the equidistance of one of the combs (eg. comb 1), it is necessary to generate

40 2. Equidistance of the generated comb modes Optical frequency (THz) Reference comb modes Radio frequency (MHz) Microresonator comb modes 100 MHz Po w e r Po w e r ≈1 THz f₀rf f₁rf f₂rf f₀rf f₁rf f₂rf a b

Figure 2.3: Principle of multiheterodyne beat note generation. Panel a. Superimposing two frequency combs with different repetition rates (in this case 1 THz and 100 MHz) leads to the generation of beat notes between the comb modes. The lowest frequency beat notes (Panel b) denoted frf

0 , f1rf and f2rf can be measured with a fast photodiode.

Note that the frequency axis in panel a is broken in order to display several beat notes simultaneously. Approximately 10,000 reference comb modes are in between two adjacent microresonator comb modes.

beat notes of at least three modes of comb 1 with adjacent modes of comb 2. Thus, the additional beat notes of the neighboring modes of comb 1 are defined as fBk:

fBk =|fceo2+mk·frep2−fceo1−(n+k)·frep1| (2.5)

with mk =

fceo1+ (n+k)·frep1−fceo2

frep2

. (2.6)

Figure 2.4 shows the beat notes of four adjacent modes of comb 1 with the neighboring modes of comb 2. In this figure, the repetition rate of comb 1 is fixed at 110 Hz, while the repetition rate of comb 2 is varied. Obviously, the relative frequency of the beat notes depends on the repetition rate of the combs, with degenerate beat note frequencies in case of the repetition rate of comb 1 corresponds to an integer multiple of the repetition rate of comb 2. This also follows from equations 2.4 and 2.6. For beat notes at equal frequencies, it is necessary that fBk−fB0 = 0 ∀ k ∈N. Without loss of generality we assume that the

comb mode of comb 2 is at higher frequency than the neighboring mode of comb 1, which allows for neglecting the absolute value infB0 and fBk. Consequently we obtain

0=! fBk−fB0 = (mk−m0)·frep2−k·frep1

=

fceo1−fceo2+ (n+k)·frep1

frep2

−

fceo1−fceo2+n·frep1

frep2

2.2 Multiheterodyne equidistance measurement principle 41 9.8 10 10.2 10.4 10.6 10.8 11 11.2 0 2 4 6

Repetition rate comb 2, frep2 (Hz)

B ea t no te fr eq ue nc y (Hz ) 9.8 10 10.2 10.4 10.6 10.8 11 11.2 0 2 4 6

Repetition rate comb 2, frep2 (Hz)

B ea t no te fr eq ue nc y (Hz ) a b

Figure 2.4: Multiheterodyne beat notes between two frequency combs with different repetition rate. The repetition rate of comb 1 is fixed at frep1 = 110 Hz, while the

repetition rate of comb 2 is varied (horizontal axis). For convenience, the carrier envelope frequency of both combs is set to zero. The vertical axes show beat notes of four adjacent modes of comb 1 with the nearest neighboring modes of comb 2. Panel a shows beat notes of modes n = 2,3,4,5 of comb 1 with the respective modes of comb 2. Panel b shows beat notes at higher frequencies generated by the modes n= 5,6,7,8 of comb 1. In case of the repetition rate of comb 1 being an integer multiple of the repetition rate of comb 2, the four beat notes are at the same frequency (frep1= 110 Hz is an integer multiple of

frep2 = 10 andfrep2= 11). The shaded regions depict the repetition rate ranges in which

equidistant beat notes can be observed. This is evident in the vicinity of frep2 = 10 and

frep2 = 11. At the outer parts of the shaded region, the beat notes are not equidistant

in this graph, but can still be changed to be equidistant by varying the carrier envelope offset frequency of one of the combs.

42 2. Equidistance of the generated comb modes

It is easy to show that equation 2.7 is satisfied in casefrep1 =`·frep2with`being an integer

value. In figure 2.4 this is the case for frep2 = 10 andfrep2= 11 and the four adjacent beat

notes are equidistant in the vicinity of these repetition rates. An important value is the range in which equidistant beat notes can be observed. This corresponds to the range in which the difference between the beat note frequencies is smaller than half the repetition rate of the comb with smaller mode spacing. The difference between two beat notes ∆fB

close to a degenerate point with frep1 =`·frep2 is given as

∆fB= (mk−m0)·∆frep2 !

< frep2

2 . (2.8)

Inserting mk and m0, we obtain the maximum change ∆frep2 of the repetition rate frep2

that still allows for observation of equidistant beat notes as ∆frep2 ! < f 2 rep2 2k_·frep1 (2.9) with k being the number of simultaneously observed beat notes. Additionally we assume that one of the carrier envelope offset frequencies can be tuned by more than 50% of the smaller repetition rate and thatfrep2 < frep1 as defined before. The shaded region in figure

2.4 depicts the repetition rate ranges in which equidistant beat notes can be observed according to equation 2.9.

Figure 2.5 shows four adjacent beat notes as they are expected between a microresonator based frequency comb with a mode spacing of frep1 = 1 THz and a reference comb with

frep2= 100 MHz. The four beat notes are originating from the 200th to the 203rd mode of

the microresonator comb (which would experimentally correspond to a wavelength around 1500 nm). The lower panel shows the response of the beat note frequencies to a variation of the repetition rate of the reference comb. The plotting range on the horizontal axis corresponds to frep2 = frep1/` with ` = 10,000 and ` = 9,999. Consequently, the beat

notes are at degenerate frequencies at the low and high frequency end of the plot. The upper panels show zooms into the data at different frequencies in between these degenerate points. Here, it can be seen that it is likely to not experimentally observe equidistant beat notes between the two frequency combs unless the repetition rate of the reference comb is adapted to an adequate value according to equation 2.9.