electro optic effect experiment

TP Optics in Heidelberg

Part 2: electro-optic effect ¶.

Some materials change their optical properties when they are brought into an electric field. To use this effect one needs an electro-optic medium, such as some cubic crystals; they called electro-optical modulators (EOM). In that case, the refractive index \(n\) of the medium becomes a function of the applied field \(n=n(E)\) . Since the variations of \(n\) with \(E\) are only small, \(n(E)\) can be expanded in a Taylor series about \(E=0\) :

Using the coefficients

and neglecting higher order terms, one can rewrite \(n(E)\) as

Typical values for \(r\) are \(10^{-12}\) to \(10^{-10}\) m/V. It is sometimes convenient to use the electric permittivity instead of the refractive index: \( \eta=\varepsilon_{0}/\varepsilon \) . With

this yields

For \(r \gg s\) we obtain the linear electro-optic effect or Pockels Effect, where the refractive index depends linearly on the applied field. In that case the equations are reduced to \(\eta=\eta_{0}+rE\) and

Where \(r\) is called the Pockels coefficient.

Similarly, when \(r \ll s\) , we can see from Eq. (3) that we obtain a quadratic dependence of refractive index on the electric field. This is known as the Kerr effect, which will not be studied here.

Since the electro-optic crystals are generally birefringent, the effect of the index of refraction on the light is governed by the relative orientation of the electric field, the crystal axes and the beam path. Thus, the Pockels coefficient \(r\) in Eq. (2) is actually only one entry from a Pockels coefficient-tensor.

Since the index ellipsoid, the indicatrix, can now no longer be written with just the principal axes, we have to revert to its general form:

We thus obtain for the different components:

The entries in the \(r\) matrix are constrained by the crystal symmetry. The typical magnitudes for the \(r_{ij}\) -coefficients are \(10^{-12}\) m/V. The material used in the practicum is LiNbO \(_{3}\) . It is a uniaxial crystal with a crystal symmetry \(3m\) and it has indices of refraction \(n_{1}=n_{2}=n_{o} , n_{3}=n_{e}\) . The Pockels Coefficients for LiNbO \(_3\) are:

Where the first matrix shows the general case for crystals with \(3m\) symmetry. However, the Pockels coefficients are very hard to measure reliably and they can be affected strongly by contaminants inside the crystal. Therefore, the above numerical values are only indicative 1 .

For the case of a crystal without an applied field, the indicatrix is given by

Depending on the direction of the applied electric field and the direction of the light through the crystal, several distinct situations are possible. Each assembly shows a different behaviour.

One of the most widespread orientations is with the Pockels cell having a longitudinal field, i.e. applied along the extraordinary axis, \(z\) , and the light also travelling along that axis. This means that the light passes through transparent electrodes. In a cell with a transverse field the electric field can be applied along the ordinary axis, \(x\) , and the light then travels along the extraordinary axis, the optical axis \(z\) .

The Pockels cell of this practicum, however, has the electric field applied along the extraordinary axis \(z\) , with the light travelling along one of the ordinary axes, the \(y\) -axis. In this case, the applied electric field \(\vec{E} = E_z\) modifies the indicatrix to:

This equation can be cast in the form of Eq. (5)

with the modified ordinary and extraordinary indices of refraction \(n_{o}'(E)\) and \(n_{e}'(E)\) given by

With the approximation \( 1/\sqrt{1+x} \approx 1-\tfrac{1}{2} x \) for small \(x\) , we can indeed rewrite these indices of refraction in the manner we have seen before in Eqs. (2) and (4) :

The Pockels Cell ¶

An electro-optic crystal between two capacitor plates is called an electro-optical modulator. When the crystal exhibits the Pockels effect it is called a Pockels cell.

For light with a wavelength \(\lambda\) travelling through a medium of length \(L\) with an index of refraction \(n\) , the incurred phase in the medium is given by:

Since each ordinary and extraordinary axis, has a different index of refraction, \(n_o\) and \(n_e\) , respectively. Light travelling through a birefringent crystal experiences a phase retardation \(\Delta\Phi\) between the two polarizations equal to

With a Pockels cell one can manipulate both \(n_o\) and \(n_e\) by the strength of the electric field, as outlined above. The phase retardation \(\Delta\Phi\) due to a Pockels cell with an applied field \(E\) is thus given by $ \( \Delta \Phi(E)=\frac{2\pi}{\lambda} \left(n_e'(E)-n_o'(E)\right)L. \) $

using the approximations of Eqs. (6) this becomes:

With the applied voltage \(V=Ed\) we thus obtain that the retardation is given by Eq. (8) :

are the retardation for \(V=0\) and the so called half-wave voltage, respectively. At \(V=V_\pi\) the relative phase retardation between the two polarizations is \(\pi\) , causing the Pockels cell to act as a half-wave plate.

Transverse Intensity Modulation ¶

One can use a Pockels Cell to modulate the intensity of light. As we have seen, the relative phase retardation between the two polarizations is variable. For intensity modulation, one needs to put the Pockels cell between two crossed polarizers, placed at \(45^\circ\) with respect to the optical axes of the crystal. This set-up has an intensity transmittance \(T = \sin^2 (\Phi/2)\) . With equation (7) the transmittance of the device then is a periodic function of V:

The device can also be used as a linear modulator if the system is operated in the region near \( T(V)=1/2\) .

Optical Activity and Faraday-Effect ¶

Optically active media rotate the plane of polarization of linear polarized light. As described above, one can treat linear polarization as a superposition of two circular polarized waves with the opposite direction of rotation. In an optically active medium these two circular waves have different phase velocities, and thus the resulting linear polarized wave turns its plane of polarization. Natural optical activity appears in chiral media, such as solutions of dextrose or lactic acid. For these media, the direction of rotation does not depend on the direction of propagation through them; the behaviour is symmetric.

Some media become optically active when an axial magnetic field is applied, which is known as the Faraday effect. In this case the direction of rotation does depend on the direction of propagation, because the B-Field determines a spatial axis and thus breaks symmetry. The angle by which the polarization is rotated depends on the length of the medium \(L\) , on the magnetic field strength \(|\vec{B}|\) and on the so-called Verdet constant \(\upsilon\) :

The Verdet constant is a function of the wavelength: $ \( \upsilon=-\frac{\pi\gamma}{\lambda n} \) $

where the magnetogyration coefficient \(\gamma\) is a material constant of the medium.

Fig. 5 A Faraday isolator or optical diode ¶

The Faraday effect can be used to build an optical isolator. An optical isolator consists of a medium sensitive to the Faraday effect in an axial magnetic field. The set-up is enclosed by two polarizers with an angle of \(45^\circ\) to each other, as shown in figure Fig. 5 . In an optical isolator the first polarizer defines a plane of linear polarization of the light incident on the optically active medium inside the isolator. The medium rotates the plane of polarization by \(45^\circ\) clockwise, so that the light can pass the rear polarizer.

Light entering the isolator from the other side will first be polarization filtered by the rear polarizer. The plane of polarization is then rotated again by \(45^\circ\) by the medium, this time counter-clockwise. The result is that the polarization is finally rotated perpendicular to the front polarizer, thus it is absorbed. In this way the optical isolator is a one-way device for light, so that is also known as an optical diode. Note that the length of the active medium and the magnetic field determine the amount by which the polarization is rotated, so that optical isolators need to be designed to the wavelength of the used light in order to work properly.

Experiment ¶

In this part you will get a feeling of how an electric field (a voltage) can alter the optical properties of a crystal, in the sense that the refractive index of the material is changed. In the first few experiments you will measure the change of the index of refraction directly. In the second set of experiments, you will study the extent of the manipulation by its effect on the polarization state of light transmitted through the crystal.

The electric field is usually applied in the direction of the optical axis of the crystal. There are two possibilities for the direction of the light: parallel or perpendicular to the electric field. In the set-up used in this lab-course, the electric field is along the optical axis and the laser beam is incident perpendicular to the field. Take a closer look at the device; you can clearly see the (copper) electrodes against the crystal. (Don’t do this when the laser is on…)

Since the electro-optic effects are only rather small, one needs a high voltage to drive the Pockels cell. The high voltage power supply used to drive the Pockels cell can be controlled by the dials on its front, see Fig. 6 .

Fig. 6 Front view of the high voltage power supply used to drive the Pockels cell. ¶

The preferred way during the practicum, however, is to control the high voltage supply using a control voltage, applied to the analog input on the front panel of the power supply. A periodic input signal will cause an accordingly varying output voltage, e.g. you can sweep the voltage on the Pockels cell using a sawtooth wave form. The toggle-switch determines whether the power supply is controlled by the dial-settings ( Int ) or by the input voltage ( Ext ). Note, however, that the output voltage or current can never exceed the internal control settings (as set with the dials), even when using the analog input: make sure that you set the output range correctly. The output voltage and current are indicated on the displays of the power supply.

An input voltage between 0V and \(\sim 10\) V at the analog input results in an output voltage between 0V and \(\sim3\) kV. The voltage applied to the Pockels cell should not exceed 2kV. A voltage range between 0V and 1.8kV produces sufficient data to evaluate.

Characterization of the high voltage supply / Familiarization with the set-up ¶

During the experiments you will need to measure the desired optical signal with a photodiode against the voltage on the Pockels cell. The oscilloscope cannot handle the high voltage, but you can measure the control voltage on the analog input of the high voltage supply. Therefore, you need to know the voltage amplification factor of the high voltage power supply. The power supply is built for low frequency modulation, so either use a low frequency modulation or the output of the low voltage power supply.

First step : Measure the amplification factor of the high voltage amplifier / power supply.

Switch off the high voltage power supply. Leave the power supply connected to the filter-box, but disconnect the Pockels cell from the filter-box in order no to harm the cell inadvertently. The filter-box remains connected to discharge the high-voltage output and to let you check whether the current-settings of the high voltage power supply are not too low. To disconnect the cell from the filter-box: disconnect the cables on the filter-box side so that you can measure the entire set-up without high voltage risks, such as short-circuiting etc. due to having open cables next to a steel table.

Next, connect the dc power supply to the analog input of the high voltage power supply. Set the toggle switch to Ext .

Since the oscilloscope cannot take more than 300V, you cannot measure the high voltage signal directly; you will have to rely on the indicators on the display of the power supply.

Evaluate your measurements in your log book (Is the amplification linear? Can you use your data points or do they scatter too much? What limits should you set on the input voltage the not to harm the Pockels cell? etc.) before proceeding with the next step.

Second step : Set up the function generator and the oscilloscope for the following series of experiments

Connect the function-generator to the oscilloscope and display a full period on the screen with a frequency of \(\sim0.6 - 1\) Hz.

Adjust the amplitude and offset of the function generator correctly, so that the Pockels cell will survive ( \(V_{\mathrm{output}}=0-1800\) V).

Hint: make your adjustments of the function generator at a high frequency ( \(\sim\) kHz) to have an accurate feedback on your changes. Don’t forget to switch back to low frequencies when actually measuring.

Mach-Zehnder interferometer ¶

To determine whether light undergoes a phase shift and to which extent, one needs a set-up that compares the phase of the original laser beam with the phase of the light transmitted through the Pockels cell. Here, a Mach-Zehnder interferometer is useful. A Mach-Zehnder interferometer consists of two non-polarizing ( Why? ) beam splitters and two (totally reflecting) mirrors, see Fig. 7 . Since the two beams travel independently and spatially separated through the interferometer, phase shifts can be introduced to a single beam, without affecting the other one. The result of the phase shift can be determined by the intensity output. Due to the two separated beams, a Mach-Zehnder Interferometer is difficult to align, but its high sensitivity to phase shifts makes it useful for various applications.

Fig. 7 Set-up for phase modulation within a Mach-Zehnder Interferometer ¶

The intensity at one output port of the interferometer is \(I_{\mathrm{out}}\) is related to the incident intensity \(I_{\mathrm {in}}\) by

where \(\phi=\phi_1-\phi_2\) is the phase difference between the light beams passing through the branches 1 and 2. For the transmittance we have:

If the Pockels cell is placed in branch 1, we get

Yielding the following relation between \(\phi\) and the applied voltage:

where the constant \(\phi_0=\left(\phi_1-\phi_2\right)_0\) includes the path difference between the two branches. (In this set-up \(\phi_0 = 0\) , but in other books you might find other types of Mach Zehnder Interferometers.)

We finally obtain for the transmittance:

First step: Determine the directions of the optical axes of the Pockels cell (see also chapter on Part 1 and the table above). Construct a Mach-Zehnder Interferometer with the electro-optical modulator in one arm.

Set up the first beam splitter and the mirrors. It is recommended to have the beams all lying in the same, horizontal, plane. If this is impossible, the original laser beam might not be parallel to the optical table, or the first beam splitter is tilted. The former can be corrected by using a so-called ‘’dog leg’’ before entering the interferometer: use two mirrors to create a Z-shaped laser beam path, which allows you to align the laser beam in every possible direction. The latter problem can be corrected with the screws of the mounting of the beam splitting cube.

Next, adjust the mirrors so that both laser beams cross (again, preferably in the same horizontal plane) at a certain point, preferably with the beam paths perpendicular to each other.

Insert the second beam splitter. The point where the two beams cross has to be at the beam splitting interface within the cube. You will have a good approximation of this when the beams enter the cube at the same position (in the center of the surface, ideally) and/or if they seem to overlap directly after the cube in both exit ports of the interferometer.

Now, adjust the second beam splitter so that both beams in at least one exit port overlap. At a proper alignment you will be rewarded with an interference pattern in the exit beams. It is helpful to use a lens to widen up the beam, so that you can detect the interference pattern more easily.

To measure the phase difference, adjust the interferometer so that you see \(\sim 3 - 5\) interference maxima in the laser beam. Widen up the interference pattern with a lens so that the fringe spacing is approx. 2mm at the position of the photodiode and project only the maximum of a single fringe ( Why? ) onto the photodiode (use an iris).

In general: Keep in mind that you can only influence reflected beams, not those transmitted by beam splitters. Also note that a large interferometer is more sensitive to noise (vibrations from walking around etc.).

Now set up the measurement

Make sure you have determined the correct position/orientation of the Pockels cell with respect to the polarization of the incident light.

Make sure that you do not saturate the photodiode ( \(V_{\mathrm{out}}<3\) V).

Measure the output intensity as a function of the applied voltage for both crystal axes.

Compare the result for both \(\pm90^\circ\) axes. What effect does a rotation of \(180^\circ\) of the Pockels cell have?

Evaluation ¶

Determine the half-wave voltage \(U_\pi\) for both crystal axes.

Determine the Pockels coefficients \(r_{13}\) and \(r_{33}\) .

Just for fun: connect the loudspeaker to the photodiode and tap lightly on the table next to the interferometer; can you explain what you hear and what you see in the interference pattern?

Polarization Manipulation / Intensity Modulation ¶

The Pockels cell is a birefringent crystal, with both the fast axis and the slow axis voltage dependent, the device acts as an arbitrary-wave retarder. To understand this fact we have to observe the phase difference between both perpendicular planes of linear polarization. As explained in the theory section, this phase difference is given by

The effect of the Pockels cell on the polarization state of transmitted light can be studied in a manner equivalent to the experiments of Part 1 . You can measure the intensity of light transmitted by a pair of crossed polarizers with the Pockels cell in between. Here, you will measure the intensity behind the analyzing polarizer with a photodiode as a function of the high voltage applied to the Pockels cell.

Fig. 8 Experimental Set-up for Transverse Amplitude Modulation ¶

Construct the set-up outlined in Fig. 8 , with the Pockels cell placed between two crossed polarizers with their transmission axes at an angle of \(45^\circ\) to the optical axis of the crystal. Why does the Pockels cell need to be at \(45^\circ\) ? Make sure you have determined the correct orientation of the Pockels cell. Make sure you will not saturate the photodiode.

Measure the transmitted power on the photodiode as a function of the applied high voltage on the Pockels cell for both orientations \(\pm 45^\circ\) . Record your data on the computer.

Use the plots of transmitted power vs. applied voltage for both orientations \(\pm 45^\circ\) to determine the half-wave voltage \(V_\pi\) .

What is the difference between the \(\pm 45^\circ\) cases?

If the first intensity maximum is not at 0V: why is that?

Calculate \(V_\pi^{\mathrm{calc}}\) for the Pockels cell, using Eq. (9) and your previously measured values of \(r_{13}\) and \(r_{33}\) .

Calculate \({V_\pi^{\mathrm{calc}}}'\) directly from your measured \(\left( U_\pi \right)_e\) and \(\left( U_\pi \right)_o\) .

Compare your calculated and measured results to the specifications: \(V_\pi \approx 380\) V.

Linear Amplitude Modulation ¶

Consider the set-up of your previous experiment, see Fig. 8 . For a certain range of voltages, the intensity output of the setup shows an almost linear dependency with the applied voltage. This means that it is possible to linearly modulate the intensity of the transmitted light and thus transmit an arbitrary signal optically. For this, one has to modulate the applied voltage to the Pockels cell with the desired signal. For maximum signal amplitude modulation, you have to set up the Pockels cell at this particular voltage, then also a low-voltage signal can suffice to give an appreciable intensity variation. What is the necessary high voltage offset? At this voltage, the Pockels cell acts equivalently to another optical element. Which element is that?

Fig. 9 Voltage range for Linear Amplitude Modulation ¶

To test the possibility of linear amplitude modulation, and to get an idea of the possible response-time of the Pockels cell, use the set-up of the previous experiment, see Fig. 8 . Use the ‘internal’ setting of the high voltage supply to apply a DC offset voltage to the Pockels cell. Use the function generator for a high frequency ( \(\sim\) kHz) signal. Add both signals via the filter-box in the set-up; now the Pockels cell will ‘see’ both voltage signals. Compare the function generator / input signal with the photodiode signal on the oscilloscope; optimize the amplitude response of the photodiode (in other words: optimize the modulation efficiency of your set-up) by varying the offset voltage for the Pockels cell.

What happens with the transmitted signal at different offset voltages? Why is that?

At which points can you improve the signal-to-noise ratio? e.g. should you amplify the laser power, the high voltage, the modulation amplitude and/or other parts of the set-up?

Now replace the signal of the function generator with a radio signal and connect the output to a loudspeaker instead of to the oscilloscope. First check with the loudspeaker whether the radio (web-radio from the lab-computer) is turned on and tuned to a radio station. Connect the photodiode to the loudspeaker and listen to the effect of changing the offset voltage on the Pockels cell and varying the orientation of the last polarizer.

Describe and explain your observations.

The fast response of electro-optic modulators to electric field changes is used in LCD-displays and similar devices. Electric fields then switch the orientation of long polymer chains between crossed polarizers to vary the light transmission through the pixel.

Another source gives for LiNbO \(_3\) : \(r_{13}=9.6\) , \(r_{22} = 6.8\) , \(r_{33} = 30.9\) , \(r_{51} = 32.6\) . All values in [pm/V].

Part 1: Polarization optics

Part 3: Acousto-Optic Effect

Encyclopedia … combined with a great Buyer's Guide !

Encyclopedia > letter P > Pockels effect

Pockels Effect

Author: the photonics expert Dr. Rüdiger Paschotta

Definition : the phenomenon that the refractive index of a medium exhibits a modification which is proportional to the strength of an applied electric field ( linear electro-optic effect )

DOI : 10.61835/4j0    Cite the article : BibTex plain text HTML

The Pockels effect (first described in 1906 by the German physicist Friedrich Pockels) is the linear electro-optic effect , where the refractive index of a medium is modified in proportion to the applied electric field strength. This effect can occur only in non-centrosymmetric materials. The most important materials of this type are crystal materials such as lithium niobate (LiNbO 3 ), lithium tantalate (LiTaO 3 ), potassium di-deuterium phosphate (KD * P), β-barium borate (BBO), potassium titanium oxide phosphate (KTP), and compound semiconductors such as gallium arsenide (GaAs) and indium phosphide (InP). A relatively new development is that of poled polymers, containing specifically designed organic molecules. Some of these polymers exhibit a huge nonlinearity, with nonlinear coefficients which are an order of magnitude larger than those of highly nonlinear crystals.

Mathematically, the Pockels effect is best described via the induced deformation of the index ellipsoid, which is defined by

in a Cartesian coordinate system. An electric field can now change the coefficients according to

with the electro-optic tensor components . Note that the first index ( ) runs from 1 to 6 in this contracted notation, where e.g. corresponds to the y-z component.

Usually, only some of the coefficients are nonzero, depending on the crystal symmetry and the orientation of the coordinate system with respect to the crystal axes. For example, for lithium niobate (LiNbO 3 ) or lithium tantalate (LiTaO 3 ), which belong to the symmetry group 3m, the non-zero coefficients for the commonly used coordinate system are , , , = . For application of these materials e.g. in Pockels cells for electro-optic modulators , the largest tensor element ( ) is often used. Its magnitude is of the order of 30 pm/V for LiNbO 3 , with some wavelength dependence. Most other nonlinear crystal materials (e.g. BBO) have significantly lower electro-optic coefficients of a few pm/V, whereas some electrically poled polymers exhibit substantially higher values than LiNbO 3 .

The equation describes the change of , rather than directly the change of the refractive index. As the index changes are usually small, the approximation

(based on a first-order Taylor expansion) is often used.

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What in the crystalline structure determines how strong the Pockels effect is?

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The symmetry of the crystal structure is crucial: with a too high symmetry, there is no Pockels effect at all. Further, the crystal orientation matters a lot, and the bonding structure. However, I don't know a simple criterion to judge the strength of the Pockels effect in a material.

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Electro-Optic Effects in Crystals: Pockels Linear Electro-Optic and Kerr Quadratic Electro-Optic Effects

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• T. S. Narasimhamurty 2  

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The development of the laser has largely been responsible for the tremendous growth of interest during the past decade and a half in the study of the electro-optic and nonlinear optical properties of solids. These optical phenomena are applied in fabricating a large number of optical devices such as tunable narrow band interference polarizing monochromators, light modulators, light beam deflectors, frequency shifters, and secondharmonic generators. The electro-optic phenomena as described in this book deal entirely with the effect of an electric field on the index of refraction of materials in the solid state.

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Narasimhamurty, T.S. (1981). Electro-Optic Effects in Crystals: Pockels Linear Electro-Optic and Kerr Quadratic Electro-Optic Effects. In: Photoelastic and Electro-Optic Properties of Crystals. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-0025-1_8

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Journal of the Optical Society of America B

• pp. 840-853
• • https://doi.org/10.1364/JOSAB.11.000840

Theory of and experiment with the electro-optic effect in internal reflection near the critical angle

Douglas L. Butler, Geoffrey L. Burdge, and Chi H. Lee

Author Affiliations

Douglas L. Butler, 1 Geoffrey L. Burdge, 1 and Chi H. Lee 2

1 Laboratory for Physical Sciences, 8050 Greenmead Drive, College Park, Maryland 20740 USA

2 Department of Electrical Engineering, University of Maryland, College Park, Maryland 20742 USA

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• Electrooptical effects
• Light beams
• Optical properties
• Phase modulation
• Phase shift
• Total internal reflection
• Original Manuscript: January 21, 1993
• Revised Manuscript: November 2, 1993
• Published: May 1, 1994
• Full Article
• Figures ( 17 )
• Equations ( 33 )
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The electro-optic effect in internal reflection near the critical angle can directly modulate the reflectance of an optical beam with microwave electrical signals. The measured change in reflectance is 1.3 × 10 −6 V rms −1 cm for each applied microwave electric field, and modulation has been detected with input microwave signals as small as 10 mV rms . We present the theoretical explanation of this interaction in two complementary pictures, one picture based on nonlinear-optical theory and one based on electro-optics. Our experiments with fundamental optical and electrical properties match the predictions of both theories. The most salient feature of the electro-optic effect in internal reflection is that the strength of the modulation peaks sharply at the critical angle.

© 1994 Optical Society of America

Douglas L. Butler, Geoffrey L. Burdge, Chi H. Lee, and H. J. Simon Opt. Lett. 17 (16) 1125-1127 (1992)

Mary Carmen Peña-Gomar and Augusto García-Valenzuela Appl. Opt. 39 (28) 5131-5137 (2000)

Fuzi Yang and J. R. Sambles J. Opt. Soc. Am. B 11 (4) 605-617 (1994)

S. H. Han and J. W. Wu J. Opt. Soc. Am. B 14 (5) 1131-1137 (1997)

W. E. Moerner, S. M. Silence, F. Hache, and G. C. Bjorklund J. Opt. Soc. Am. B 11 (2) 320-330 (1994)

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Liquid crystals and electro-optic modulation.

March 30, 2020 English Posts , Light 8,485 Views

Abstract : in this Post we show how, by using an easily available liquid crystal cell by disassembling low cost equipment, it is possible to create an electro-optical modulator based on the polarization of light. Then we characterized our DIY modulator by carrying out transmittance, polarization and response speed measurements.

Introduction

Liquid crystals are a class of organic compounds with particular liquid-crystalline properties , discovered in 1888 by the Austrian botanist Friedrich Reinitzer. Reinitzer noticed that by heating cholesterol benzoate this first became opaque, and then cleared up as the temperature gradually increased. Once cooled, the liquid turned bluish and then crystallized.

In practice, these substances do not pass directly from the liquid to the solid state, but in particular conditions they are able to organize themselves in intermediate phases ( mesophases ) which present characteristics of both the crystalline and liquid solid state. This dualism justifies the term by which these compounds are indicated: liquid crystals. Liquid crystals (LC) are therefore a state of matter which has intermediate properties between those of conventional liquids and those of solid crystals. For example, a liquid crystal can flow like a liquid, but its molecules can be oriented similarly to the crystal. There are many different types of liquid crystal phases, which can be distinguished by their different optical properties .

Liquid crystals have found many applications in physics, chemistry, electronics and life sciences. Among these applications there are many devices that exploit the electro-optical effect of LC materials such as LC display, LC light modulator, LC optical modulator and LC optical switch.

In this post we want to show the operation of an LC cell used as a light modulator.

Principle of Operation

Most of the liquid crystals are made up of elongated molecules which, normally, are arranged with the major axis parallel to those of the molecules nearby. It is possible to control the alignment of the molecules if the liquid crystal is placed on a finely corrugated surface : if the corrugations are parallel, the molecules are also arranged parallel to each other. An LCD (liquid crystal display) consists of a liquid crystal between two finely corrugated surfaces , the corrugations of one surface are perpendicular to those of the other surface. If the molecules close to a surface have a direction, those close to the other surface have a perpendicular direction, the intermediate ones are rotated in the intermediate direction, in practice the molecules organize themselves in a helical structure , these devices are called twisted nematic type. The light passing through the device changes its polarization following the orientation of the molecules; then, passing through the liquid crystal, it is rotated 90° (fig A) . When an electric field is applied to the liquid crystal, the molecules, which are polarized, arrange themselves vertically and then the light passes without undergoing the rotation of the polarization direction (fig B) .

As can be seen from the figure above, the liquid crystal is placed between two transparent electrodes, through which the electric field E is applied. By inserting a polarizing filter before to the cell input and an analyzer filter to the cell output, it is possible to exploit the polarization rotation effect to create an optical switch .

the Liquid Crystal Cell

Our Electro-Optic polarization modulator is based on a liquid crystal cell. The liquid crystal cell is harvested from a low-cost auto-darkening welding mask filter (figure below). This is similar in construction to an LCD display but the purpose is different, the auto-darkening filter has a light sensor that drives the filter dark when it detects the welding arc, in order to protect the user’s eyes.

Of course the mask must be disassembled to remove the filter from its enclosure. The purple glass (green on the other side) is a filter that cuts out infrared and ultraviolet emissions, this filter also cuts out some of the light reaching our light detector and it must be removed, this can be done with ease, lifting it with a sharp razor. Without this filter the unbiased optical attenuator passes around 30% of the incoming light and darkens to allow only 1% when driven with a voltage of 5V. It is also possible to remove the back-side polarizer film using a sharp knife : the remaining device form now a voltage-controlled polarization filter. The image below shows the mask dismantled with LC cell, filter and the electronics (it can be kept because you never know).

Experimental Setup

The experimental setup is quite simple. We used our optical bench on which we placed a 632nm He-Ne laser, our liquid crystal cell and a photodiode photometer. The following image shows all the equipment.

The two images below show the detail of the laser, the rotatable analyzer filter and the photodiode.

Experimental Measures

With our experimental apparatus we have made a series of measures on the behavior of the liquid crystal cell.

Liquid Crystal Transmittance

The first test was the measurement of light transmission (transmittance) as a function of the voltage applied to the cell electrodes. The graph below shows the result: the threshold voltage is approximately 1V and the optical switch is completed for values ​​greater than 2.5V, for which the transmittance is less than 2%. A voltage step of 5V therefore ensures the transition from 100% transmittance to 0% transmittance.

Voltage Controlled Polarizer

The electro-optical modulation that is carried out with the liquid crystal cell is based on the rotation of the polarization plane of the light, we can therefore think of the LC cell as a device that rotates the polarization according to the applied voltage. We therefore measured the rotation angle produced as a function of the applied voltage. The graph is presented below: you go from an angle of 90° in the absence of voltage to an angle equal to 0° for voltages greater than 2.5V when the molecules are aligned according to the direction of the electric field. For voltage values ​​between 1V and 2V the relationship between angle and voltage is well approximated by a linear trend , as seen in the second graph.

Liquid Crystal Response Speed

An interesting measure to do is the evaluation of the response times of the LC cell to a voltage step applied to the electrodes. The experiment consists in applying a voltage pulse to the electrodes of the LC cell and measuring the intensity of the transmitted light with the photodiode in order to evaluate the times of variation of the transmittance. We know that the LC cell is an intrinsically slow device with rather long switching times. It is however interesting to make this measurement and try to make physical observations, even if only of a qualitative nature. The impulse lasts 100ms and amplitude of 5V. The results are presented in the following layouts.

From the details on the rising and falling edges it emerges that the switch-off time is about 6ms , while the switch-on time is much longer, worth about 40ms . In the “switch-off” process, the molecules are forced to align according to the electric field, while in the “switch-on”, when the electric field is reset, the molecules take their natural position. It is natural that the alignment of the molecules forced by the electric field is faster than their subsequent internal rearrangement, the latter process according to the oscilloscope trace assumes the characteristics of an exponential process : fast in the first section, slower in the part the final. We can assume that these slow times are mainly due to the viscosity of the liquid crystal and depend on the temperature , the applied electric field and the type of molecule (with greater or lesser polarity ).

In our LC cell the maximum frequency of use is only 13Hz, as shown in the following diagram.

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Gamma Spectroscopy with KC761B

Abstract: in this article, we continue the presentation of the new KC761B device. In the previous post, we described the apparatus in general terms. Now we mainly focus on the gamma spectrometer functionality.

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Large impact of strain on the electro-optic effect in (Ba, Sr)TiO 3 thin films: Experiment and theoretical comparison

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Shinya Kondo , Tomoaki Yamada , Alexander K. Tagantsev , Ping Ma , Juerg Leuthold , Paolo Martelli , Pierpaolo Boffi , Mario Martinelli , Masahito Yoshino , Takanori Nagasaki; Large impact of strain on the electro-optic effect in (Ba, Sr)TiO 3 thin films: Experiment and theoretical comparison. Appl. Phys. Lett. 26 August 2019; 115 (9): 092901. https://doi.org/10.1063/1.5117218

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(001)-epitaxial (Ba 0.5 Sr 0.5 )TiO 3 (BST) thin films with different magnitudes of compressive strain were fabricated on SrRuO 3 /SrTiO 3 substrates by pulsed laser deposition, and their electro-optic (EO) properties were characterized by modulation ellipsometry at different temperatures. All fabricated films showed an increased paraelectric-to-ferroelectric phase transition temperature upon compressive strain and revealed c -domain structures in the ferroelectric phase. We experimentally clarified that the EO properties of compressively strained BST thin films are enhanced toward the phase transition temperature modified by the strain. The experimental results were compared with the theoretical prediction based on a phenomenological thermodynamic model. Although the measured EO coefficient r c was less than that theoretically predicted, the experimentally observed strain effect on the EO properties is in good qualitative agreement.

All of the coefficients used in the thermodynamic calculation for BST are listed in Ref. 19 .

The refractive indices of BST films should change with temperature and misfit strain. However, in this evaluation, we regard n as 2.2 from Ref. 7 since the effect of the change in n on EO coefficients is negligible.

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A novel equivalent combined control architecture for electro-optical equipment: performance and robustness.

1. Introduction

2. problem solve, 2.1. tracking issues in electro-optical equipment, 2.2. tracking performance of equivalent composite control, 2.3. robustness of equivalent composite control, 2.4. impact of equivalent composite control on the velocity loop.

• Considering a 2 ∈ 0.134 , 0.25 and the amplification effect of differentiation (differencing) on high-frequency noise, equivalent composite control imposes quite high demands on the resolution of the velocity gyroscope.
• It is necessary to design filters to suppress high-frequency noise. A high bandwidth cannot effectively suppress the high-frequency noise of the gyroscope, while a low bandwidth essentially filters out useful signals. θ ˙ d t represents a typical random signal, and classical filters struggle to achieve good results.

2.5. Tracking Experiment

3. conclusions, author contributions, institutional review board statement, informed consent statement, data availability statement, conflicts of interest.

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Liu, Y.; Zheng, Y.; Chen, M.; Chen, J.; Wang, W. A Novel Equivalent Combined Control Architecture for Electro-Optical Equipment: Performance and Robustness. Appl. Sci. 2024 , 14 , 6708. https://doi.org/10.3390/app14156708

Liu Y, Zheng Y, Chen M, Chen J, Wang W. A Novel Equivalent Combined Control Architecture for Electro-Optical Equipment: Performance and Robustness. Applied Sciences . 2024; 14(15):6708. https://doi.org/10.3390/app14156708

Liu, Yang, Yulong Zheng, Mo Chen, Jian Chen, and Weiguo Wang. 2024. "A Novel Equivalent Combined Control Architecture for Electro-Optical Equipment: Performance and Robustness" Applied Sciences 14, no. 15: 6708. https://doi.org/10.3390/app14156708

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• Published: 29 July 2024

Voltage-controlled nonlinear optical properties in gold nanofilms via electrothermal effect

• Changjian Lv 1 ,
• Fanchao Meng   ORCID: orcid.org/0009-0004-7837-7294 1 ,
• Linghao Cui 1 ,
• Yadong Jiao 1 ,
• Zhixu Jia   ORCID: orcid.org/0000-0002-9857-9772 1 ,
• Weiping Qin   ORCID: orcid.org/0000-0002-5763-5846 1 &
• Guanshi Qin 1  

Nature Communications volume  15 , Article number:  6372 ( 2024 ) Cite this article

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• Mode-locked lasers
• Nanophotonics and plasmonics
• Nonlinear optics

Dynamic control of the optical properties of gold nanostructures is crucial for advancing photonics technologies spanning optical signal processing, on-chip light sources and optical computing. Despite recent advances in tunable plasmons in gold nanostructures, most studies are limited to the linear or static regime, leaving the dynamic manipulation of nonlinear optical properties unexplored. This study demonstrates the voltage-controlled Kerr nonlinear optical response of gold nanofilms via the electrothermal effect. By applying relatively low voltages (~10 V), the nonlinear absorption coefficient and refractive index are reduced by 40.4% and 33.1%, respectively, due to the increased damping coefficient of gold nanofilm. Furthermore, a voltage-controlled all-fiber gold nanofilm saturable absorber is fabricated and used in mode-locked fiber lasers, enabling reversible wavelength-tuning and operation regimes switching (e.g., mode-locking—Q-switched mode-locking). These findings advance the understanding of electrically controlled nonlinear optical responses in gold nanofilms and offer a flexible approach for controlling fiber laser operations.

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Introduction.

Gold nanostructures (GNs), such as nanofilms, gratings, nanoparticles, and metamaterials, have been widely exploited in optics because of their unique surface plasmon resonance (SPR) properties. The SPR can strongly enhance linear and nonlinear optical responses around specific wavelengths on demand, leading to the demonstration of unprecedented possibilities for extreme light concentration and manipulation, which makes GNs become excellent platforms for various applications in nonlinear optics 1 , 2 , 3 , quantum optics 4 , 5 , nano-optics 6 and bio-therapy 7 , 8 , 9 . Recently, various approaches (e.g., electrical, thermal, optical, and mechanical stimulus, etc.) for dynamically controlling the optical properties (linear or nonlinear regime) of gold nanostructures have been developed for advancing photonics technologies 10 , 11 , 12 , 13 , 14 , 15 . Among those approaches, electrical tuning is attractive owing to its easy control, and the corresponding devices can be integrated into optoelectronic devices. Up to now, electrical tuning has been widely investigated for dynamically controlling the optical properties of the gold/dielectric hybrid nanostructures, and has achieved great breakthroughs, such as electrically controlled nonlinear generation of light with plasmonics 16 , 17 , resonant plasmonic micro-racetrack modulators with a bandwidth of 176 GHz 18 . However, the electrical activity of those devices is derived from the dielectric materials rather than the gold itself. It is because the bulk carrier density ( n B ) is quite large in metals (e.g., gold). To change the surface carrier density ( n S ) significantly and realize electrical tuning of SPR properties in metals, one needs to achieve very small thickness ( t ) values ( n S  =  n B t ) 19 , 20 , 21 . The difficulty of making continuous metal films with sufficiently small t over a large area prevented the demonstration of electro-optic tunable SPR in metals. Very recently, Maniyara et al. fabricated ultrathin gold films (UTGFs) with a sufficiently low nanometric thickness via a novel deposition technique and further demonstrated the dynamic tuning of optical properties (linear regime) in UTGFs by ion-gel gating. SPR peaks were shifted over hundreds of nanometers and amplitude-modulated by tens of percent through gating using relatively low voltage, which was attributed to the modification of optical damping in UTGFs 19 .

Despite the electrical gating technique, thermo-optical tuning is another effective and cost-efficient approach for optical modulation. The temperature dependence of gold permittivity has been widely studied 22 , 23 , 24 , 25 , 26 . The increase of temperature in GNs results in the change of the temperature-dependent damping constant γ leading to the collateral effect of a change in permittivity Δ ε , which consequently influences the interaction with light and modifies the optical response of the system 27 . The thermo-optical dynamics of gold films have undergone detailed study through direct spatiotemporal imaging, which provides valuable insight into the interplay between electrons and phonons in gold 24 , 25 . This is important for designing nanoscale thermal management in nano-optoelectronic devices. A large optical modulation of ~30% was demonstrated in 1-nm-thick gold grating by elevating the electron temperature via ultrafast optical pumping 22 . Although electrical and thermal modulation has shown impressive achievements, the previous studies primarily focused on the modulation of linear optical properties. Whereas, dynamically manipulating Kerr nonlinear optical response of gold nanostructures remains unexplored.

In this paper, we demonstrated the voltage-controlled Kerr nonlinear optical response of gold nanofilms via electrothermal effect. The nonlinear absorption coefficient and the nonlinear refractive index of gold nanofilms were reduced by 40.4% and 33.1%, respectively, through electrical Joule heat with relatively low voltages (~10 V). Furthermore, voltage-controlled all-fiber gold nanofilm saturable absorbers were fabricated and used for constructing a mode-locked fiber laser. By changing the applied voltage, the central wavelength of the mode-locked pulses can be reversibly tuned over ~26 nm. And a reliable switching between continuous-wave mode-locking and Q-switched mode-locking regimes can be achieved by simply changing the applied voltage.

Voltage-controlled linear and nonlinear optical response of gold nanofilm

We first illustrate the voltage-controlled linear optical response of the gold nanofilm deposited on quartz. The device was fabricated by the physical vapor deposition method (details in Methods), as shown in Fig. 1a . The physical deposition of a gold film typically follows the Volmer-Weber growth mode due to its weak adhesion with the substrate, resulting in a noncontinuous film with discrete islands and random crevices 19 , 28 , 29 , 30 , 31 . The deposition parameters, such as vacuum condition, deposition rate, and substrate temperature, are critical to the surface morphology as well as the optoelectrical properties of the deposited film 30 , 31 , 32 . Moreover, introducing a seed layer (e.g. Cu) is critical for modifying the wetting behaviors of the gold on the substrate and reducing the surface roughness 19 . We fabricated various thicknesses of gold nanofilms, both unseeded (without Cu seed) and seeded with a Cu seed layer (see Supplementary Fig.  1 ), the seeded gold nanofilms show smoother surfaces and more homogeneous morphologies. Therefore, to improve the quality of gold nanofilm, a Cu seed layer was introduced in our experiment. In particular, a 1 nm copper seed layer followed by a 15 nm gold layer was deposited on the quartz to form a bilayer nanofilm (hereafter termed as gold nanofilm), with an area of 17 mm × 3.5 mm. The 1 nm copper seed layer forms energetically favorable nucleation sites for the incoming gold atoms, thus facilitating a lower sheet resistance and optical loss of the device 19 , 33 , 34 . Scanning electron microscopy (SEM) image of the gold nanofilm in Fig.  1b shows accretion islands connected to form a semicontinuous network. The root mean square (RMS) roughness (R q ) of the nanofilm was measured to be ~1.08 nm via an atomic force microscope (AFM).

a Schematic of voltage-controlled gold nanofilm deposited on the quartz. b SEM image and c AFM pattern of the gold nanofilm. d Film temperature ( T temp ) as a function of applied voltages ( U ), inset: heat maps of the nanofilm at 1, 6, and 10 V. e Experimental absorption spectra of the gold nanofilm at different applied voltages. f Relative absorbance change (at 2 μm) as a function of applied voltages. Black dots: experimental data, black circles: FDTD simulation. g FDTD simulated absorption spectra of the gold nanofilm at different applied voltages. Inset: geometrical structure of the gold nanofilm derived from SEM image.

Surface roughness plays a major role in determining the resistivity in ultrathin metal films 33 . We connected electrodes to the film boundary with silver paste. When a direct current (DC) voltage of 10 V was applied, the seeded gold nanofilm exhibited a higher temperature due to its lower resistivity compared to the unseeded one at equivalent thicknesses (see Supplementary Fig.  2 ). Besides, the seeded gold nanofilm shows a homogeneous heat distribution, while the unseeded film exhibits local heating near the electrode. This further confirms the critical role of the Cu seed layer in realizing high-quality gold nanofilms. Particularly, the total resistance between two electrodes of the 15 nm seeded gold nanofilm is ~21 Ω. Figure  1d shows the film temperature as a function of the applied voltages. As the applied voltage was increased from 0 to 10 V, the temperature of the nanofilm rose from ~299 to 353 K due to the occurrence of the electrothermal effect induced by electrical Joule heat.

The electrothermal-dependent optical absorption spectra of the 15 nm seeded gold nanofilm (hereafter the gold nanofilm all refers to the 15 nm gold nanofilm seeded with a 1 nm Cu seed layer, unless otherwise specified) were characterized via a UV-vis-NIR spectrophotometer (300 − 2500 nm). As shown in Fig.  1e , the absorption peak of nanofilm was located at ~992 nm, which is attributed to the localized SPR as a signature of the presence of isolated metallic islands 35 , 36 , 37 . The broad plasmon band extends into the infrared region with frequency-independent absorption from ~1700 to 2500 nm 38 , 39 , ensuring enhanced light-matter interaction. We note that the intensity and frequency of the absorption peak depend on the size, shape, and density of the islands forming the film 40 , 41 , 42 , which can be easily tuned by adjusting the film thickness (see Supplementary Fig.  6 ). A thinker film contributes to an increase in grain size and density, consequently leading to a red-shift of the absorption peak. By applying electrical voltage along the nanofilm, a monotonic decrease in absorbance was observed. The relative absorbance change is defined as |Δ α  | / α 0  = |  α U  −  α 0  | / α 0 , where α U and α 0 represent the absorbance of the sample with and without applying voltages, respectively, and U represents the applied voltages. It was found that the |Δ α  |  /α 0 can be reduced by 5.9% (at 2 μm) as the applied voltage was up to 10 V, as shown in Fig.  1f . To understand the tunable nature of the plasmons, the gold dielectric function was modeled by the temperature-dependent Drude model. As the film temperature increases with applied voltages, the associated increased electron-phonon scattering rate leads to an increase in the Drude damping constant. This results in a significant increase of the imaginary part of gold permittivity, which in turn suppresses the plasmonic resonance 23 , 43 , 44 , 45 , 46 . Despite the amplitude modulation in SPR, the resonance peak also blue-shifts to ~968 nm, which is related to the thermal-dependent electron density and effective electron mass (see Supplementary Section  IV ) 23 , 46 . The dependence of the imaginary part of the gold permittivity on temperature is governed by two competing mechanisms 23 : electron-phonon scattering and changes in the morphology of the gold grains induced by grain boundary movements. As the temperature is elevated from room temperature to 200 °C (473 K), the increasing electron-phonon scattering predominantly influences the gold permittivity and consequently, the optical properties. Above the threshold temperature of 200 °C, the gold grains start to move and merge together, leading to the size and shape change of the gold grains 23 . However, the highest temperature induced by Joule heating in our experiment (353 K) remains significantly below the threshold temperature for grain boundary movement (473 K) 23 . As a result, no obvious morphology changes were observed even after applying a voltage up to 10 V (see Supplementary Fig.  9 ). Therefore, the thermal-induced increase in electron-phonon scattering, and consequently the damping constant, is the dominant factor contributing to the observed effect. To further confirm the temperature-dependent SPR modulation in our case, we simulated the absorption spectra of the gold nanofilm by the 3D FDTD method (Lumerical FDTD Solutions). The morphology of the gold nanofilm was derived from the SEM image, and the results in Fig.  1g show a noticeable resonant mode at the wavelength of ~1060 nm. By adopting the voltage-dependent Drude damping constant in the simulation, we were able to reproduce the tunable SPR intensity of the gold nanofilm. The simulated results were found to agree well with the experimental data, verifying that the SPR absorption of the gold nanofilm could be modified via electrical-induced Joule heating. In previous studies 19 , 22 , significant optical modulation was achieved in the ultrathin gold films (thickness of several nm), whereas limited modulation was observed in thicker films (typically ≥ 10 nm). Our system emphasizes the critical role of Joule heating-induced temperature elevation in optical modulation, which mainly depends on the resistance of the film (see Supplementary Fig.  2 ). In this regard, further improvement of the fabrication method is expected to diminish the resistance of ultrathin gold nanofilms, which may improve the optical modulation capabilities.

Furthermore, the voltage-controlled nonlinear optical response of the gold nanofilm was investigated by the Z-scan measurement, which allows direct characterization of the nonlinear absorption coefficient ( β ) and nonlinear refractive index ( n 2 ) 47 , 48 . Both open- and closed-aperture measurements of the gold nanofilm with applying voltages were performed with a 1990 nm femtosecond fiber laser (see Supplementary Section  V ). Figure  2a shows the normalized open-aperture Z-scan transmission ( T ) curves of the gold nanofilm under different voltages. Obviously, the transmission increases as the film approaches the focus ( z  = 0, corresponding to maximum laser intensity), indicating the observation of saturable absorption. This can be attributed to the ground state bleaching of the plasmon, which occurs due to intraband electron excitation 49 , 50 . The values of nonlinear absorption coefficients were deduced from the curves by data fitting (Eq.  3 , in Methods) and were plotted versus voltages in Fig.  2b (black squares, left axis). A monotonic decrease of the nonlinear absorption coefficient can be observed as the applied voltage increases, which gives a 40.4% reduction of β as the voltage is up to 10 V. The normalized transmittance obtained from the closed-aperture Z-scan measurement is shown in Fig.  2c , which has been divided by the corresponding open-aperture Z-scan transmittance. The peak-valley configuration indicates that the nonlinear refractive index is negative, which results in self-defocusing at high input intensities. The negative n 2 reveals that the main contribution is due to the free electrons in sp band 51 , 52 . From Fig.  2b (red squares, right axis), we found that the nonlinear refractive index reduces by 33.1% as the applied voltage increases to 10 V. Upon the increase in the applied voltages, the decreased linear absorption of the gold nanofilm leads to a reduction of the local fluctuation of electron density, thus resulting in a decrease in the nonlinear absorption coefficient and the nonlinear refractive index.

a Open-aperture Z-scan data of the gold nanofilm on quartz measured for different applied voltages. Solid lines are theoretical fittings. b Experimental nonlinear absorption coefficients ( β , in black, left axis) and nonlinear refractive indexes ( n 2 , in red, right axis) as a function of applied voltage, which are calculated from Z-scan data. c Closed-aperture Z-scan data. d Theoretical calculation of voltage-dependent β (in black, left axis) and n 2 (in red, right axis) based on the anharmonic oscillator model. e Nonlinear transmittance ( T ) at different voltages derived from P-scan measurement. f Modulation depth (Δ T , in black, left axis) and saturation intensity ( I sat , in red, right axis) as a function of applied voltage derived from P-scan data. Error bars represent standard deviation.

To quantitatively explain the voltage-dependent nonlinear optical properties of the gold nanofilm, we treat the plasmon in the gold nanofilm as a classical anharmonic oscillator 53 , 54 . The third-order Kerr-type nonlinear susceptibility χ ( 3) of gold can be expressed as 54 , 55

where e is the electron charge, ω p is the plasmon frequency, m e is the effective electron mass, \(b={\omega }_{0}^{2}/{d}^{2}\) characterizes the strength of nonlinearity, and d stands for the atomic dimension of gold. The factor \(D(\omega )={\omega }_{0}^{2}-{\omega }^{2}-2i\gamma \omega\) contains the resonance frequency ω 0 and the damping constant γ . Evidently, the damping constant γ plays a critical role in determining the behavior of χ (3) 3 , 56 . Figure  2d shows the calculated voltage dependence of the nonlinear absorption coefficient and the nonlinear refractive index of the gold nanofilm (for details, see Methods). It can be seen that both β and n 2 decrease with the increasing of applied voltages, which agrees well with the experimental results.

The voltage-dependent saturable absorption property of the gold nanofilm was then investigated via the P-scan measurement 48 , 57 (see Supplementary Section  V ). The measured transmittance as a function of the incident power is shown in Fig.  2e , which was fitted with a two-energy-level model (see Supplementary Section  VII ). The fitted modulation depth and saturation intensity as functions of the applied voltage are shown in Fig.  2f . As the applied voltage was increased from 0 V to 10 V, the modulation depth Δ T reduced from 4.53% to 3.08%, and the saturation intensity I sat increased from 8 to 9.66 MW/cm 2 . This is because the increased Joule heating leads to a decrease in linear absorption coefficient ( α 0 ) and a stronger electron-phonon scattering. As a result, the modulation depth which is proportional to the linear absorption coefficient (Δ T ∝ α 0 , see Supplementary Section  VII ), decreases, while the saturation intensity which is inversely proportional to linear absorption cross-section and relaxation time ( I sat  =  ħω/2σ 0 τ r , where σ 0 is linear absorption cross-section and τ r is relaxation time), increases. To the best of our knowledge, this is the first demonstration of voltage-controlled nonlinear optical properties in gold nanofilms via electrothermal effect.

Note that the Cu seed layer itself exhibits negligible linear and nonlinear optical responses (see Supplementary Section  X ), while the observed notable electrothermal optical modulation primarily originates from the gold nanofilm. However, the employing of a (1 − 2 nm) Cu seed layer remains critical for achieving efficient electrothermal modulation, which contributes to a smoother surface, lower resistance, and stronger SPR absorption of the fabricated gold nanofilm.

Voltage-controlled all-fiber gold nanofilm device

As has been shown that the voltage-controlled gold nanofilm deposited on quartz manifests significant electrothermal modulation in both linear and nonlinear responses. Utilizing these electrically tunable features, the voltage-controlled all-fiber gold nanofilm saturable absorber (SA) was fabricated and the schematic of the device is shown in Fig.  3a . The 15 nm seeded gold nanofilm was evaporated on the flat surface of the side-polished fiber (SPF) with the same procedure as that for the quartz. Two electrodes were electrically attached at each side of the polished boundary by silver paste.

a All-fiber gold nanofilm SPF SA, total film thickness: ~16 nm (1 nm Cu/15 nm Au), the distance between fiber core boundary and polishing surface: ~2 μm. V + : positive voltage, GND: power ground. b Thulium-doped fiber laser cavity. WDM wavelength-division multiplexing, TDF thulium-doped fiber, ISO isolator, PC polarization controller, OC optical coupler, and DC source direct current source.

The voltage-controlled linear absorption spectra of the gold nanofilm-SPF device were measured by using a super-continuum light source. As illustrated in Fig.  4a , the device shows broadband absorption from 600 nm to 2000 nm owing to the SPR absorption of the gold nanofilm. It is interesting to compare the absorption spectra of the fiber device (Fig.  4a ) with that of the normal-incidence case (Fig.  1e ). It can be seen that for the fiber device, the interaction below 1750 nm seems to be significantly reduced. This can be understood by considering the wavelength dependence of the mode-field distribution for the fiber device. Specifically, the light interacts with the gold nanofilm via the evanescent field of the guided mode, and the intensity of the electric field at the gold nanofilm is strongly wavelength-dependent, which decreases rapidly with decreasing wavelength. Since the linear absorption (interaction) in evanescent mode is proportional to the integral of the electric field intensity at the film over the film width 58 , the interaction was significantly reduced for wavelengths below 1750 nm as shown in Fig.  4a . As a DC voltage was applied to the device (in the same direction as the propagating light), the absorbance at longer wavelengths dramatically decreased, which was attributed to the reduction of the SPR absorption of the gold nanofilm with the increase of the applied voltage (shown in Fig.  1e ). Moreover, the multiple reflections of light propagation along the polished fiber axis lead to a significant enlargement of the effective interaction area and length 59 . This results in a remarkable reduction of optical absorption at 2 μm, as depicted in Fig.  4b (black squares), with a decrease of ~36.2% at 10 V. It is worth mentioning that the increase of the overall transmission through the fiber device also contributes to a reduction of the insertion loss of the all-fiber device. Specifically, at 2 μm, the insertion loss was decreased by ~3 dB as the voltage increased to 10 V.

a Linear absorption ( α ) spectra of the gold nanofilm-SPF device at different applied voltages. b Relative absorbance change (Δ α / α 0 , in black) and insertion loss (in red) as a function of applied voltage. c Nonlinear transmittance ( T ) of the all-fiber device at different applied voltages. d Modulation depth (Δ T , in black) and saturation intensity ( I sat , in red) as a function of applied voltage. Error bars represent standard deviation.

The voltage-controllable saturable absorption of the gold nanofilm-SPF device was characterized by the balanced twin-detector method. As shown in Fig.  4c, d , as the electrical voltage applied to the all-fiber device increases from 0 V to 10 V, the modulation depth gradually decreases from 17.6% to 13.3%, while the saturation intensity increases from 0.29 to 0.5 MW/cm 2 . This is in accordance with the results obtained from the gold-quartz device. However, more significant modulations were achieved from the all-fiber device owing to the enhanced light-matter interaction.

It is known that the modulation depth and saturation intensity of an SA are critical to pulse shaping dynamics in mode-locking fiber laser systems 60 , 61 , 62 , 63 , 64 . To demonstrate the powerful modulation function of the voltage-controlled device, we constructed a thulium-doped mode-locked fiber laser (TDFL), which utilized the gold nanofilm-SPF as an SA (GFSA), as illustrated in Fig.  3b . The mode-locked operation without applied voltages was centered at 1964.9 nm, with a pulse duration of ~482 fs and a signal-to-noise ratio (SNR) of ~64 dB (see Supplementary Fig.  13 ). The insertion loss of the SA device is ~7.35 dB at 0 V, which includes the saturable (linear) absorption loss α 0 and the non-saturable loss α ns (see Supplementary Eq.  4 ). When used in a mode-locked fiber laser, a low non-saturable loss α ns and a high saturable absorption loss (or modulation depth) α 0 are usually favored 65 , 66 . Specifically, a low non-saturable loss can increase the laser slope efficiency 65 , while a high saturable absorption loss facilitates reliable self-starting of mode-locking and leads to short pulse duration 65 .

With the other cavity parameters (e.g. pump power and PC state) fixed, we applied voltages to the GFSA device along the fiber axis. Interestingly, an increase in the voltage resulted in a gradual red-shift of the mode-locking wavelength, as shown in Fig.  5a . It is observed that the central wavelength of the mode-locked laser shifted from 1964.9 to 1991.5 nm as the applied voltage increased from 0 to 7 V, which was mainly ascribed to the reduced linear loss of the GFSA at higher voltages. To explain the wavelength-tuning mechanism of the mode-locked fiber laser, we carried out numerical simulations based on a lumped scalar model (for details, see Methods). In the simulation, the linear losses and saturable absorption parameters of the SA at each applied voltage were derived from experimental data (Fig.  4a, d ). Figure  5b shows the tunable mode-locked spectra obtained from numerical simulations. The simulated mode-locked wavelength shifts from 1965.76 to 1988.64 nm when the voltage was increased from 0 V to 7 V, which agrees well with the experimental results. Further controlled simulations were conducted which fixed either linear loss or nonlinear saturable absorption. These simulation results reveal that the wavelength tunability results from both the change in linear loss and nonlinear saturable parameters of the GFSA, with the linear loss change emerging as the dominant effect.

a Experimental optical spectra, b numerically simulated optical spectra, and c experimental autocorrelation trace of the mode-locking operation at different applied voltages. Experimentally measured d central wavelength ( λ center ), e spectral 3-dB bandwidth (Δ λ 3-dB ), and f pulse duration ( T width ) of the mode-locking pulses as a function of applied voltage.

The mode-locked wavelength can be tuned reversibly, as shown in Fig.  5d (and Supplementary Fig.  14 ). The stronger intracavity nonlinearity as the voltage increased, contributes to the broadening of the spectral 3-dB bandwidth, as shown in Fig.  5e . Moreover, the pulse width decreased from 482 to 375 fs as the voltage increased from 0 V to 7 V (Fig.  5c, f ), which is in accordance with the broadening of the spectral bandwidth. Notably, the mode-locking repetition rate was reduced by ~3 kHz as the voltage was increased to 7 V (see Supplementary Fig.  15 ). The reduction in repetition rate can be attributed to the red-shift of the mode-locked wavelength, which in turn causes a decrease in the group velocity of the pulse.

A reversible switching between two operation regimes was observed as the voltage was further increased to 8 V. We note that the laser kept operating at the continuous wave mode-locking (CWML) regime when the voltage was below 7 V. However, as the voltage was increased to 8 V, the laser switched to the Q-switched mode-locking (QSML) regime as shown in Fig.  6 . The transition can be clearly seen from the measured spectrum shown in Fig.  6a . It was a typical mode-locked soliton spectrum with Kelly sidebands on both sides when the applied voltage was 7 V, and as the voltage was set to 8 V, the measured average spectrum evolved into a bell-shaped profile with a central wavelength of 1965.3 nm. Intriguingly, by switching the applied voltage between 7 V and 8 V, the operation regimes can be switched between CWML and QSML. The operation regime of a mode-locked laser critically depends on the interplay between the gain saturation and the saturable absorption. It has been demonstrated both theoretically 60 and experimentally 63 , 64 that by adjusting saturable absorption parameters, it is possible to switch between different pulse operation regimes, such as CWML, Q-switching (QS), and QSML in fiber lasers. The stability criterion for CWML against QSML is given as 60 , 67

where E p is the intracavity energy, E sat,L is the saturation energy of the gain medium, E sat,A is the saturation energy of the SA, and Δ T is the modulation depth of the SA. Equation  2 indicates that Q-switched instability occurs when the intracavity pulse energy is below the critical value E p,c  = ( E sat,L E sat,A Δ T ) 1/2 . When the voltage was increased from 7 V to 8 V, both the measured modulation depth and the non-saturable loss decreased, while the saturation energy increased. The operation-regime switching may be attributed to the increasing saturation energy of the saturable absorber, which decreased E p below the critical value E p,c .

a Optical spectra of CWML and QSML by driving 7 V/8 V on/off voltage. QSML regime characteristics: b pulse train oscillogram and c radio frequency spectrum at 8 V. d Pulse width (in black, left axis) and repetition rate (in red, right axis) of the Q-switched envelope, and e output power (in red, right axis) and pulse energy (in black, left axis) of the QSML laser as a function of applied voltage.

In the QSML regime, the Q-switched envelope exhibits a 2.56 μs temporal width, which contains a number of ultrashort mode-locked pulses as shown in Fig.  6b . The measured radio frequency (RF) spectrum of the QSML pulses is shown in Fig.  6c . The fundamental frequency peak of the mode-locked pulses was at 24.8 MHz. Multiple sideband frequency components with a frequency interval of 111 kHz can also be observed, which corresponds to the repetition rate of the Q-switched envelope. The QSML sustained as the driving voltage was increased up to 10 V. By changing the voltage from 8 to 10 V, the repetition rate of the Q-switched envelope could be tuned from 111 to 118.5 kHz, and the envelope temporal width reduced from 2.56 to 2.48 μs, as plotted in Fig.  6d , which is typical for the Q-switched behavior. The average output power of the laser also increased by increasing the driving voltage. It can be expected that this flexible voltage-controlled switching is very attractive in nonlinear frequency conversion, precise fabrication, and supercontinuum generation 60 , 68 .

In comparison, we also fabricated a voltage-controlled all-fiber 15 nm-thick gold nanofilm SA without a Cu seed layer, which was subsequently inserted into the TDFL as a replacement of the original GFSA. Despite enabling mode-locking and demonstrating wavelength-tunability, this laser with unseeded SA exhibited a narrower tuning range (see Supplementary Fig.  21 ) compared to the original one implemented with a seeded SA. The limited tuning range can be attributed to the high surface roughness and large resistance of the unseeded gold film. Moreover, a laser with such an unseeded SA tended to undergo unstable operation and it also exhibited no operation-regime switching. Therefore, a gold film at an appropriate thickness (15 nm here) grown with a copper seed layer will be favored to exhibit useful optical modulation. Such films exhibit reduced roughness and resistance, narrow grain-size distribution, and homogeneous morphology 30 , 32 , ensuring high Joule heat generation, stable working temperature, and uniform temperature distribution.

Although gold has been demonstrated ultrafast nonlinear optical response 24 , 69 , 70 , 71 , the response time of the voltage-controlled GFSA was measured to be ~10 s (see Supplementary Fig.  24 ) due to the slow thermal equilibrium of the electrothermal device 72 , 73 . It is useful to compare the performance of several related optical modulators, which utilized acousto-optic (AO), electro-optic (EO), thermo-optic (TO), photothermo-optic (PTO), or electrothermo-optic (ETO) effect (see Supplementary Table  1 ). The commercial ones, including acousto-optical modulator (AOM) and electro-optical modulator (EOM), exhibit ultrafast response time and high extinction ratios. Although the fiber-coupled AOM and EOM can be easily integrated into fiber optical systems, these devices often require external RF drivers which increases system complexity and cost. Thermo-optical (TO, PTO, ETO) modulators, primarily graphene-based, exhibit strong temperature dependence of optical conductivity and notable thermal effects due to the exceptionally small electronic heat capacity 22 . This often leads to impressive modulation efficiency (see Supplementary Table  1 ). However, most of those thermo-optical modulators focused on manipulating the linear optical response, there has been limited exploration into actively controlling nonlinear optical properties, especially for operating at the mid-infrared range (2 μm). In comparison, our voltage-controlled GFSA device offers facile modulation of both linear and nonlinear optical properties, and has the advantage of cost-efficiency, easy fabrication and operation, which allows for easy switching of operation-regime in mode-locked fiber lasers. Moreover, by depositing a gold film onto a microfiber, the response time may be reduced to milliseconds or even tens of microseconds due to the decreased device size and the reduced thermal equilibrium time 74 , 75 , 76 .

We have developed a gold nanofilm device that can actively control its optical response by modifying the damping factor through electrothermal engineering. However, despite the electrothermal effect, other factors such as carrier doping or morphology evolution may also be responsible for such an optical modulation in the gold nanofilm. To confirm that the electrothermal effect is the dominant effect in the optical modification, we applied backward voltages to the nanofilm fiber device. A red-shift of mode-locking wavelength from 1965.4 to 1991.2 nm was also obtained (see Supplementary Section  IX ), which is quite similar to the forward voltage experiment. It means that the direction of applied voltage has no influence on the mode-locking evolution. To further confirm that the electrothermal effect is the dominant factor for the optical modulation, we performed experiments by directly controlling the temperature of the film without applying any voltages (see Supplementary Section  IX ). Similar mode-locking evolutions can still be obtained. The reversible manner of mode-locked dynamics can exclude the impact of electrothermal-induced morphology evolution of gold. These results suggest that dissipative heating is the dominant effect, which modifies the linear optical response and saturable absorption properties of the gold nanofilm. Note that the Joule heating would also elevate the temperature of the SPF underneath the gold nanofilm, which would induce the change of optical properties of the optical fiber itself and therefore may contribute to the observed effect. To clarify this point, we inserted an additional bare SPF into the mode-locked fiber laser cavity and changed the temperature of the bare SPF by heating while keeping the temperature (voltage) of the GFSA fixed (see Supplementary Section  IX ). A negligible mode-locking wavelength shift was observed, which indicates that the temperature-induced optical properties variation of the SPF itself has a negligible impact on the observed electrothermal optical modulation.

Though Maniyara et al. have shown relatively large wavelength tuning (∆ λ/λ  = 13.6%) and linear transmission modulation (14.4%) in ultrathin gold ribbons by ion-gel gating 19 , such operations require complicated fabrication techniques which may be particularly challenging to realize in all-fiber systems. Compared with the electrical gating based lithography of ultrathin gold ribbons 19 or single-layer nanostructures 20 , our method is certainly facile and user-friendly in device fabrication and operation. Additionally, by using other metal nanofilms (such as silver and copper) with remarkable SPR properties, good electrically and thermally conductive, comparable or more excellent electrothermal-induced optical modulation may be obtained.

To summarize, we have demonstrated the electrothermal-induced and voltage-controllable Kerr nonlinear optical response of the gold nanofilm device, and show its application in manipulating mode-locking operations. The electrothermal-induced effective modification of the damping factor contributes to dynamically tunable optical nonlinearity in the gold nanofilm, which enables a wide range modulation in the nonlinear absorption coefficient and nonlinear refractive index. We also realized an all-fiber SA device by depositing the gold film on a side-polished fiber, which shows controllable large optical absorption variation (36.2%) and nonlinear modulation depth alteration (4.3%) at a low operation voltage of 10 V. Furthermore, by incorporating this electrically controllable all-fiber SA in a fiber laser cavity, flexible manipulation of the operation states of the mode-locked laser were realized by simply adjusting the applied voltages. This allowed for reversible wavelength-tuning and even switching among operation regimes (e.g., mode-locking—Q-switched mode-locking). This work substantially broadens the research scope of optical nonlinearity in GNs and represents a key technique for electrothermal tunable optical nonlinear processes, which will immediately find enormous applications, particularly in electrically controlled nonlinear optics and optoelectronic information processing in modern photonic waveguide systems.

Device fabrication

Voltage-controlled gold nanofilm was fabricated by depositing gold on quartz via the physical vapor deposition (PVD) method. The device actually consists of metallic gold/copper bilayer film. A 1 nm thick copper layer (17 mm × 3.5 mm) was first deposited on the quartz with a deposition rate of 0.6 Å/s as the seed layer. Then, a 15 nm thick gold film was grown on the copper layer with a deposition rate of 0.3 Å/s at a pressure of 1.4 × 10 −5 mbar. Two metal wires were bonded to the film boundaries by silver paste to construct an integrated circuit, as shown in Fig.  1a .

Voltage-controllable all-fiber gold nanofilm device was fabricated with a similar process, while the gold was deposited on the flat surface of side polished fiber (SPF). The SPF (purchased from Phoenix Photonics LTD) has a polished length of 17 mm and the distance from the polished surface to the fiber core boundary is about 2 μm. A 1 nm thick copper layer was first deposited on the flat polishing surface of the SPF with a deposition rate of 0.6 Å/s as the seed layer. Then, a 15 nm thick gold film was deposited on the copper layer with a deposition rate of 0.3 Å/s at a pressure of 1.4 × 10 −5 mbar. Two metal electrodes were electrically attached at both sides of the polished boundary by silver paste.

FDTD simulation

We utilized commercially available FDTD Solutions (Lumerical Inc.) to simulate the absorption spectra of the gold nanofilm deposited on the quartz substrate. The geometrical structure of the gold nanofilm was derived from the SEM image, as shown in the inset of Fig.  1g . The optical parameters of the gold nanofilm were determined based on the calculated voltage (temperature)-dependent gold permittivity and the substrate parameters were obtained from the SiO 2 of the software. The gold nanofilm was illuminated from the top by the normally incident (backward along the z -axis) plane wave, which is linearly polarized in the x - y plane. The transmitted optical field was recorded by a frequency domain power planar monitor, placed inside the quartz. To reduce the simulation time, periodical boundary conditions were used in the x - y plane simulation area. Perfect matching layers were applied at the top and bottom boundaries of the simulation area to prevent any reflections. The transmittance spectra of the nanofilm were achieved from the transmitted optical field monitor. The absorption spectrum of the film was then calculated as – ln( T ) 56 , where T is the optical transmission through the nanostructure.

Z-scan measurements

The voltage-controlled third-order nonlinear coefficients ( β and n 2 ) of gold nanofilm on quartz were characterized using the well-developed Z-scan technique at different applied voltages (see Supplementary Section  V ). A pulsed fiber laser centered at 1990 nm, with a pulse duration of 150 fs and a repetition rate of 48 MHz was used as the incident laser source. For open-aperture Z-scan experiments, the normalized transmittance T ( z ) can be theoretically written as 47 :

where q 0 = βI 0 L eff , L eff  = [1–exp( –αL )]/ α represents the effective thickness of the sample and α is the linear absorption coefficient, I 0 is the peak irradiance at the beam waist, x  =  z / z 0 and z 0  =  πw 0 2 / λ is the Rayleigh range of the Gaussian beam, where w 0 denotes the beam waist at the focus and λ is the operating wavelength. The nonlinear coefficients β from 0 to 10 V were fitted to be −10 × 10 −5 , −9.97 × 10 −5 , −9.69 × 10 −5 , −9.3 × 10 −5 , −8.23 × 10 −5 , and −5.98 × 10 −5 mW −1 , respectively.

In the closed-aperture measurements, the variations in the transmission correspond to both nonlinear absorption and nonlinear refraction. Thus, the closed-aperture data must be divided by the open-aperture data to obtain the pure nonlinear refractive index 47 . Then the theoretical expression of the divided Z-scan data can be fitted with the formula 47 :

By fitting with the experimental data, the nonlinear refractive indexes from 0 to 10 V were calculated to be −10 × 10 −12 , −9.88 × 10 −12 , −9.49 × 10 −12 , −8.79 × 10 −12 , −8.07 × 10 −12 , and −6.73 × 10 −12 m 2 W −1 , respectively.

Anharmonic oscillator model

The third-order nonlinear susceptibility of gold was modeled by a classical anharmonic oscillator, where we assumed a centrosymmetric structure of gold and ignored the second-order nonlinearity. The third-order Kerr-type susceptibility of gold can be expressed as Eq.  1 . The Z-scan experimental results of the nonlinear refractive index ( n 2 ) and nonlinear absorption coefficient ( β ) were related to the real and imaginary part of χ (3) 71 , 77 . By considering the electrothermal-induced damping constant, the analytical solution of voltage-dependent n 2 and β can be expressed as:

where ε 0 is the permittivity of free space, c is the speed of light, N is the electron density and n 0 is the linear refractive index. γ ( U ) is the voltage-dependent damping constant, where γ 0 is a constant for gold.

Fiber laser cavity

The fiber laser cavity was constructed using a commercial 1570 nm erbium-doped fiber laser as the pump source, which was launched into the cavity through a 1570/1980 nm wavelength-division multiplexing (WDM). A 0.4 m long thulium-doped fiber (TDF, Nufern, SM-TSF-5/125) served as the gain medium, with group velocity dispersion (GVD) of −20 ps 2 /km at 1980 nm. To ensure unidirectional light propagation, a polarization-independent isolator (PI-ISO) was added to the cavity. A polarization controller (PC) was used to adjust the polarization state. A 10 dB optical coupler (OC) was adopted to extract 10% of the laser and the rest continued propagating in the cavity. The fiber tails of all components were standard single-mode fiber (SMF 28e) with GVD of −67 ps 2 /km at 1980 nm. The total length of the ring cavity was ~8.4 m and the net dispersion was calculated to be −0.57 ps 2 . Finally, our voltage-controlled GFSA was inserted into the cavity where the laser output properties such as optical spectrum, pulse duration and repetition rate were monitored at different applied voltages.

Fiber laser simulations

Numerical simulations are based on a lumped scalar model. The pulse propagation in different fiber segments in the cavity is modeled by the nonlinear Schrödinger equation (NLSE) 78 , 79 :

where A  =  A ( z , t ) is the pulse envelope, z is the propagation coordinate, t is the co-moving time, g is the fiber gain, β 2 is the GVD parameter, and γ NL is the nonlinear parameter. At 2 μm, β 2 (SMF) = − 67 ps 2 /km and γ NL (SMF) = 1.1 × 10 −3 W −1 m −1 , and β 2 (TDF) = − 20 ps 2 /km and γ NL (TDF) = 2.6 × 10 −3 W −1 m −1 . The gain term g is non-zero only in the TDF segment and is modeled as 79 , 80 :

where g 0  = 14.4 m −1 is the small-signal gain. E  = ∫ |  A | 2 dt is the intracavity pulse energy, E sat is the saturation energy of the gain fiber, Ω is the detuned angular frequency, and Ω g corresponds to a 113.89 THz gain bandwidth. The saturable absorption is modeled by a transmittance function which is given as 78 , 79 :

where Δ T ( U ) is the modulation depth of the SA at a certain voltage U , P ( t ) = | A ( z , t )| 2 is instantaneous power, and P sat ( U ) is the saturation power of the SA. Experimentally measured saturable absorption parameters at voltages of 0, 2, 4, and 6 V (Fig.  4d ) were used in the numerical simulations, and the corresponding saturable absorption parameters at 1, 3, 5, and 7 V were interpolated. Then the wavelength-dependent linear transmittance of the GFSA (Fig.  4a ) at each voltage was implemented in the frequency domain. The simulation used 2 13 data points, a 200 ps time window, and adaptive spatial steps. The initial field was seeded at the input to the TDF through a (random phase) one photon per mode 81 .

Data availability

All data generated in this study are available within the manuscript and the Supplementary Information. Source data are provided with this paper.

Code availability

The simulation code used in this manuscript is available from the corresponding author upon request.

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Acknowledgements

This research was supported by the National key research and development program of China (2020YFB1805800 (G.Q. and Z.J.)), National Natural Science Foundation of China (NSFC) (Grant Nos.62090063 (G.Q.), 62075082 (Z.J.), U20A20210 (Z.J.), 61827821 (Z.J.), U22A2085 (G.Q.), 62235014 (F.M.), 62205121 (F.M.)), and the Opened Fund of the State Key Laboratory of Integrated Optoelectronics (G.Q., Z.J. and F.M.).

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G.Q., F.M., and C.L. conceived the idea and designed the experiments. C.L. built the systems and performed the experiments. F.M. and C.L. developed and performed the numerical simulations. G.Q., F.M., and C.L. performed data analysis, interpretation of the results, and preparation of the manuscript. L.C. and Y.J. assisted in building the systems. Z.J. and W.Q. contributed to data analysis, reviewing, and editing. All authors contributed to data analysis and the preparation of the manuscript.

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Lv, C., Meng, F., Cui, L. et al. Voltage-controlled nonlinear optical properties in gold nanofilms via electrothermal effect. Nat Commun 15 , 6372 (2024). https://doi.org/10.1038/s41467-024-50665-7

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DOI : https://doi.org/10.1038/s41467-024-50665-7

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