DE102021112017A1 - Process for active stabilization of the effective length of optical time and frequency transmission links - Google Patents

Abstract

The invention relates to a method for actively stabilizing the optical path length of optical time and frequency transmission paths, with the steps: (a) modulation of a modulator signal with a modulator frequency (fc) onto an optical input frequency (fa) of input light (20) , which is introduced into a reference wave input (12), (b) emitting light with the modulated optical frequency (fa+fc) onto a transmission path, in particular into an optical fiber (16) leading to a useful output (18), and (c ) Determining an HF beat frequency (fr) between the input frequency (fa) of reference light (20') and a reception frequency (frück) of return light (30) which is reflected at the useful output (18) and through the light guide (16) was returned, according to the invention (d) mixing the HF beat frequency (fr) with a local oscillator frequency (fl) in a superheterodyne receiver, so that an intermediate frequency (fg = fr- fl) arises, (e) digital Bes tuning an intermediate frequency phase (φg) of the intermediate frequency (fg) and (f) generating the modulator signal based on the intermediate frequency phase (φg) so that a change in the optical path length of the light guide (16) is compensated for.

Description

The invention relates to a method for actively stabilizing the effective length of optical time and frequency transmission links according to the preamble of claim 1. According to a second aspect, the invention relates to an optical signal transmission device according to the preamble of claim 8.

In order to be able to use an optical frequency at a useful output spaced apart from the frequency source, light guides, in particular glass fibers, are usually used. The optical path length that the input light covers on its way to the useful output can fluctuate over time if the light guide changes. This is particularly the case when the light guide--as provided according to a preferred embodiment--has a length of at least 10 meters, in particular at least 100 meters.

A cause for such a change can be, for example, temperature fluctuations, air pressure fluctuations, vibrations or changes in the composition of the environment around the light guide. A change in the optical path length changes the frequency and/or the phase at the useful output. It is therefore desirable to compensate for influences on the optical path length in the light guide.

From the DE 10 2008 062 139 B4 is known to generate a correction signal from a beat frequency between the input light and the return light at the useful output, by means of which the frequency received at the useful output is corrected. The disadvantage is that 2π phase errors, so-called cycle slips, can occur, i.e. the effect that the receiver experiences a phase shift of 2π at the useful output. This leads to an error in the transmission of the optical time or frequency signal.

Another disadvantage of known methods is the comparatively high sensitivity to interference.

The object of the invention is to improve the transmission of optical frequencies.

The invention solves the problem by a method having the features of claim 1. It is advantageous if the modulator signal is generated on the basis of the intermediate frequency phase by means of direct digital synthesis. Alternatively, the modulator signal can be generated by analog synthesis.

The invention also solves the problem by a method having the features of claim 3 or the features of claims 1 and 3. In this case the beat frequency phase φ _r is preferably the intermediate frequency phase φ _g of an intermediate frequency f _g and is determined by (a) mixing the RF beat frequency f _r with a local oscillator frequency _fl in a heterodyne receiver to produce the intermediate frequency, and (b) digitally determining an intermediate frequency phase φ _g of the intermediate frequency f _g .

According to a second aspect, the invention solves the problem by an optical signal transmission device having the features of claim 8. The signal generator is then preferably designed to generate the modulator frequency by means of direct digital synthesis. Alternatively, the signal generator is designed to generate the modulator frequency by means of analog synthesis.

The invention also solves the problem by an optical signal transmission device having the features of claim 10 or the features of claims 8 and 10. In this case, the receiver is preferably a heterodyne receiver that is designed to digitally determine a phase difference between the reference light and Return light that was reflected at the useful output and returned through the light guide.

In particular, the invention uses superheterodyne detection of phase differences between the input light and the return light.

In the context of the present description, a modulator signal is understood to mean, in particular, an electrical signal. The modulation can be acousto-optical or electro-optical. HF stands for high frequency.

The advantage of the solution according to the invention is that phase disturbances, ie variations in the optical path length between the useful output and the reference wave input, can usually be compensated for with a high bandwidth.

In addition, the sensitivity to fluctuations in the optical power at the reference wave input and/or the useful output, which is generally low compared to many known methods, is advantageous.

It is also advantageous that—according to a preferred embodiment of the invention—the signal strength of the HF beat frequency can be detected. It is then possible to identify those time intervals where there was not a sufficiently large signal such that interference is not corrected could become. In other words, the time intervals in which it was possible to compensate for the change in the optical path length of the light guide can be determined. The determination of these time intervals represents a preferred embodiment of the method according to the invention. It is then possible to transmit the optical frequency with a small degree of uncertainty in this corresponding interval.

It is also favorable that the local oscillator frequency f _l can be selected in such a way that the intermediate frequency f _g is above a transition frequency of the analog electrical 1/f noise, in particular above 100 Hertz. The choice of such a local oscillator frequency represents a preferred embodiment of the method. In this case, the transmission of the optical frequency is largely insensitive to this 1/f noise.

It is favorable if the local oscillator frequency f _l is generated by direct digital synthesis. It is favorable if the high-frequency local oscillator is connected to a fed-back frequency source and generates the local oscillator frequency f _l using the frequency of the frequency source. Such high-frequency local oscillators provide a low level of uncertainty when generating the HF frequency. Since the optical frequency is at least three orders of magnitude larger, the contribution of the uncertainty of the generation of the HF frequency to the uncertainty of the optical time and frequency signal at the useful output is extremely small.

According to a preferred embodiment, the method comprises the step of splitting the input light by means of a beam splitter and reflecting the split light beam by means of a reflector element, so that reference light is produced, with the determination of the RF beat frequency being a superimposition of the reference light and the return -Light on a detector element includes.

Alternatively, the method may include the steps of (a) splitting the input light using a beam splitter and (b) reflecting the split light beam using a reflector element to form the reference light.

It is favorable if the beam splitter and the reflector element are rigidly connected to one another by means of a connecting structure. The path between the beam splitter and the reflector element on the one hand and the light guide on the other hand as well as the beam splitter and the detector element form a Michelson interferometer. This one has strongly asymmetrical arms. Due to the fact that the beam splitter and the reflector element are rigidly coupled to one another, the reference distance is kept constant with little uncertainty.

The reference light can also be generated in other ways; it is advantageous if the reference light is in a fixed phase relationship to the input light.

It is particularly favorable if the beam splitter, the reflector element and the connecting structure are kept at a predetermined temperature. In other words, the beam splitter, the reflector element and the connection structure are thermally stabilized. It is favorable when temperature fluctuations are less than 20 mK/s.

It is favorable if the determination of the intermediate frequency phase comprises the following steps: (a) low-pass filtering of a mixture signal which is produced by mixing the RF beat frequency with the local oscillator frequency, (b) digitizing the low-pass filtered mixture signal and (c) determining the Intermediate frequency phase using a phase detection device, in particular a quadrature demodulator or a lock-in detector. Since the local oscillator frequency is known with little uncertainty, the mixed signal also contains the phase information with little uncertainty. The frequency signal containing the difference in the phase of the input light and the return light at the detection element is obtained by the low-pass filtering. The intermediate frequency phase can be determined from this signal with high accuracy using the phase detection device.

The modulator signal has a modulator frequency f _c and a modulator frequency phase φ _c . Generating the modulator signal based on the intermediate frequency phase preferably includes the step of controlling a direct digital synthesizer by means of a controller, in particular a PI controller, based on the intermediate frequency phase to a target phase. This allows the frequency and phase of the modulator signal to be controlled so that optical path length changes are quickly compensated. The control characteristics of the method avoid 2π phase errors (cycle slips) even for high-frequency phase fluctuations.

The local oscillator frequency is preferably between f _l =1 megahertz and f _l =100 gigahertz. Thus, the intermediate frequency f _g is preferably between 100 hertz and preferably at most 30 megahertz. Such frequencies can easily be digitized.

The frequency f _c of the modulator signal is preferably between 1 megahertz and 100 gigahertz.

It is favorable if the method includes the step of modulating the input light at the useful output with a second modulator signal. The second modulator signal is preferably also generated by direct digital synthesis. Because the input light is modulated with the second modulator signal only at the useful output, it is ensured that light at a frequency that results from both modulators was reflected at the useful output.

If a second modulator signal is used, its frequency is preferably in the range below 100 gigahertz and preferably above 1 megahertz.

The receiver is preferably a heterodyne receiver. The superheterodyne receiver is designed in particular to digitally determine a phase difference e between the reference light and return light that was reflected at the useful output (18) and fed back through the light guide (16).

The heterodyne receiver preferably has (a) a detection element arranged to mix the input light with the return light so that an RF beat frequency exists, (b) a high frequency local oscillator for generating a local oscillator frequency, (c) a mixer for mixing the RF beat frequency and the local oscillator frequency to produce an intermediate frequency, and (d) phase detection means for determining an intermediate frequency phase of the intermediate frequency. The signal generator generates the modulator signal, in particular the modulator frequency, preferably based on the intermediate frequency phase.

It is favorable if the superheterodyne receiver has a digital lock-in detector for determining a phase difference in the form of double the phase difference between the return light and the reference light. The signal generator preferably has a direct digital synthesizer. The direct digital synthesizer is preferably controlled by means of a controller, in particular a PI controller, based on the phase difference e.

The frequency transmission device preferably has a beam splitter for splitting the input light and/or a reflector element for reflecting the split light beam so that reference light is produced. It is favorable if the beam splitter and the reflector element are rigidly coupled to one another by means of a connection structure. This reduces the measurement uncertainty and/or the structure. For the same reasons, a temperature stabilization unit arranged to keep the beam splitter, the reflector element and the connection structure at a predetermined temperature is advantageous.

The optical frequency transmission device can have a second modulator, which is arranged at the useful output and designed to modulate a second modulator frequency f _d . The second modulator frequency f _d is preferably at most 100 GHz.

According to a preferred embodiment, the optical frequency transmission device comprises an optical frequency source which has a frequency instability of at most 10 ^-13 , in particular 10 ^-14 , preferably 10 ^-15 , each based on a determination time of one second.

• 1 ein Schaltbild der erfindungsgemäßen optischen Frequenzübertragungsvorrichtung mit dem Durchführen eines erfindungsgemäßen Verfahrens.

The invention is explained in more detail below with reference to the attached drawing. while showing

• 1 a circuit diagram of the optical frequency transmission device according to the invention with the implementation of a method according to the invention.

1 shows an optical frequency transmission device 10, which has a reference wave input 12, a modulator 14, a transmission path 16, for example in the form of an optical fiber, and a useful output 18. The method also works for free jet distances.

The reference wave input 12 is designed to introduce input light 20, which is usually in the form of a laser beam. It is possible, but not necessary, for the reference wave input to have a connection piece 22.1, by means of which a feed light guide 24 can be connected.

The feed light guide 24 is connected, for example, to a frequency standard (not shown), in particular an optical atomic clock.

The modulator 14 is arranged behind the reference wave input 12 in a light propagation direction L and is designed to modulate the input light 20 with a modulator signal. The input light 20 has an input frequency f _a . and an input phase φ _a . The modulator signal has a modulator frequency f _c and a modulator phase φ _c .

Transmission in an optical fiber 16 is preferably transmission in a glass fiber. The light guide 16 is connected to the modulator 14 for guiding light emitted from the modulator 14 . A length l ₁₆ of the light guide 16 is preferably at least 10 centimeters, in particular at least 3 meters, in especially at least 30 meters, more preferably at least 300 meters. As a rule, the length l ₁₆ is less than 40,000 km.

The useful output can also have a connection piece 22.2, by means of which a discharge light guide 26 can be connected, but this is not necessary. The only decisive factor is that light can emerge from the light guide 16 in such a way that it can be used for another application.

The optical frequency transmission device 10 also includes a beam splitter 28, by means of which the rear light 30 can be coupled out. Back light 30 is light that has run from the modulator 14 through the light guide 16 and has been reflected by a reflector 19 at the useful output 18 .

The beam splitter 28 also allows reference light 20 ′ to be generated from the input light 20 . For this purpose, part of the input light 20 is reflected by the beam splitter 28 onto a reflector element 33 . The beam splitter 28 and the reflector element 33 are rigidly coupled to one another by means of a connection structure 31 shown schematically.

The return light 30 is fed to a heterodyne receiver 32 which has a detection element 34, a radio frequency local oscillator 36, a mixer 38 and filter 46, an analog to digital converter 48 and a phase detection device 40.

The return light 30 is superimposed on the detection element 34 with reference light 20 ′, which was decoupled from the input light 20 at the beam splitter 28 , for example.

The optical frequency transmission device 10 may include a second modulator 42, but this is not necessary. If the second modulator 42 is present, then it changes the frequency of the useful light 44 at the useful output 18 by a second modulator frequency f _d . The useful light 44 emerging from the useful output 18 has a useful light frequency f _b $${\text{f}}_{\text{b}}={\text{f}}_{\text{a}}+{\text{f}}_{\text{c}}+{\text{f}}_{\text{i.e}}.$$

It should be noted that the formula does not include a frequency component for any interference, as this is eliminated as described below.

The superimposition of the reference light 20' and the return light 30 on the detection element 34 results in a number of mixed frequencies. One of the mixing frequencies is the HF beat frequency f _r mit $${\text{f}}_{\text{right}}=2\left({\text{f}}_{\text{c}}+{\text{f}}_{\text{n}}+{\text{f}}_{\text{i.e}}\right).$$

This contains an interference frequency f _n which is caused by a change in the optical path length of the transmission path 16, in particular the glass fiber. The phase angle φ _r of the HF beat frequency f _r is $${\mathrm{\phi }}_{\text{right}}=2\left({\mathrm{\phi }}_{\text{c}}+{\mathrm{\phi }}_{\text{n}}+{\mathrm{\phi }}_{\text{i.e}}\right).$$

mit der Zwischenfrequenz-Phase $${\mathrm{\phi }}_{\text{g}}={\mathrm{\phi }}_{\text{r}}-{\mathrm{\phi }}_{\text{i}}=2\left({\mathrm{\phi }}_{\text{c}}+{\mathrm{\phi }}_{\text{n}}+{\mathrm{\phi }}_{\text{d}}\right)-{\mathrm{\phi }}_{\text{I}}.$$

After mixing with the local oscillator frequency f _l , a frequency called an intermediate frequency f _g results as one of the mixing frequencies $${\text{f}}_{\text{right}}={\text{f}}_{\text{right}}-{\text{f}}_{\text{i}}=2\left({\text{f}}_{\text{c}}+{\text{f}}_{\text{n}}+{\text{f}}_{\text{i.e}}\right)-\text{fi}$$

with the intermediate frequency phase $${\mathrm{\phi }}_{\text{G}}={\mathrm{\phi }}_{\text{right}}-{\mathrm{\phi }}_{\text{i}}=2\left({\mathrm{\phi }}_{\text{c}}+{\mathrm{\phi }}_{\text{n}}+{\mathrm{\phi }}_{\text{i.e}}\right)-{\mathrm{\phi }}_{\text{I}}.$$

The local oscillator frequency f _l is adjusted in such a way that in the present case an intermediate frequency of f _g ~10 to 10,000 kilohertz results. However, higher intermediate frequencies are also possible. The intermediate frequency f _g is preferably above the maximum phase change speed dφ _n /dt to be expected, which is introduced by the light guide 16 . In addition, the intermediate frequency f _g is preferably below twice the sum of the frequencies of the modulator signal. It preferably applies $${\text{f}}_{\text{G}}\le 2\left({\text{f}}_{\text{c}}+{\text{f}}_{\text{i.e}}\right).$$

The intermediate frequency f _g is digitized by an analog/digital converter 48 . The analog-to-digital converter 48 emits a signal that digitally describes the intermediate frequency f _g and the intermediate frequency phase φ _g . This signal is supplied to a phase detection device 40 which is formed by a lock-in detector 52 in the present case. The lock-in detector 52 allows a phase difference e mit to be determined $$\text{e}={\mathrm{\phi }}_{\text{G}}-{\mathrm{\phi }}_{\text{G}\text{,target}\text{.}}$$

gilt.The phase difference e is the difference between the actual intermediate frequency phase φ _g and a desired phase φ _g,soll . The target phase is chosen so that $$\text{e}={\mathrm{\phi }}_{\text{G}}-{\mathrm{\phi }}_{\text{G}\text{.target}}={\mathrm{\phi }}_{\text{f}}-{\mathrm{\phi }}_{\text{f}}-{\mathrm{\phi }}_{\text{G}\text{.target}}=2{\mathrm{\phi }}_{\text{n}}$$

is applicable.

The phase difference e is fed to a controller 54 of a signal generator 50 . The controller 54 generates a control signal which is fed to a direct digital synthesizer 56 of the signal generator 50 will be. The direct digital synthesizer 56 generates the modulation frequency f _c of the modulator signal by direct digital synthesis.

It is favorable if the direct digital synthesizer 56, the lock-in detector 52, the high-frequency local oscillator 36 and optionally the second modulator 42 are derived from a common frequency source 58 in a phase-locked manner.

The detection element 34 can have a photodiode. Alternatively, the high-frequency local oscillator 36 can be formed by a divider which divides a standard frequency f ₅₈ of a frequency source 58 by a predetermined fixed divider. Again alternatively, the high-frequency local oscillator 36 can be designed as a phase-locked loop (PLL) with a voltage-controlled oscillator, which uses the normal frequency f ₅₈ of the frequency source 58 as a reference.

The superheterodyne receiver 32 can also be designed as a single-sideband receiver with both quadratures.

The beam splitter 28, the detection element 34 and the connecting structure 31 can be kept at a predetermined temperature by means of a temperature stabilization unit 60 shown schematically.

Reference List

10

frequency transmission device

12

reference wave input

14

modulator

16

light guide

18

useful output

19

reflector

20

entrance light

22

fitting

24

feed light guide

26

derivation light guide

28

beam splitter

30

Taillight

31

connection structure

32

overlay receiver

33

reflector element

34

detection element

36

High Frequency Local Oscillator

38

mixer

40

phase detection device

42

second modulator

44

useful light

46

filter

48

Analog to digital converter

50

signal generator

52

lock-in detector

54

controller

56

Direct digital synthesizer

58

frequency source

60

temperature stabilization unit

φa

input phase

φc

modulator phase

φd

second modulator phase

φg

intermediate frequency phase

φg,set

target phase

φn

spurious frequency phase

φr

beat frequency phase

dφn/dt

phase change rate

e

phase difference

fa

optical input frequency

fb

useful light frequency

FC

modulator frequency

fd

second modulator frequency

fg

intermediate frequency

fl

local oscillator frequency

fn

interference frequency

fr

HF beat frequency

f58

normal frequency

l16

length of the light guide

L

direction of light propagation

QUOTES INCLUDED IN DESCRIPTION

This list of documents cited by the applicant was generated automatically and is included solely for the better information of the reader. The list is not part of the German patent or utility model application. The DPMA assumes no liability for any errors or omissions.

Patent Literature Cited

• DE 102008062139 B4 [0004]

Claims (14)

Method for actively stabilizing the optical path length of optical time and frequency transmission paths, with the steps: (a) modulating a modulator signal with a modulator frequency (f _c ) onto an optical input frequency (f _a ) of input light (20), the is introduced into a reference wave input (12), (b) emission of light with the modulated optical frequency (f _a + f _c ) onto a transmission path, in particular into a light guide (16) which leads to a useful output (18), and ( c) Determination of an HF beat frequency (f _r ) between the input frequency (f _a ) of reference light (20') and a reception frequency (f _back ) of back light (30) which is emitted at the useful output (18 ) was reflected and returned through the optical fiber (16), characterized by the steps of: (d) mixing the RF beat frequency (f _r ) with a local oscillator frequency (f _l ) in a heterodyne receiver such that an intermediate frequency (f _g = f _r - f _l ) arises, (e) dig itales determining an intermediate frequency phase (φ _g ) of the intermediate frequency (f _g ) and (f) generating the modulator signal based on the intermediate frequency phase (φ _g ) such that a change in the optical path length of the light guide (16) is compensated. procedure after claim 1 , characterized in that the modulator signal (f _c ) is generated on the basis of the intermediate frequency phase (φ _g ) by means of direct digital synthesis. Method for actively stabilizing the effective length of optical time and frequency transmission links according to the preamble of claim 1 , characterized by the steps of: (a) determining a beat frequency phase (φ _r ) of the RF beat frequency (f _r ), and (b) controlling the modulator frequency (f _c ) from the beat frequency phase (φ _r ) using direct digital synthesis , so that a change in an optical path length of the light guide (16) is compensated. Method according to one of the preceding claims, characterized by the steps of: (a) dividing the input light (20) by means of a beam splitter (28) so that reference light (20') is produced, the determination of the HF beat frequency (f _r ) superimposing the reference light (20') and the return light (30) on a detection element (34), or (b) splitting the input light (20) by means of a beam splitter (28) and reflecting the split light beam by means of a reflector element (33), so that the reference light (20') is created. A method according to any one of the preceding claims, characterized in that determining the intermediate frequency phase (φ _g ) comprises the steps of: (a) low-pass filtering a composite signal obtained by mixing the RF beat frequency (f _r ) with the local oscillator frequency (f _l ) arises, (b) digitizing the low-pass filtered mixture signal and (c) determining the intermediate frequency phase (φ _g ) by means of a phase detection device (40). Method according to one of the preceding claims, characterized in that the generation of the modulator signal with the modulator frequency (f _c ) and the modulator frequency phase (φ _c ) based on the intermediate frequency phase (φ _g ) includes the step of controlling a direct digital synthesizer (56) by means of a controller (54) based on the intermediate frequency phase (φ _g ) _to a target phase (φ target ). Optical signal transmission device (10) with (a) a reference wave input (12), (b) a modulator (14), which is arranged behind the reference wave input (12) in the direction of light propagation (L), for modulating a modulator frequency (f _c ) onto an optical input frequency (f _a ) of input light (20) launched into the reference wave input (12), (c) a light guide (16) into which light coming from the modulator (14) is launched, (d) a Useful output (18) by means of which light can be decoupled from the optical waveguide, and (e) a reflector arranged at the useful output (18) for reflecting light from the optical waveguide (16), characterized by (f) a heterodyne receiver (32) which is formed is for digitally determining a phase difference (e) between the reference light (20') and return light (30) reflected at the useful output (18) and returned through the light guide (16), and (g) a signal generator ( 50) to generate the Modulat or frequency (f _c ) based on the phase difference (e). Optical signal transmission device (10) after claim 7 , characterized in that the signal generator (50) for generating the modulator frequency (f _c ) is designed by means of direct digital synthesis. Optical signal transmission device (10) according to any one of the preceding Claims 7 or 8th or according to the preamble of claim 7 , characterized by (a) a receiver configured to determine a beat frequency phase (φ _r ) an RF beat frequency (f _r ) between the input frequency (f _a ) of reference light (20') and a reception frequency (f _back ) of back light (30) reflected at the useful output (18) and returned through the optical fiber (16), and (b) a signal generator (50) for generating the modulator frequency (f _c ) from the phase difference (e) by direct digital synthesis. Optical signal transmission device (10) after claim 9 , characterized in that the receiver is a heterodyne receiver (32) which is designed to digitally determine a phase difference (e) between the reference light (20') and the return light (30) which reflects at the useful output (18). and returned through the light guide (16). Optical signal transmission device (10) according to one of Claims 7 , 8th or 10 , characterized in that the heterodyne receiver (32) includes (a) a detection element (34) arranged to mix the input light (20) with the return light (30) to produce an RF beat frequency (f _r ). , (b) a high frequency local oscillator (36) for generating a local oscillator frequency (f _l ), (c) a mixer (38) for mixing the RF beat frequency (f _r ) and the local oscillator frequency (f _l ) so that an intermediate frequency ( f _l ) arises, and (d) a phase detection device (40) for determining an intermediate frequency phase (φ _g ) of the intermediate frequency (f _g ), and that (e) the signal generator (50) determines the modulator frequency (f _c ) based on the Intermediate frequency phase (φ _g ) generated. Optical frequency transmission device (10) according to one of Claims 8 , 10 or 11 , characterized in that (a) the heterodyne receiver (32) has a digital lock-in detector (52) for determining a phase difference (e) in the form of twice the phase difference of the return light (30) to the reference light (20') and that (b) the signal generator (50) has a direct digital synthesizer (56) and (c) a controller (54), in particular a PI controller, which uses the direct digital synthesizer (56). of the phase difference (e). Optical frequency transmission device (10) according to one of Claims 7 until 12 , characterized by (a) a beam splitter (28) for splitting the input light (20), (b) a reflector element (33) for reflecting the split light beam so that reference light is produced, (c) the beam splitter (28) and the reflector element (33) are rigidly coupled together by means of a connection structure (31), and by (d) a temperature stabilization unit (60) arranged to support the beam splitter (28), the reflector element (33) and the connection structure (31) at a given temperature. Optical frequency transmission device (10) according to one of Claims 7 until 13 , characterized by (a) a second modulator (42) which is arranged at the useful output (18) and designed to modulate a second modulator frequency (f _d ), (b) the second modulator frequency (f _d ) being at most 100 GHz. Pl/bro-be

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