Supercontinuum generation with photonic crystal fibers

Transcription

1 Experiment 46 Supercontinuum generation with photonic crystal fibers Supercontinuum refers to a continuum of optical frequencies, or light with a very broad spectral bandwidth. This is also the reason that it is informally known as white light. The first observations of frequency broadening that led to the generation of supercontinuum were reported in crystals and glasses by Alfano and Shapiro almost half a century ago [1, 2]. Since then, supercontinuum generation has been investigated in a wide variety of nonlinear materials. Broad spectral beams often ranging from UV to IR with high spatial coherence have been obtained thanks to the major advances in laser technology, especially ultrashort pulsed laser systems [3]. Due to these unique properties, supercontinuum sources carry applications in many fields, including fluorescence microscopy, frequency metrology, and optical coherence tomography. Before the invention of the photonic crystal fiber by Prof. P. St. J. Russell [4], the generation of intense supercontinuum was possible only with the help of highly-specialized laser systems and extremely-high peak powers. Today, the nonlinear properties of photonic crystal fibers (PCFs) enable the implementation of white light sources using relatively-simpler lasers and lower intensities. This makes it possible to apply supercontinuum in wide areas of current research. Their significance is highlighted by the Nobel Prize in 2005 for Theodor Hänsch, who used a white light source for the high-precision determination of frequencies 1. In this experiment, we shall generate supercontinuum by pumping a solid core photonic crystal fiber [3 6] with intense pulses at a wavelength of 1064 nm and spectrally examine the output. Due to a variety of nonlinear optical processes occurring inside the fiber, new frequency components are produced. These new frequencies interact with each other, spectrally broadening the pulse as it propagates through the PCF. Using different (pump) laser intensities, we shall try to identify single nonlinear processes. For the measurement, we shall use a spectrometer the operation of which is based on optical Fourier transform. We shall find that the generated continuum encompasses almost the entire visible and near infra-red region. 1 Theoretical background This section introduces some basic physical concepts on which the experiment is based. It constitutes a rather broad overview and appropriate literature [7 9] must be referenced to gain a deeper insight into the concepts. As such, a majority of these topics will also be covered in the lecture Experimental Physics 3: Optics and Quantum Phenomena (Experimentalphysik 3: Optik und Quantenphänomene). Although it is not required to know all formulae and derivations by heart, you should obtain a basic understanding of the physical principles before performing this experiment. At the end of this manual (section 3, to be precise), some questions are marked with a sign and in blue colour. You must submit their answers ( ID: weisslicht@physik.uni-erlangen.de) two days before the actual experiment has been scheduled. 1.1 Plane waves, instantaneous frequency, and dispersion The wavefronts surfaces of constant phase of a plane wave, assumed to be travelling in the z direction, are infinite planes parallel to the xy plane. Although it is not practically feasible to produce a true plane wave, approximations of actual waves to plane waves considerably simplify the explanation of many optical phenomena. Light being an electromagnetic wave, the simplest expression for the electric field associated with a laser pulse at spatio-temporal coordinates z, t is: E(z, t) = E 0e ı(ωt kz+ϕ) + c.c., (1) where E 0 is the amplitude, ω is the optical frequency, k = k is the wavenumber 2, and ϕ [0, 2π] denotes the (constant) phase shift of the field. Deviations of actual waves from plane waves can be captured through relations such as E 0 E 0(z, t) and ϕ ϕ(z, t). Assuming the latter case, i.e. the phase shift is indeed a function of time, one can calculate the instantaneous frequency by differentiating the global phase Γ(z, t) = kz ωt + ϕ(z, t), ω inst = ϕ(z, t) (kz ωt + ϕ(z, t)) = ω, (2) t t 1 More at 2 i.e., the magnitude of the corresponding wave vector k. 1

2 Supercontinuum generation in photonic crystal fibers 2/10 where the wavenumber k has been assumed to be independent of time. The phenomenon in which the phase and/or group velocity of an optical field depends on the frequency ω (or equivalently, wavelength λ) is called chromatic dispersion. A mathematical description of chromatic dispersion of second and higher order can be given using the Taylor expansion of the wavenumber k as a function of the frequency ω k(ω) = k 0 + k ω (ω ω0) k 2 ω (ω 2 ω0) k 6 ω (ω 3 ω0) (3) around a central frequency ω 0. The empirical dependence of the refractive index n of a medium on the frequency of the propagating light, i.e., n n(ω) is specified by the Sellmeier equation [9]. Using the dispersion relation k(ω) = n(ω) ω c, (4) and the appropriate Sellmeier equation, one can numerically calculate the dispersion of various orders for a medium at a given frequency. 1.2 Fundamentals of nonlinear optics The Drude-Lorentz model describes a material as an ensemble of oscillators. In the case relevant here, these oscillators are driven by an optical field due to which a polarization P (r, t) is induced in the medium. As long as the intensity of the light field is low and/or the interaction of light with the medium is very weak, the following equation P (r, t) = ε 0χ (1) E(r, t), (5) suffices to describe the linear response of the medium. Here, ε 0 is the vacuum permittivity and the quantity χ (1) is known as the linear susceptibility. However, if the interaction of the medium with the optical field is quite strong or if the intensity of the light field is very high, the linear equation (5) cannot correctly describe the induced polarization. In that case, the polarization induced in the medium is given by the following nonlinear equation: P (z, t) = ε 0 ( χ (1) E(z, t) + χ (2) E 2 (z, t) + χ (3) E 3 (z, t) +... ). (6) where χ (j) is the j th order susceptibility. Note that we have replaced E(r, t) by E(z, t) to simplify the expression. For instance, it can be evaluated directly using equation (1) for the plane wave. Furthermore, both the field and all the susceptibilities are considered as scalar quantities here. In reality, the electric field must be treated as a vector and χ (j) becomes a (j + 1)-rank tensor [7]. Finally, the dependence of the nonlinear susceptibilities on the optical frequency ω of the incoming field is not explicitly shown. Nonetheless, equation (6) captures the essence of nonlinear optics. It shows a time-varying polarization that acts as a source of new components of the electromagnetic field 3. Depending on the intensity distribution I(z, t) = E(z, t) 2 through the material, a multitude of nonlinear processes occur. The resulting (new) components of the field interact amongst themselves as well as with the existing components inside the medium, giving rise to further new components. This generally leads to a spectral broadening and only a detailed and comprehensive analysis (performed at the output of the medium) can normally make it possible to identify single processes. The values of the susceptibilities naturally depend on the composition of the material. The structure of the material also plays a central role, especially concerning the nonlinear susceptibilities. For example, in the centrosymmetric assembly of molecules in an isotropic medium, χ (2) is zero [7]. In general, the magnitude of the nonlinear susceptibility decreases with increasing orders and in most cases, the response of a material is dominated by the first few orders (χ (j) for j 4). Figure 1 illustrates some possible interactions of the input and output electric fields for the second and third order nonlinearities. Note that in the χ (2) processes, the frequencies ω 1 and ω 2 are both necessarily less than ω 0. The connexion between susceptibility and optics can perhaps be understood better by the equation n(ω) = 1 + χ(ω). (7) that shows the relationship between the refractive index of the medium and susceptibility. For χ(ω) χ (1) (ω), one obtains the (normal) refractive index which governs physical principles such as the Snell s law. 3 An expression for which can be obtained through Maxwell s equations [7]. 2

3 Supercontinuum generation in photonic crystal fibers 3/10 Figure 1: Second and third order nonlinear optical processes. a) Two new frequencies ω 1 and ω 2 can be generated from ω 0 due to the χ (2) nonlinearity (ω 1 + ω 2 ω 0). b) The inverse process is also possible, i.e., fields at two exisiting frequency components ω 1 and ω 2 may result in a field at new frequency (ω 0 ω 1 +ω 2). c) The interaction due to χ (3) nonlinearity can be most generally described by interaction of four frequencies Optical Kerr Effect One of the most important nonlinear optical effects is due to the light-intensity dependent modification of the refractive index n of the medium. n(i) = n 0 + n 2 I (8) Here n 0[n 2] is the linear[nonlinear] refractive index of the medium. At a given λ, n 0 can be computed using Sellmeier relations while n 2 can be derived from the third order nonlinearity χ (3) (ω). For a glass fiber (or more formally, silica-based standard optical fiber), n 2 = cm 2 /W. If the intensity is a temporal function 4, or I I(t), then the refractive index of the material as seen by the optical pulse of the light is also a function of time. The optical Kerr effect in glass fibers, such as the photonic crystal fiber in our experiment, drives phenomena that can lead to the modification of the optical spectrum of the input light beam. These include self phase modulation, four wave mixing, modulation instability, and soliton formation. We study two of these below in brief Self phase modulation The phenomena of self phase modulation (SPM) is essentially a time-dependent phase shift induced on an intense pulse as it traverses the nonlinear medium. It can be explained using the concept of instantaneous frequency and dispersion; see subsection 1.1. By substituting the latter into the former, and using the fact that the material exhibits the Kerr effect given by equation (8), one can derive an expression for the instantaneous frequency as a function of the intensity of the input pulse. In other words, dependent on the temporal intensity profile of the optical pulse, the instantaneous frequency inside the pulse changes as it propagates through the medium. As the pulse emerges out of the medium, the overall change in the spectrum is proportional to the nonlinear index n 2 and the length L of the medium. Since optical fibers essentially provide L in the km range, the effects of SPM on the propagating pulses can be fairly significant Four wave mixing In this experiment we are especially interested in the case in which two incident fields are converted into two output fields at other frequencies, as also shown in figure 1(c). Such a process may most generally be called four wave mixing (FWM) and just like SPM, it arises from the third order nonlinearity as well. In FWM, the interaction depends on the relative phases of the participating beams. To elaborate, the following phase-matching conditions need to be satisfied: k3 + k 4 = k 1 + k 2 (9) ω 3 + ω 4 = ω 1 + ω 2 (10) One can visualize this process as the momentum and energy conservation of single photons 5. In case both incident photons are contained within the same laser pulse (sometimes called the pump), one obtains the case of degenerate four wave mixing. Figure 2 illustrates this situation. 1.3 Photonic crystal fibers Light propagates in optical fibers due to total internal reflection. Conventional optical fibers consist of an inner core surrounded by a solid cladding. To ensure total internal reflection at a given wavelength λ, the 4 which is naturally obtained in case of pulsed lasers. 5 Simply multiply all the variables in the equations by the Planck constant ħ. 3

4 Supercontinuum generation in photonic crystal fibers 4/10 Figure 2: Schematic representation of degenerate four wave mixing. Two identical incident photons are converted into two output photons with different frequencies and momenta. refractive index of the core is slightly higher than that of the cladding, i.e. n core(λ) n clad (λ). Optical fibers also offer simultaneously a long interaction length and a small mode area which effectively means that the atoms in the core see a high intensity. This can be exploited to generate nonlinear effects in fibers, as indicated by equation (6). It is mainly the unique cladding structure that makes photonic crystal 6 fibers different from conventional fibers. The solid and continuous cladding is replaced by a microstructure containing an array of air holes that run along the length of the fiber parallel to its core in PCFs. To produce photonic crystal fibers, hollow silica glass tubes are stacked and fixed in the first step. The stacked assembly is called preform and its overall structure can be chosen with some flexibility. The preform is heated in several steps and drawn out. Air pressure keeps the holes in the structure from collapsing. This stack and draw procedure is explained in more detail on the website of the Russell division at the Max Planck Institute for the Physics of Light [10]. The photonic crystal fiber used in our experiment is the so-called solid core fiber, which means it has a solid glass core. The reason for choosing glass is due to its easy availability, good mechanical strength, and transparency in the visible and IR regimes. The air hole structure in the cladding (due to the hollow tubes, as explained above) leads to a reduction of the effective refractive index. Smaller core radii can then be used while the fiber still supports only the fundamental mode, i.e., it is an endlessly single mode fiber [6]. This can be verified by the so-called V parameter which conforms to the normalized frequency and which should be between 1 V 2.4 so that the fiber supports only the fundamental mode [8]. It is given by: with ρ fiber denoting the radius of the fiber core Nonlinear effects in PCFs V = 2π fiber λ ρ n 2 core (λ) n 2 clad (λ), (11) Nonlinear effects that lead to a spectral broadening, or the supercontinuum generation, are highly dependent on the dispersion of the media. Clever dispersion design can be used for significantly reducing the input power requirements. During the production of PCFs, characteristics such as the zero dispersion wavelength can be tuned by choosing an appropriate preform structure and other process parameters. This makes it easy to obtain fairly bright supercontinuum light from PCFs by pumping them with pulses having peak powers of the order of a few kws instead of MWs. Due to glass being an isotropic material, the second order susceptibility χ (2) = 0 as mentioned in subsection 1.2. However, electric-quadrupole and magnetic-dipole moments or defects/impurities inside the fiber core may lead to χ (2) 0 or very weak second order nonlinear effects [8]. For instance, a tiny frequency doubling more formally called second harmonic generation may be observed. In addition, some self phase modulation is also expected from these kind of glass fibers. The dominant process for the spectral broadening however comes from four wave mixing; see subsection As an optical pulse propagates through the photonic crystal fiber, its spectrum can be broadened by a factor of 1000 or more. 2 Experiment 2.1 Short description of the setup The optical setup shown in figure 3 can be assumed to be divided in two main blocks which are discussed below. We briefly describe the role of some of the components in these blocks here and explain them in more detail in subsections 2.2 and The name may indicate, as is also sometimes wrongly believed, that the material used in manufacturing of PCFs is of a crystalline nature. However, this is not the case. 4

5 Supercontinuum generation in photonic crystal fibers 5/10 4-f configuration power meter head & controller power (μw) P2 slit BPF P1 M4 M2 irises MO in PCF MO out M3 NDF telescope knife edge M1 laser Figure 3: Tabletop view of the setup for generating and measuring supercontinuum. Various abbreviated optical elements are MO: microscope objective, NDF: neutral density filter, BPF: band pass filter (colour filter), P: prism, M: mirror, PCF: photonic crystal fiber. 1. Generation of the supercontinuum: A pulsed laser is used for pumping the photonic crystal fiber (PCF) in this experiment. The collimated laser beam runs through a telescope that serves to change the beam waist. It can optionally also pass through a neutral density filter (NDF). The beam is finally diverted towards the PCF with the help of two mirrors (M1 and M2). Using a microscope objective (MO in), the laser beam is coupled into the PCF. As explained in section 1, the supercontinuum itself is generated due to different nonlinear processes inside the PCF. 2. Measurements of the spectra: After having propagated through the fiber, the supercontinuum is outcoupled and re-collimated by another microscope objective (MO out). There is an option of inserting different band pass filters (BPFs) into the setup. Also, the beam waist can be measured with the help of a knife edge mounted on a sliding table. The white light is guided into the measurement apparatus a spectrometer with the help of two mirrors (M3 and M4). The spectrometer used in this experiment is based on the 4 f configuration that consists of two identical prisms (P1 and P2), two identical lenses and a slit installed on a motorized stage. After the prism P2, the power at a certain wavelength selected by the slit can be measured using a power meter. In addition to the optical setup, a computer system is also used to control the operation of the power meter and motorized stage simultaneously. To elaborate, a program running on the computer can be used for controlling the position of the slit while recording the optical power measured by the power meter. This can then be later stored into a file. 2.2 Procedure Generation of the supercontinuum Here we describe the first part of the setup that covers in detail on how to obtain the broadband spectra from the PCF. Apart from introducing some general rules and tricks, we shall also outline the safety procedures to be observed during the experiment. In case you are unsure at any step, you must ask the supervisor of the experiment Obtaining the laser output The laser used for pumping the PCF operates at a central wavelength of 1064 nm, repetition rate 8.5 khz and optical pulse width of 660 ps. Internally, this laser itself is pumped by a laser diode at a wavelength of 810 nm. The laser can be switched on simply by turning the key attached to the laser controller box and pressing the ON (red) button. As and when the laser is running, the occupants of the room must wear laser-safety goggles and the doors of the room must remain closed. 5

6 Supercontinuum generation in photonic crystal fibers 6/10 a) b) power (µw) power meter ND filter laser M2 M1 I1 I2 x z y MO in Figure 4: a) Characterization of the neutral density filters. b) Beam walking using a pair of mirrors and iris diaphragms to focus into the fiber. The objective lens for incoupling is a fixed part of a movable 3D translation stage (the fiber tip, not shown in the figure, is mounted on the movable part). The position of this fiber tip can be adjusted in the x, y or z directions with the three screws of the 3D stage. The laser output can be observed by placing the IR-sensitive card in the beam path. The output power of this laser, as measured directly by a power meter set to measure at 1064 nm is 50 mw. The power meter head in this experiment can also maximally measure 50 mw. Since a higher optical power could damage the sensor element in the power meter, it is advised not to keep the power meter directly in the path of the laser beam for too long Setting up the telescope A telescope is utilized in this experiment to change the beam waist or diameter obtained at the laser output 7. This is necessary to ensure a maximal coupling of light into the PCF. The telescope must be placed in beam path as shown in figure 3. As a general rule of thumb, the central portion of an optical component must be employed in/for the transmission/reflection of light. For instance, you must try to place both the lenses (that form the telescope) in such a way that the optical beam passes through them as close as possible to their respective centers. Also, while handling any optical component, for instance to adjust its position, take care not to touch the glass part with your bare fingers Characterization of the neutral density filters Before performing this step, make sure the wavelength setting at the power meter is 1064 nm. If not, adjust it using either the software or the power meter controller. Place the power meter after one of the chosen neutral density filters, as shown in figure 4(a), and measure the output power while characterizing the measurement accuracy, i.e., noting down the level of power fluctuations. Repeat this procedure for all the other ND filters. During the transition from one NDF to another, remember to block the laser beam Coupling light into the PCF The iris diaphragms I1 and I2 facilitate guiding the beam so that it enters the microscope objective lens optimally. Assuming the beam hits the mirrors M1 and M2 near their respective centers, you can adjust the beam path using the beam walking technique as illustrated by figure 4(b). During this entire procedure, you must only open or close an iris diaphragm but not change its position in the x, y or z directions. The IR-sensitive card and/or the power meter can be used to judge the power of the beam passing through an iris. For the beam walking procedure: 1. Make the aperture of the iris diaphragms I1 and I2 as small as possible. 2. Using only the first mirror M1, steer the beam in the x and y directions so that it passes through the slightly-open aperture of I1 maximally. 3. Open I1, and use the second mirror M2 to steer the beam so that it passes through I2 maximally. 4. Repeat steps one to three above until the beam passes through both I1 and I2 optimally at the same time, i.e., any further iterations do not increase the power observed after a closed iris. Now open both I1 and I2 fully and place the power meter after the outcoupling microscope objective MO out and maximise the power reading by aligning the fiber tip w.r.t. the wavefront of the optical beam. For this 7 The divergence of the laser itself is corrected by the lens directly at the output of the laser. 6

7 Supercontinuum generation in photonic crystal fibers 7/10 x y z power (µw) PCF MO out knife edge power meter Figure 5: Knife edge measurement. The smallest unit on the micrometer screw corresponds to a movement of 10 µm in the x direction. purpose, the position of the fiber tip relative to the objective lens MO in can be carefully controlled using the three adjusting screws of the 3D translation stage. Be very careful not to disturb the fiber while adjusting the screws. Once you have obtained a few mws of power that does not seem to increase any further by using the 3D stage alone, repeat the beam walking step described before. In this manner, you can ensure that a maximum coupling of light into the fiber is obtained. Once a relatively high coupling ratio 8 is obtained, you can stop. The output beam should be visible on the white portion of the IR card Knife edge measurements Mount the power meter and the knife edge assembly near the outcoupling microscope objective MO out as shown in figure 5. The initial placement of the knife edge must not be too far from the beam but should not also result in a loss of power. By rotation of the screw located at the base of assembly, move the blade in the direction of the beam. You should measure until you cannot detect any power, i.e., until you measure only the background light. Movement in steps of 100 µm or smaller is recommended (least count of the micrometer screw = 10 µm). Using the power measurements obtained from this part of the experiment, you can estimate the diameter of the laser beam. 2.3 Procedure Measurements on the spectra As shown in figure 6 and already mentioned, we use a 4 f configuration consisting of two prisms, two lenses and a slit (installed on a motorized stage) for doing the spectral analysis of the supercontinuum generated in the PCF. The term 4 f essentially denotes the total distance between the two prisms in this configuration, where f is the focal length of the identical lenses 9. Note that the configuration is symmetric in nature. In the following, we recommend that you switch off the room lights whenever you want to record a measurement trace. If you prefer to have the lights on during measurements, you should definitely cover the spectrometer with the metal screens available. In addition, you should then record a full spectrum with the laser blocked. This trace gives you the background level which must be subtracted from all the measurement traces recorded otherwise Calibration of the spectrometer (zero point setting) Put the 650 nm band pass filter after the objective MO out while noting the correct transmission direction of light. Observe the (brilliant red) spot after the prism P1 on a white card. By turning the rotatory platform on which this prism is mounted, you can translate the spot on the card in the x direction. You should adjust the angle between the outgoing light and the incoming light so that their relative deflection is as small as possible. This puts the prism in the required angle of deviation or θ = θ min as shown in figure 6. Although this may be performed by moving the knobs of M4, we recommend that only the prism be rotated. Perform the same step for P2 (remove the slit, if present, to let the beam propagate to P2). Once the minimum angle of deviation is set for P2, put the power meter in the beam path after P2 and set the wavelength of the power meter to 650 nm. This can be done through the panel interface of the program. Now insert the 50 µm slit onto the motorized stage and move it in the x direction with the help of the software 10. You can initially choose coarser steps to bring the slit as close as possible to the light spot. The 8 the power at the output of the fiber divided by the power at the input. 9 which implies that the distance between either of the two lenses and the slit is also exactly one focal length. 10 Make sure that the setting Full is chosen in the Parameters panel of the software interface. 7

8 Supercontinuum generation in photonic crystal fibers 8/10 θ P2 slit P1 f f f f M4 power (μw) power meter x y z band pass filter M3 Figure 6: Setting up and basic operation of the spectrometer. Once the spectrometer is calibrated, the motorized stage and the power meter are controlled by the computer. The inset on the left shows the deviation of a ray as it passes through the prism. step size and slit speed can be controlled using the dropdown menu in the Slit panel. Negative distances in the program correspond to a movement of the slit towards smaller wavelengths. As and when some of the beam begins to pass through the slit, you would observe increasing powers at the power meter stationed after P2. By choosing smaller or finer steps, you can then make sure that the beam passes through the slit optimally. This position corresponds to the zero point of the spectral measurements. You can set this in software by pressing the Set zero point button. After this, the motion of the motorized stage is synchronized with the wavelength setting of the power meter so that the latter estimates the correct power at the wavelength selected by the moving slit. Measure the transmission spectra for the other two band pass filters (at 550 nm and 850 nm). Then take another trace without any filters. This is the complete supercontinuum and you should be able to identify at least two more peaks here: one at 1064 nm and another at 810 nm; refer subsection for more details. In this manner, you would basically obtain five pairs of values: the x coordinate of the well-defined peaks and the corresponding powers. This is helpful in the mapping from x to λ for plotting the spectra Miscellaneous spectral measurements Insert the five ND filters (after the telescope as shown in figure 4(a)) with the lowest values one by one and measure the full spectrum. The motivation behind these measurements, as explained in section 1.2, is that different intensities may result in different nonlinear processes. This could bring in a stark variation of the supercontinuum. Block the laser beam and exchange the 50 µm slit with the 10 µm slit. Set the option on Detail in the Parameters of the software interface. This option will shift the starting point of the motorized stage to an appropriate value. It will also, as the name suggests, make finer steps so that the 10 µm slit can resolve the peaks around the pump wavelength of 1064 nm. This can facilitate, for instance, the identification of four wave mixing processes. Before recording a trace, it is advised that you re-check the calibration of the spectrometer by using the 650 nm band pass filter as was explained in the previous subsection. If the calibration is correct, record the trace. Otherwise, repeat the steps explained in the previous subsection. For the penultimate step of this experiment, insert five ND filters with the highest values one by one and measure the respective spectra. Record at least two more spectra with different combinations of the ND filters. As the last step, remember to copy the files from the PC. Also, write down the values, e.g., the focal lengths of the lenses that make the telescope, which may be of importance in the evaluation that you need to present in the report. Ask your supervisor if you are not sure whether you have noted down all the values. 3 Discussion and report writing In the report, you should analyse the data from the various measurements taken during the experiment. You can use a polynomial fit to convert from the x data to λ (refer subsection 2.3.1) and then plot the spectra 8

9 Supercontinuum generation in photonic crystal fibers 9/10 from all measurements. These include the full and detail measurements, with or without neutral density filters, and/or band pass filters. You should discuss the results in addition to reflecting upon the questions below. Remember that the questions with signs (additionally in blue colour) form the prelab work: their answers must be submitted at least two days before the day the actual experiment has been scheduled ( ID: The remaining questions must be addressed in the final report. 3.1 Pulsed laser optics What is the difference between the carrier frequency of a pulse and its instantaneous frequency? What effects does dispersion have on a light pulse? Keep in mind that a light pulse has a relatively broad spectrum (compared to, for instance, a monochromatic light field). How may the widths of an optical pulse envelope in the frequency and time domain be connected? Given an average power 11 P avg = 50 mw and the repetition rate f Rep = 8.5 khz of the pulse train from a laser at a central wavelength of λ = 1064 nm and a pulse width of τ = 0.66 ns, calculate the peak power of this pulse? You may assume the temporal shape of the pulses to be rectangular. If the pulse in the previous question travelled through a PCF with radius ρ fiber = 5 µm, what was the peak intensity seen by the atoms inside the PCF? Assume the PCF cross-section to be a perfect circle. 3.2 Nonlinear optics Calculate explicitly the polarization induced in a medium during a nonlinear process of the second order when two monochromatic fields come upon the medium. What, as a consequence, happens in the case of nonlinear processes of the third order? Think of the possible limits to the validity of the Sellmeier equation. What effect may the energy and momentum conservation have on the possibly-generated frequencies in the four wave mixing process (subsection 1.2.3)? Plot the variation of power observed with the different ND filters used in subsection Compare the measured spectra obtained in subsection with reference to these different input powers and try to identify different nonlinear processes. 3.3 Beam and Fiber optics What is meant by the term numerical aperture (NA) in a physical sense? What is the NA of 1) a glass fiber 2) a microscope lens? relationship between them to achieve a good coupling. Also think about what should be the What are the Gauss q parameters? What is the ray transfer matrix formalism for Gaussian beams? Compare the basic structure of photonic crystal fibers with that of (conventional) single mode fibers. Determine the diameter of the laser beam from your knife edge measurements. What are the different ways to define the diameter of a light beam? Which way did you choose? Why do you get a Gaussian beam profile at the exit of the fiber? The divergence of the beam corresponds to the maximum numerical aperture of the photonic crystal fiber. From this, calculate the NA of the fiber and compare it with the NA of the microscope lens of Calculate the beam diameter in the focus (at the endface of the PCF) from the estimated diameter of the laser beam and the focal length (f = +8 mm) of the microscope objective MO out and the divergence of the beam with the help of Gauss q parameters. Assuming the beam diameter before being coupled into the fiber (at MO in i.e.) is the same as your measurement, calculate the divergence of the laser. 11 e.g., the measurements by a power meter yield the average power. See subsection for example. 9

10 Supercontinuum generation in photonic crystal fibers 10/ Spectrometry As described in subsection 2.3.1, the prisms are used in the configuration of the minimum angle of deviation. How can the refractive index of the prism be calculated from this configuration? What are the advantages of using the 4 f configuration in your opinion? What would happen if the distance between one of the lenses and the slit was not equal to the focal length f? Calculate the connection between wavelength and deflection angle in the prism with the help of the Sellmeier coefficients for SF 11 glass. What follows from the graph for the resolution of the spectrometer? Calculate the resolution of the spectrometer used in the experiment in the green and in the infrared wavelength range. References [1] R. R. Alfano and S. L. Shapiro, Emission in the Region 4000 to 7000 ÅVia Four-Photon Coupling in Glass, Physical Review Letters 24, (1970). [2] R. R. Alfano and S. L. Shapiro, Observation of Self-Phase Modulation and Small-Scale Filaments in Crystals and Glasses, Physical Review Letters 24, (1970). [3] J. M. Dudley, G. Genty, and S. Coen, Supercontinuum generation in photonic crystal fiber, Reviews of Modern Physics 78, (2006). [4] P. Russell, Photonic Crystal Fibers, Science 299, (2003). [5] J. K. Ranka, R. S. Windeler, and A. J. Stentz, Visible continuum generation in air-silica microstructure optical fibers with anomalous dispersion at 800 nm, Opt. Lett. 25, (2000). [6] W. Wadsworth, N. Joly, J. Knight, T. Birks, F. Biancalana, and P. Russell, Supercontinuum and fourwave mixing with Q-switched pulses in endlessly single-mode photonic crystal fibres, Opt. Express 12, (2004). [7] R. W. Boyd, Nonlinear Optics (Academic Press, 2008), 3rd ed. [8] G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 2006), 4th ed. [9] RP Photonics (better content compared to Wikipedia), [10] Fabrication of photonic crystal fibers at MPL, topics/fabrication.html. 10