# The Influence of the External Signal Modulation Waveform and Frequency on the Performance of a Photonic Forced Oscillator

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Porous Silicon Fabrication

#### 2.2. Photonic Bandgap Structure

#### 2.3. Photodyne Fabrication

#### 2.4. Experimental Setup

#### 2.5. Oscillations Theoretical Model

_{0}with the electric field polarized in the y-direction with a magnitude of:

_{i}and B

_{i}are the complex amplitudes of the electric field in each region of the structure. The 0 label is for the amplitudes in the air region. The wave vectors are represented by k

_{i}at different regions on the structure in the x-direction. β is the wave vector in the z-direction and is given by $\omega {n}_{0}\mathrm{sin}\left({\theta}_{0}/\mathrm{c}\right)$, where ${n}_{0}$ and ${\theta}_{0}$ are the refractive index and angle of incidence of the air region, c is the speed of light and ω is the light angular frequency. The wave vectors in the x-direction are given by $\omega {n}_{0}\mathrm{cos}\left({\theta}_{i}/\mathrm{c}\right)$, where ${n}_{i}$ and ${\theta}_{i}$ are the refractive index and angle of incidence of region I, the latter given by ${\theta}_{i}={\mathrm{sin}}^{-1}({n}_{0}\mathrm{sin}{\theta}_{0}/{n}_{i})$. By using a similar formalism as presented in [9,10,11,12,13,14,15,16], it can be shown that for lossless dielectrics the surface force density only exists in the x-direction and is given by:

_{i}and B

_{i}and their phases φ

_{i}can be calculated by using the well-known transfer matrix method [2].

_{light}for any waveform signal used here, as well as an inverse relation between the amplitude of oscillations and the damping coefficient h. The electromagnetic force per mass unit parameter controls the total energy provide to the photodyne. This energy controls the maximum amplitude of the oscillations. Therefore, for any waveform, the oscillation amplitude depends on the combination of these parameter values. That is, the maximum theoretical mechanical power used by the photodyne is given by the combinations of these three parameters.

## 3. Results

^{2}while h and ${n}_{light}$ took different values in each example.

_{light}has been normalized for the representation).

## 4. Discussion and Conclusions

^{−3}equals 497,600 Hz, almost half a MHz. These range of frequencies are in the correct range found elsewhere in photonic crystals [17,18] and they could be 0.5 × 10

^{6}faster that the frequencies we found in our experiments. Moreover, since GAWBS is a thermal effect [18], in the past, we have done some control experiments to rule out heating effects from laser illumination. To measure the influence of laser fluctuations and temperature changes on the possible generation of mechanical oscillations in our device, a laser illuminated one photodyne device for five minutes and the vibrometer measured any possible oscillation. The result only showed a 1/f

^{a}noise signal with no peaks within the frequency band of interest (1 Hz–64 Hz), thus if GAWBS were presented they could certainly produce some mechanical oscillations but clearly it was not the case [15]. Finally, it is known that GAWBS could modulate the existence of localized photonic states [17] and, since the electromagnetic force is always maximized when a localized state is excited, a photodyne with no localized states could not maximize the electromagnetic force. Nevertheless, theoretical calculations showed that the electromagnetic force is still high (500 times higher than any current optical tweezer) even in the case where there are no localized states. Consequently, from all these arguments, GAWBS cannot explain the nature of the oscillations we have found in this work and in the past [13,14,15] but they may contribute as noise within our measurements. For future work, experimental studies on measuring GAWBS could be included. This can be interesting for sensor applications since the GAWBS peaks are highly sensitive to strain and temperature.

## Supplementary Materials

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

- Dholakia, K.; Spalding, G.; MacDonald, M. Optical tweezers: The next generation. Phys. World
**2002**, 15, 31–35. [Google Scholar] [CrossRef] - Ashkin, A.; Dziedzic, J.M. Optical trapping and manipulation of viruses and bacteria. Science
**1987**, 235, 1517–1520. [Google Scholar] [CrossRef] [PubMed] - Higurashi, E.; Ohguchi, O.; Tamamura, T.; Ukita, H.; Sawada, R. Optically induced rotation of dissymmetrically shaped fluorinated polyimide micro-objects in optical traps. J. Appl. Phys.
**1997**, 82, 2773–2779. [Google Scholar] [CrossRef] - Li, J.M.; Dong, T.L.; Shan, G.J. Enhancement of electromagnetic force by localized fields in one-dimensional photonic crystal. Prog. Electromagn. Res. M
**2009**. [Google Scholar] [CrossRef] - Gomez-Medina, R.; San José, P.; García-Martín, A.; Lester, M.; Nieto-Vesperinas, M.; Sáenz, J.J. Resonant radiation pressure on neutral particles in a waveguide.
Phys. Rev. Lett.
**2001**, 86, 4275–4277. [Google Scholar] [CrossRef] [PubMed] - Mizrahi, A.; Horowitz, M.; Schächter, L. Torque and longitudinal force exerted by eigenmodes on circular waveguides. Phys. Rev. A
**2008**. [Google Scholar] [CrossRef] - Povinelli, M.L.; Ibanescu, M.; Johnson, S.G.; Joannopoulos, J.D. Slow-light enhancement of radiation pressure in an omnidirectional
reflector waveguide. Appl. Phys. Lett.
**2004**, 85, 1466. [Google Scholar] [CrossRef] - Soukoulis, C.M. Photonic Band Gaps and Localization, 1st ed.; Springer: New York, NY, USA, 1993; ISBN 978-0-306-44494-4. [Google Scholar]
- Yeh, P. Optical Waves in Layered Media; Wiley-Interscience: Hoboken, NJ, USA, 2005; ISBN 978-0-471-46990-2. [Google Scholar]
- Joannopoulos, J.D.; Johnson, S.G.; Winn, J.N.; Meade, R.D. Photonic Crystals: Molding the Flow of Light; Princeton University Press: Princeton, NJ, USA, 2011; ISBN 978-0-691-12456-8. [Google Scholar]
- Soukoulis, C.M. Photonic Crystals and Light Localization in the 21st Century; Springer Science & Business Media: New York, NY, USA, 2012; ISBN 978-0-792-36948-6. [Google Scholar]
- Antonoyiannakis, M.I.; Pendry, J.B. Electromagnetic forces in photonic crystals. Phys. Rev. B
**1999**. [Google Scholar] [CrossRef] - Lugo, J.E.; Doti, R.; Sanchez, N.; de la Mora, M.B.; del Rio, J.A.; Faubert, J. The bifoil photodyne: A photonic crystal oscillator. Sci. Rep.
**2014**. [Google Scholar] [CrossRef] [PubMed] - Lugo, J.E.; Doti, R.; Sanchez, N.; Faubert, J. Inducing forced and auto-oscillations in one-dimensional photonic crystals with light. J. Nanophoton.
**2014**. [Google Scholar] [CrossRef] - Sanchez-Castro, N.; Palomino-Ovando, M.A.; Estrada-Wiese, D.; del Río, J.A.; de la Mora, M.B.; Doti, R.; Lugo, J. EA photonic self-oscillator based on porous silicon. Mesopourous Biomater.
**2016**. [Google Scholar] [CrossRef] - Mizrahi, A.; Schachter, L. Electromagnetic force son the dielectric layers of the planar optical Bragg acceleration structure. Phys. Rev.
**2006**. [Google Scholar] [CrossRef] - Krishnamurthy, S.; Santos, P.V. Optical modulation in photonic band gap structures by surface acoustic waves. J. Appl. Phys.
**2004**, 96, 1803. [Google Scholar] [CrossRef] - Elser, D.; Sylvestre, T.; Maillotte, H.; Mélin, G.; Laude, V. Guided acoustic wave Brillouin scattering in photonic crystal fibers. J. Phys. Conf. Ser.
**2007**, 92, 012108. [Google Scholar] [CrossRef] - EETIMES Phase Noise and Jitter—A Primer for Digital Designers. Available online: https://www.eetimes.com/document.asp?doc_id=1277196 (accessed on 16 March 2018).

**Figure 1.**(

**a**) Photonic device used in the experiment. Yellow and blue layers represent the Psi multilayer, while light blue layer is the air defect in the middle of the structure. (

**b**) The measured transmittance of the complete multilayered device of porous silicon (top) has been compared with the theoretical model (bottom). The band gaps coincide in all cases, specifically in the selected working region (around 633 nm), for the refractive index of ${n}_{1}=1.1$ and ${n}_{2}=2$ and thicknesses of ${d}_{1}=335\text{}\mathrm{nm}$ and ${d}_{2}=438\text{}\mathrm{nm}$. The transmittance line of the multilayer used in the experiment is the black line.

**Figure 3.**(

**a**) Experimental setup: Photodyne (1), Neutral filter wheel (2), infrared band-pass (3), He-Ne Laser (4), mechanical chopper (5), vibrometer (6), photocell (7), computer (8), oscilloscope (9), function generator (10), and lineal polarizer (11). The drawing in the orange circle shows the mounted photodyne. A picture of the multilayers produced by SEM can be seen in the blue circle. (

**b**) An example of ideal waveforms of the signals used in the experiments. From top to bottom: sinusoidal, rectangular and triangular.

**Figure 4.**Photodyne movement velocity (

**a**,

**b**) and Fourier spectrum (

**c**,

**d**). Comparison of the theoretical (orange) and experimental results (blue). Rectangle signal, input frequency of 5 Hz. Calculated value of h = 41.4 and n

_{ligth}= 0.42.

**Figure 5.**Photodyne movement velocity (

**a**,

**b**) and Fourier spectrum (

**c**,

**d**). Comparison of the theoretical (orange) and experimental results (blue). Sinusoidal signal, input frequency of 10 Hz. Calculated value of h = 27.1 and n

_{ligth}= 0.50.

**Figure 6.**Photodyne movement velocity (

**a**,

**b**) and Fourier spectrum

**(c**,

**d**). Comparison of the theoretical (orange) and experimental results (blue). Rectangle signal, input frequency of 15 Hz. Calculated value of h = 33.2 and n

_{ligth}= 0.5.

**Figure 7.**Behavior of the photodyne displacement (in nanometers) against three different types of waveforms. The parameter n

_{light}has been normalized for the representation.

**Figure 8.**Maximum power of the oscillation movement against the three external frequencies and waveforms.

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**MDPI and ACS Style**

Sánchez-Castro, N.; Palomino-Ovando, M.A.; Estrada-Wiese, D.; Valladares, N.X.; Del Río, J.A.; De la Mora, M.B.; Doti, R.; Faubert, J.; Lugo, J.E. The Influence of the External Signal Modulation Waveform and Frequency on the Performance of a Photonic Forced Oscillator. *Materials* **2018**, *11*, 854.
https://doi.org/10.3390/ma11050854

**AMA Style**

Sánchez-Castro N, Palomino-Ovando MA, Estrada-Wiese D, Valladares NX, Del Río JA, De la Mora MB, Doti R, Faubert J, Lugo JE. The Influence of the External Signal Modulation Waveform and Frequency on the Performance of a Photonic Forced Oscillator. *Materials*. 2018; 11(5):854.
https://doi.org/10.3390/ma11050854

**Chicago/Turabian Style**

Sánchez-Castro, Noemi, Martha Alicia Palomino-Ovando, Denise Estrada-Wiese, Nydia Xcaret Valladares, Jesus Antonio Del Río, Maria Beatriz De la Mora, Rafael Doti, Jocelyn Faubert, and Jesus Eduardo Lugo. 2018. "The Influence of the External Signal Modulation Waveform and Frequency on the Performance of a Photonic Forced Oscillator" *Materials* 11, no. 5: 854.
https://doi.org/10.3390/ma11050854