Design, implementation and characterization of a pulse stretcher to reduce the bandwith of a femtosecond laser pulse

Transcription

1 BACHELOR Design, implementation and characterization of a pulse stretcher to reduce the bandwith of a femtosecond laser pulse ten Haaf, G. Award date: 2011 Link to publication Disclaimer This document contains a student thesis (bachelor's or master's), as authored by a student at Eindhoven University of Technology. Student theses are made available in the TU/e repository upon obtaining the required degree. The grade received is not published on the document as presented in the repository. The required complexity or quality of research of student theses may vary by program, and the required minimum study period may vary in duration. General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. Users may download and print one copy of any publication from the public portal for the purpose of private study or research. You may not further distribute the material or use it for any profit-making activity or commercial gain

2 Eindhoven University of Technology Department of Applied Physics Coherence and Quantum Technology (CQT) Design, Implementation and Characterization of a Pulse Stretcher to Reduce the Bandwidth of a Femtosecond Laser Pulse G. ten Haaf CQT Supervisors: prof.dr.ir. O.J. Luiten dr.ir. E.J.D. Vredenbregt ir. W.J. Engelen

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4 Abstract Ultrafast electron diffraction enables the study of physical, chemical and biological processes on a very short time- and length scale. To do this, a large transverse coherence length of the electron bunch is required. Therefore ultracold electron bunches have to be created. This can be done by an ultracold plasma in an accelerator structure. Here the electrons are created by near threshold photo ionization. The excess energy created by this process determines the temperature, so also the transverse coherence length, of the electron bunch. To make this excess energy as low as possible, the bandwidth of the laser pulse has to be very low. For a temperature of 10 K, the bandwidth of the laser pulse has to be 0.24 nm. This thesis describes the design, implementation and characterization of a pulse stretcher to create laser pulses with a bandwidth of 0.24 nm. The pulse stretcher contains a grating to spatially separate the wavelengths. After this grating the light goes through a lens, which converts the angular dispersion of the grating to a linear dispersion. A slit is then placed after the lens to cut out the target bandwidth. The slit that has to be used to cut out a 0.24 nm bandwidth, is one with a width of 250 µm, which is a realizable size. The efficiency of a setup like this would be 1-2%. The shape of the spectrum of the transmitted light, depends on the longitudinal and transverse position of the slit and on the slit width. A model is made to check what the spectrum would look like as a function of these parameters. The most important conclusion is that the spectrum becomes more sharply peaked around the central wavelength if the slit is placed closer to the lens. It will have a more triangle like shape instead of a block form. If the slit is placed in the focal point of the lens, the spectrum will have the straightest edges, but they are not completely straight because of instrumental broadening. iii

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6 Contents 1 Introduction 1 2 Grating Theory Normal diffraction gratings Blazed diffraction gratings Angular dispersion Setup Grating configuration Beam path Beam projection Group velocity dispersion Two level system Focusing the beam Lens choice Model Gaussian laser spot Instrumental broadening Boundaries Intensity as a function of wavelength Results of the model Longitudinal slit position Transverse slit position Slit width Measurements Dispersion Longitudinal distance Efficiency Conclusion and outlook 35 Appendices 37 v

7 Contents A Additional measurements 37 B Alignment 41 vi

8 Chapter 1 Introduction Ultrafast electron diffraction is a technique to investigate structures on a atomic level and a ultrashort timescale. When an electron beam is sent through a sample, the resulting diffraction pattern can give insight in the atomic structure. A requirement for the electron beam is that the transverse coherence length is larger then the lattice constant of the sample. This means that the electron temperature has to be very low. A state-of-the art electron source at the CQT-group creates femtosecond electron bunches by photoemission in a photogun[1]. The transverse coherence length of these bunches is 2 nm. This coherence length is large enough for simple lattices, such as a gold, but if one wants to study a protein crystal, with a typical lattice spacing of 3 nm, a higher transverse coherence length has to be achieved. One solution is to make an electron source that creates colder atoms. At the CQT-group such a source has been developed, namely an ultracold plasma (UCP) in an accelerator structure. The working principles of this source are described in reference [2]. The ultracold plasma is created in two stages, first a cloud of Rb-atoms is cooled and trapped by a so called magneto-optical trap (MOT). The second stage is the creation of the ultracold plasma itself. Therefore two lasers cross each other at the position of the atom cloud. The first one excites the atoms to a higher energy level, from which it can be brought to the ionization threshold with a 480 nm laser. Now an electron cloud is created which can be accelerated out of the MOT by applying an electric field. The temperature of this cloud is dependent on the excess energy which is the difference between the ionization threshold, which is dependent on the acceleration voltage, and the energy of the photons. To create a cold electron bunch, the photon energy has to be just enough to be bring the electrons above the ionization threshold. In this way no excess energy is created due to the laser and the electron cloud will be as cold as possible. Since the ionization threshold is not the same for different 1

9 Chapter 1 Introduction acceleration voltages, a femtosecond laser system is needed, from which the output wavelength of the light is tunable. The femtosecond laser pulses are created by a system which is described in reference [3]. The system consists among others of an optical parametric amplifier (OPA), which is a table-top device that converts 800 nm femtosecond laser pulses in femtosecond laser pulses with an other wavelength. This device is used to create femtosecond laser pulses containing photons with just enough energy to ionize the atoms. These laser pulses are Gaussian distributions in the wavelength, so that not only photons are created which bring the electrons just above the ionization threshold, but there is allways a spread in energy. It seems logical to create an electron bunch with as little spread in energy as possible, but this is not entirely true. Due to a process called disorderinduced heating the electron bunch will heat up until its kinetic energy is in equilibrium with the potential energy. This means that the lowest possible equilibrium temperature of the electron bunch is determined by the Coulomb interaction energy of neighboring atoms. Due to this process it is useless to start with an initial electron temperature lower than 10 K [4]. Such a bunch can be created by narrowing the bandwidth of the laser pulse. To do this, a relation has to be found between the wavelength bandwidth and the initial electron temperature. The full width half maximum (FWHM) bandwidth in wavelength λ can be found by setting the FWHM bandwidth in energy E equal to the kinetic energy of the produced electrons E = 3 2 k BT i, (1.1) in which k B is the Boltzmann constant and T i is the initial electron temperature. The relation between the wavelength λ and the energy E of a photon is given by λ = hc E, (1.2) in which h is Planck s constant and c is the speed of light. So the relation between λ and E can be given by λ = dλ hc de E = E. (1.3) E2 Together with equation 1.1 and 1.2 this can be rewritten as λ = 3k Bλ 2 2hc T i. (1.4) With this equation the FWHM wavelength spread for a initial electron temperature of 10 K and a central wavelength of 480 nm, can be calculated 2

10 to be 0.24 nm. If the laser pulse is Fourier Transform limited, the Heisenberg energy-time uncertainty principle states that the pulse length is connected to the bandwidth by σ t = h 2σ E [5]. In this equation σ t and σ E are the standard deviation in respectively time and energy. This equation can be rewritten in FWHM-spreads by t = h 4 ln 2. (1.5) E This leads to an optimal FWHM pulse length of 1.4 ps. With the Heisenberg energy-time uncertainty principle and equation (1.1) it can be calculated that the 50 femtosecond laser pulses created by the tunable laser system would have a FWHM bandwidth of 6.7 nm and would create electron bunches with an electron temperature of 281 K. In reference [3] temperature measurements are discussed for electrons created by nanosecond laser pulses and femtosecond laser pulses. The expectation of these measurements was a lower electron temperature for the nanosecond pulses, because the Heisenberg uncertainty principle states that for a Fourier Transform limited pulse the bandwidth should be smaller if the pulse length is larger and a smaller bandwidth means a smaller excess energy so a lower temperature. However the measurements showed a lowest temperature of 69 K for the nanosecond pulse and 11 K for the femtosecond pulse, exactly the opposite of what was expected. To clear up what is the reason for the disagreement between theory and measurement, further research has to be done. A very interesting measurement would be a measurement in between the nanosecond and the femtosecond region. Therefore the bandwidth of the femtosecond pulse has to be narrowed, to create a picosecond pulse. This thesis describes the design, implementation and characterization of this pulse stretcher. The pulse stretcher spatially separates the wavelengths in the laser pulse with the use of a blazed grating. The theory behind such a grating is described in chapter 2. The design of the setup is discussed in chapter 3. To characterize the setup, a model is made to see what the transmission of the setup will be as a function of the wavelength of the light. This model is described in chapter 4. Finally measurements are done to see whether the model agrees with the reality. These measurements are discussed in chapter 5. 3

11 Chapter 1 Introduction 4

12 Chapter 2 Grating Theory With the use of a diffraction grating it is possible to spatially separate incoming light into its different frequency components. In this chapter the properties of a diffraction grating are discussed. More on the theory in this chapter can be found in reference [6]. In paragraph 2.1 the working principles of a normal grating are described. In the set-up that is described in this report a blazed grating is used. This kind of grating is discussed in section Normal diffraction gratings When light falls on a diffraction grating, each wavelength is diffracted with a different angle. This is due to a path-difference between light rays falling on different positions on the grating. When a ray of light falls on the grating with an angle θ i, measured from the normal to the grating, and leaves the grating with an angle θ, the total path difference between two rays of light falling on successive grooves is = 1 2 = a sin θ a sin θ i, (2.1) in which the distance between the centers of two adjacent grooves is called the grating constant a. This equation can be verified in fig In this figure a sign convention is used that an angle left from the normal of the grating is a negative angle and right from the normal is a positive angle. When the path difference between the two rays of light is equal to an integer number m of wavelengths λ there will be constructive interference of the light rays. This condition leads to the grating equation given by mλ = a(sin θ + sin θ i ). (2.2) This equation gives the angles of the maxima in the intensity distribution, but doesn t say anything about the intensity at other angles. The total 5

13 Chapter 2 Grating Theory θ i θ Δ 1 Δ -θ i θ 2 Figure 2.1 The path difference = 2-1 of light rays falling on successive grooves of a diffraction grating is determined by the angle of incidence, θ i, and the angle of the outgoing light beam with the normal of the grating, θ. a amplitude of the electric field as a function of the angle of diffraction can be calculated by making an integral of the electric field over all waves that come from one of the grooves and then add all of these integrals to get the total electric field at a certain angle of diffraction. The intensity I then is found by taking the square of the electric field amplitude. The result of this procedure [6] is given by I = I 0 ( sin β β ) 2 ( sin N α sin α where α and β are given by ) 2, (2.3) α = aπ λ (sin θ + sin θ i), (2.4) β = bπ λ (sin θ + sin θ i). (2.5) In this equation the value of b is the width of a single groove, N is the number of grooves that is illuminated by the incident light and I 0 is a constant that depends on the light source and the grating material. Two different terms can be identified in the intensity distribution, namely the diffraction term ( sin β β )2, which takes into account the diffraction of light coming from the individual grooves and the interference term ( sin N α sin α )2, which takes into account the interference between light coming from different grooves. 2.2 Blazed diffraction gratings In the diffracted light of a normal diffraction grating most of the energy will be in the zeroth order as can be seen in fig This can be understood by 6

14 2.2 Blazed diffraction gratings (sinβ/β) θ Figure 2.2 The diffraction term of the intensity distribution plotted against θ, for a normal grating (solid curve) and for a blazed grating (dashed curve). The angle of incidence θ i used for this plot is 23 o and the blaze angle is chosen 26 o 44, the grating constant a is 556 nm. The zeroth order lies at θ=-23 o and the position of the first order can be seen in fig the fact that the zeroth order normally lies exactly at the angle of specular reflection at the surface of the grating. In the set-up for stretching the laser pulse, a blazed diffraction grating is used. This kind of diffraction grating is designed to shift the energy from the zeroth order to a higher order. A blazed grating is a grating from which the surfaces of the individual grooves are tilted with a small angle called the blaze angle θ b. Through this tilt the angle of specular reflection is also shifted with the same angle θ b, when measured from the normal of the grating. Of course the angle of specular reflection doesn t change when measured from the normal of one groove. The real difference is made by the fact that for a normal grating these two normals are the same, but for a blazed grating they exactly differ an amount θ b. The intensity distribution for a blazed grating can be calculated in almost the same way as for a normal grating. The only difference is the fact that all of the individual grooves now are tilted with the blaze angle, so that all angles in the diffraction term should be measured from the normal of the groove instead of the normal of the grating. This can mathematically be done by the following coordinate transform θ i *= θ i θ b, (2.6) 7

15 Chapter 2 Grating Theory θ i * n n θ θ* θi θ b Figure 2.3 The angles with a star are measured from the normal of the groove N and the angles without a star are measured from the normal of the grating. θ*= θ + θ b, (2.7) in which the angles with a star are measured from the normal n of the groove and the angles without are measured from the normal n of the grating itself, see figure 2.3. In these formulas the same sign convention as before is used, so θ i * is again a negative angle and θ* is positive. The intensity distribution is again given by equations (2.2)-(2.5), but in the diffraction term θ and θ i are replaced by θ and θ i. Also, for a blazed grating the values of a and b are approximately the same, so b can be replaced by a. As said the energy is shifted to a higher order due to the blazing. The maximum of the intensity distribution will be at the angle of specular reflection, measured from n. This means that specular reflection is given by θ*=-θ i *. With this formula and equation 2.6 and 2.7 this leads to the angle of maximum intensity given by θ = 2θ b θ i, (2.8) so the angle of maximum intensity is shifted with an amount 2θ b. In figure 2.2 the diffraction term of the intensity distribution is plotted against θ, for a normal grating and a blazed grating. The position of the different orders due to the interference term is not changed, so the shift of the maximum in the diffraction term means that the energy is shifted from the zeroth order to a higher order. This fact can be used to spatially separate the light of different frequencies, without losing a lot of energy. 8

16 2.3 Angular dispersion Θ Figure 2.4 A plot of the first order of the intensity distribution (eq. 2.3) against θ, for different wavelengths. The distribution is scaled, so that the maximum of the distribution is set to 1. From left to right the wavelengths used for the plot are nm till nm with an increment of 0.2 nm. For this plot the following parameters are used: N =12560, θ i =23 o, θ b =26 o 44 and a=556 nm. 2.3 Angular dispersion In fig. 2.4 a plot of the intensity distribution of light with different wavelengths can be seen. The plot is zoomed in on the first order. As expected, every wavelength is diffracted at a different angle. The shape of the distributions is fully determined by the interference term, only the height is determined by the diffraction term. As the number of illuminated grooves N is increased, the distribution will become narrower. The angles between the light of different wavelengths is determined by the angular dispersion, which is the angular separation per unit range of wavelength as a function of the angle θ. The angular dispersion D is given by D = dθ dλ = m a cos θ. (2.9) 9

17 Chapter 2 Grating Theory 10

18 Chapter 3 Setup The main goal of this project is to narrow the bandwidth of a laser pulse. This is done by using a grating to spatially separate different wavelengths and then cut out the target wavelength interval with a slit. In this chapter the different aspects of the setup are discussed. The several elements used in the setup and their specifications are shown in table Grating configuration A blazed grating is specially designed to use it near first order Littrow configuration. A first order Littrow configuration is a configuration in which the light in the first order is reflected. This can be done as the incoming light falls perpendicular on the surface of a groove or in other words as the angle of incidence is θ b. To understand why this is, we take a further look on the efficiency η of the setup which is defined by η = intensity in first order intensity in all orders (3.1) The efficiency is mostly determined by the position of the different orders in the diffraction envelope. As told in chapter 2, the intensity of a certain order is only determined by the diffraction term, because this term determines the maximum of the interference peak ( ) sin N α 2 sin α in that order. So the efficiency is determined by the value of the diffraction term at the position of the first order divided by the sum of the values of the diffraction term at the position of the other orders. Now it is important to know what orders can be expected to appear for a certain angle of incidence. From equation 2.2 can be derived that θ is given by θ = sin 1 (λ/a sin θ i ) (3.2) Figure 3.1 shows a plot of the angle of diffraction θ against the angle of incidence θ i, for the orders: -2, -1, 0, 1, 2. As can be seen in fig 3.1, 11

19 Chapter 3 Setup Table 3.1 Specifications of the elements in the setup Grating Lens Mirror Thorlabs GR Grating constant 556 nm Blaze angle 26 o 44 Angular Dispersion (First Order Littrow) 1.98 mrad/nm Thorlabs N-BK7 Plano-Convex Lens Focal Length 500 mm Diameter 50.8 mm Thorlabs BB2-EO2 Diameter 50.8 mm θ 90 m=2 m=0 m= θ i m=-1 m=-2-90 Figure 3.1 A plot of the angle of diffraction θ i against the angle of incidence θ i, for the orders: -2, -1, 0, 1, 2. The wavelength of the light for this plot is 480 nm. the angle of incidence determines whether there will be a certain order or not. If we look at the positive angles of incidence it can be calculated with eq. 3.2 that there is always a zeroth order and first order. A second order only appears for angles θ i > 46 o 36 and a minus first order only appears for angles θ i < 7 o 51. The next step to derive the efficiency of the grating is to know what the blaze angle θ b of the grating is. The grating that is used in this project has a blaze angle of 26 o 44. As explained in chapter 2, this value determines at which angle of diffraction the most intensity will be found or in other words what order will contain the most energy. A graph of the diffraction term, with the effect of the blaze angle taken into account, is shown in fig Now we know the effect of the blaze angle (from section 2.2) and have the angle of diffraction as a function of the angle of incidence (with eq. 3.2) 12

20 3.2 Beam path it is possible to find the efficiency as a function of the angle of incidence. This efficiency is shown in fig Efficiency Figure 3.2 A plot of the efficiency η of the grating as a function of the angle of incidence θ i. Figure 3.2 shows that the efficiency is approximately 1 for angles of incidence near the blaze angle. So this shows that a blazed grating can best be used in Littrow configuration. 3.2 Beam path There are several conditions which the pulse stretcher setup has to satisfy. In this paragraph each subparagraph deals with one of these conditions. The conditions the setup has to satisfy are the following: 1. The incoming beam has to be projected on itself, this means that after passing the pulse stretcher the light of every wavelength has to come out parallel to the light of every other wavelength. It also means that the size of the laser spot after the pulse stretcher has to be the same as before. These conditions can t be fully satisfied, because a laser beam always has a slight divergence of itself, so it is not a complete parallel beam when it travels in the pulse stretcher. So the scope of this condition is that the laser beam is not made more divergent than it was before passing the pulse stretcher. 2. There should be no group velocity dispersion due to the pulse stretcher. This means that every wavelength has to travel the same time through 13

21 Chapter 3 Setup θ x Figure 3.3 Schematic view from the top the setup. The beam falls in from the right at the top of the picture. After the grating it is spatially separated and every wavelength has a different angle of diffraction θ. After the lens the angle of diffraction is converted to a position x, so the angular dispersion is converted in a linear dispersion. it, so that the beam is not stretched in time due to the fact that different wavelengths travel over different paths. 3. The outgoing beam has to be separated from the incoming one, to make it able to travel further to the UCP-setup. 4. The setup has to be useful for laser pulses with a central wavelength in the range nm and should be able to cut out at least a spectral range of 0.3 nm Beam projection The first condition can be satisfied by using the grating, a lens and a mirror. When the lens is placed in such a way that its focal point coincides with the laser spot on the grating, the light of different wavelengths will be parallel after passing the lens. So the lens converts an angular dispersion in a linear dispersion, see fig If the mirror is then placed after the lens and parallel to it, every wavelength is projected on itself and the beam that travels in the stretcher is almost the same as the one that travels out of it. The only difference is the fact that the out coming beam is mirrored compared to the incoming one. As said in section 3.1, the grating will work most efficiently in near first order Littrow configuration. But the grating cannot be placed exactly in first order Littrow configuration, because in that way the beam will follow the same path after the grating as before the grating. That means that the incoming beam also goes through the lens. To prevent this, the beam falls in a few degrees away from the blaze angle, so that it just 14

22 3.2 Beam path misses the lens Group velocity dispersion As can be seen in figure 3.3, the light rays of different wavelengths don t travel over the same path length, so it is not so obvious that there is no group velocity dispersion. To see whether the light of every wavelength travels the same time through the setup one has to take a look at Fermat s principle. This principle tells that a ray of light will always take the path of least time between two points [6]. In figure 3.3 it can be seen that every wavelength travels from the same point on the grating through the lens, on the mirror, back through the lens and back to the point where it came from. So every wavelength travels from the same point back to that point. This means that if the refractive index of the lens would be the same for every wavelength, Fermat s principle tells us that the light of all wavelengths will take the same time to do this and there would be no group velocity dispersion. But because the refractive index is not independent of the wavelength, there may be some group velocity dispersion, but the chosen configuration will limit this group velocity dispersion as much as possible Two level system In figure 3.3 can be seen that a setup is chosen which consists of only one grating, a lens and a mirror. A setup with the same characteristics could be made with two gratings and two lenses. The reason why a setup with only one grating is chosen, is a practical one, namely that it is easier to align. With the use of two gratings it would be very difficult to keep a (nearly) non-divergent beam because the two gratings would have to have the same angle with the optical axis of the system, while the setup in figure 3.3 has only the requirement that the lens and the mirror are parallel. Despite the practical advantage the one grating setup does also have a disadvantage, because the incoming and outgoing beam overlap each other. This makes it difficult to extract one from the other. This is solved by letting the incoming beam fall above the middle of the lens as shown in fig Because the beam travels horizontally before the lens it will go through the focal point after it. This means it will travel downwards after the lens. When the mirror now is placed a focal length away from the lens, the reflected beam will again travel horizontally when it has passed the lens for the second time, but now it is lower than the incoming beam Focusing the beam As said in subparagraph 3.2.3, the mirror is placed a focal length away from the lens because of the described two level system, but there is an other reason to do that. In figure 3.3 a simplification is made, because there is 15

23 Chapter 3 Setup f f Figure 3.4 Schematic side view of the setup. Because the beam falls in horizontally it is refracted to the focal point and reflected back to the lens such that it is again horizontal after the second pass of the lens. not taken into account that the beam has a certain size. In reality, the laser spot is not a point as assumed in figure 3.3, but has a Gaussian distribution of intensity with a standard deviation of about 1.8 mm. This would cause a major loss of energy when using a slit with a width of 0.3 mm. This problem is solved by the fact that the mirror is placed at a focal length from the lens. After the lens the beam is focused to a waist exactly at the point of the mirror, see figure 3.5. This means that at that point the standard deviation of the laser bundle is much smaller then it was before the lens. So if a slit is placed just before the mirror the effect of the size of the laser beam would be minimal. But if the slit is placed further away from the mirror, the effect of the size of the laser beam will start to play a role in the transmission of the pulse stretcher. A model for calculating the transmission of the laser pulse is presented in chapter Lens choice One of the requirements the lens has to meet is that it has to be able to convert the angular dispersion of the grating to a linear dispersion in such a way that it is possible to cut out a spectral range of 0.3 nm. In near first order Littrow configuration the angular dispersion is by eq. (2.9), 1.98 mrad/nm. With a lens with a focal length of 500 mm, this angular dispersion can be converted to a linear dispersion of 0.99 mm/nm. So to cut out a spectral range of 0.3 nm a slit with a width of 300 µm is required, which is a realizable size. The diameter of the used lens is 50.8 mm. To show that it is possible to handle laser pulses with a central wavelength in the range nm one has to take into account the size as well as the spectral spread of the 16

24 3.2 Beam path Figure 3.5 Schematic view from the top of the setup. The beam is focused to a waist after passing the lens. When a slit is placed just before the mirror all the energy of light with a wavelength centered in the slit will go through. laser beam. The approximation is made that at least two times the standard deviation in as well size as spectral spread has to be included by the lens. The standard deviation in the wavelength of the laser beam is measured to be 2.6 nm and the standard deviation in the position was discussed in the previous section. So to make the setup work at a certain central wavelength the position of this wavelength at the lens has to be at least mm =8.7 mm (which is two times the sigma in position plus two times the sigma in position times the linear dispersion) away from the border of the lens. The setup is designed so that the ray of light with a wavelength of 480 nm will travel parallel the optical axis of the lens. The central positions which a ray of light with central wavelength 472 or 490 nm will take in on the lens are approximately 8 mm and 10 mm away from this position. This shows that the setup with this lens can be used for the full wavelength range of nm, because the central positions of the light with the outer wavelengths meet the requirement stated in the former paragraph. 17

25 Chapter 3 Setup 18

26 Chapter 4 Model In chapter 3 the setup for stretching the laser pulse was described. The subject of subparagraph was the effect of the lens on the laser beam and it concluded that if the slit is placed just in front of the mirror, all power of light with a wavelength, so that its central is position between the slit boundaries, will go through the slit. In this chapter a model is described that shows what happens if the slit is placed somewhere else between the lens and the mirror. The model described in this chapter can be divided in two stages. In the first stage the intensity distribution for a certain wavelength is calculated as a function of the perpendicular x- and y- coordinates at the slit. The slit is placed at a certain position z from the lens, where the coordinate z is measured along the optical axis of the lens, see figure 4.1. In the second stage is determined what fraction of the light goes through the slit. The result of this procedure gives boundaries for what light will travel through the slit and what light does not. Finally, integration of the intensity distribution will give the total power that travels through the slit + x y z Figure 4.1 A picture of the lens and its optical axis with the used coordinate system. 19

27 Chapter 4 Model for a certain wavelength. This power is divided by the power that would have gone through if there was no slit on its way to obtain the transmission. The intensity at a certain wavelength that travels out of the pulse stretcher is calculated by multiplying the transmission by the intensity of the light with that wavelength that traveled in the pulse stretcher. 4.1 Gaussian laser spot As was mentioned in paragraph 3.2.4, the laser spot is not a point, but a Gaussian distribution. This means, not one point on the grating sends out the diffracted light, but a whole distribution of points does. The intensity of light coming from a certain point with coordinates x and y is given by a two dimensional Gaussian distribution, I e (x x 0) 2 /(2σ 2 x) (y y 0 ) 2 /(2σ 2 y). (4.1) The reference position (x 0, y 0 ) is the position on the optical axis of the lens on the grating. The standard deviations σ x and σ y are measured with a ccd camera. The result of these measurements were a standard deviation in the x-direction of σ x = (1.7±0.1) mm and in the y-direction of σ y = (1.11±0.02) mm. 4.2 Instrumental broadening In figure 2.4 a plot of the intensity distribution of the first order for several wavelengths can be seen. The intensity distribution for a certain wavelength is peaked at a certain angle of diffraction, which is given by equation (3.2). But because the number of grooves N that is illuminated by the laser spot is not infinite, the angle of diffraction is not well defined, but is given by the distribution from figure 2.4. This is called instrumental broadening [6]. In this distribution, which is given by the interference term ( sin N α sin α )2, the parameter N determines the width of the distribution. If the value of N is small the distribution is wide and vice versa. The limit of N goes to infinite will result in a Dirac delta function. The focusing of the laser beam after the lens was the subject of subparagraph In figure 3.5 it looks like the beam of one wavelength falls on the lens parallel so it is focused to a point on the mirror. This is not exactly true. Because of the instrumental broadening, the light does not fall in parallel on the lens, but has a distribution in the angle of diffraction. This means that after the lens the beam (with light of one wavelength) is not focused to a point on the mirror, but to a line, from which the width is determined by the value of N. Because the spot of the laser light is a (two-dimensional) Gaussian function, it is not obvious what the value of N is. The grooves that are further 20

28 4.3 Boundaries away from the center of the Gaussian function catch less light then the central groove, which is the groove the closest to the center of the Gaussian. So it seems logical that these grooves count less then the central groove. This idea is used to calculate a value for N that can be used in the model. First the total power that falls on the central groove is calculated. Then the total energy that falls on the grating is divided by this value. The resulting number is the value for N, used in the model. So N is given by, N (x)2 e 2σx 2 dx (x)2 a/2 a/2 e 2σx 2 dx = (4.2) 4.3 Boundaries Now we know the distribution of the light falling on the grating as well as what happens with a ray of light that falls on the grating we need to know what light is transmitted when a slit is placed between the lens and the mirror. To do this we first look at light that comes from a point on the grating which lies above the optical axis (in real life to the left of it), see figure 4.2. Because the light is reflected by the mirror it has to pass the slit two times. Of course, the light has to be between the boundaries of the slit both times to get back to the lens. The model calculates the angles of diffraction for which the light will travel through the slit for both passes of the slit. This is done by calculating a minimum angle θ 1 and a maximum angle θ 2 which define the interval of angles of diffraction that go through the slit (both times). All angles of diffraction are measured from the normal of the grating. To calculate θ 1, first the value of x 1 has to be calculated, which is the distance, measured at the lens, between the light which just goes through the slit at the upper boundary and the light with wavelength 480 nm, which travels parallel to the optical axis before the lens. The value can be determined from figure 4.2 as x 1 = x u (f z slit ) tan α. (4.3) In this equation, x u is the distance from the optical axis to the upper boundary of the slit, f is the focal length of the lens, z slit is the distance from the lens to the slit and tan α is given by tan α = x f. (4.4) In this equation x is the distance from the optical axis to the point on the grating from which the light is coming. Now θ 1 can be calculated with ( ) x1 θ 1 = θ 480 arctan, (4.5) f 21

29 Chapter 4 Model + θ1 θ480 θ2 x 1 x x 2 α x u f tanα x l (f-z slit ) tanα z slit Figure 4.2 A schematic view from the top of the pulse stretcher, which is used to determine the boundaries of what light is transmitted by the slit. 22

30 4.4 Intensity as a function of wavelength in which θ 480 is the angle of diffraction from rays of light with a wavelength of 480 nm. The parameter x 2 is the distance between the position of the ray of light at the lens and the center of the lens, for light which just goes through the slit at the lower boundary. Again the value is determined from figure 4.2 as x 2 = (x l + f tan α + (f z slit ) tan α). (4.6) In this equation x l is the distance from the optical axis to the lower boundary of the slit. The maximum angle θ 2 is given by, ( ) x2 + x θ 2 = θ arctan. (4.7) f In this paragraph the light from the grating was coming from a point above the optical axis. The boundaries for light coming from a point below the optical axis are calculated in much the same way. 4.4 Intensity as a function of wavelength Now the boundaries for light that is transmitted through the slit are known for a certain position x on the grating, a value of the transmission can be calculated. To do this the integral of the intensity distribution over the transmitted interval of the angle of diffraction is divided by the integral of the intensity distribution over all angles of diffraction. Because the diffraction term of the intensity distribution is approximately one over the full range of angles in the first order, only the interference term has to be used in the intensity distribution. So the transmission of light coming from one point one the grating, r(λ, x) is given by r(λ, x) = θ2 θ 1 ( sin Nα sin α )2 dθ π/2 sin Nα π/2 ( sin α )2 dθ. (4.8) To calculate the intensity of the laser beam as a function of the wavelength, all of the points on the grating that diffract light have to be taken into account. To obtain the total power of the light coming out of the pulse stretcher with a certain wavelength λ, an integral of the product of the transmission r(λ, x) with the intensity of the incoming light has to be taken over all points on the grating. The intensity of light traveling in the pulse stretcher setup is a Gaussian distribution in the position x multiplied by a Gaussian distribution in the wavelength. Because the intensity of the out coming light, I out is proportional to the power of the out coming light, I out can be written as, I out (λ) = C 1 e λ 2 2σ λ 2 r(λ, x)e x 2 2σx 2 y2 2σy 2 dx dy. (4.9) 23

31 Chapter 4 Model Because r(λ, x) is not a function of the coordinate y, this variable can also be integrated out the expression, which leads to I out (λ) = C 2 e λ 2 2σ λ 2 r(λ, x)e x 2 2σx 2 dx. (4.10) In these equations C 1 and C 2 are constants which among others depends on the laser source. C 2 also depends on the σ y. 4.5 Results of the model Because the integrals in equation 4.8 and 4.10 cannot be solved analytically, a numerical model is made to find the spectrum of the light after the pulse stretcher. The results of this model are shown in this paragraph. All the spectra shown in this paragraph are made for an incoming laser pulse with a central wavelength of 480 nm and a standard deviation of 2.6 nm Longitudinal slit position Figure 4.3 shows a plot of the scaled intensity against the wavelength for several longitudinal position z slit of the slit. The scaled intensity is the intensity of the out coming light I o ut divided by the intensity of incoming light with a wavelength of 480 nm. The plot shows that the width of the intensity distribution is always about the same. This can be understood by figure 4.4. It shows that light with a wavelength which can just go through the slit (two times), is the light which travels parallel to the optical axis and has a transverse position which is equal the lower (or upper, for the other side of the spectrum) boundary of the slit. So this is not dependent on the longitudinal position of the slit. Another feature shown by figure 4.3, is that the maximum in the intensity distribution becomes lower if the slit is moved towards the lens. The obvious reason for this is that if the slit is moved towards the lens, the slit is moved away from the focus of the lens, so the the spot size of the light at the point of the slit becomes larger. This means that less light will go through the slit. The peak of the spectrum is always positioned at 480 nm, because this wavelength goes through the middle of the slit. The peak becomes sharper as the slit is moved towards the lens. This can be understanded by the fact that moving away from the focus has more effect on light with its center further away from the middle of the slit Transverse slit position Figure 4.5 shows a plot of the scaled intensity against the wavelength, for several transverse positions of the slit. If the slit is shifted along the transverse axis, the wavelength of the transmitted light in the spectrum 24

32 4.5 Results of the model Scaled Intensity Wavelength (nm) Figure 4.3 A plot of the scaled intensity against the wavelength, for longitudinal positions of z slit = 500 mm, 475 mm, 450 mm, 400 mm, 300 mm, 200 mm and 100 mm. The slit width used for this plot is 0.5 mm and the slit was standing at a transverse positions of x slit =0. + Figure 4.4 Drawing of the rays of light with a wavelength which is just transmitted by the pulse stretcher. The picture shows the angle of diffraction of this ray of light is not dependent on the longitudinal position of the slit, so that also the wavelength of light which is just transmitted is not dependent on the longitudinal slit position. 25

33 Chapter 4 Model Scaled Intensity mm -1 mm 0 mm 1 mm 2 mm Wavelength (nm) Figure 4.5 A plot of the scaled intensity against the wavelength for transverse positions of the slit x slit =-2,-1,0,1,2 mm. A slit width of 0.3 mm was used for this plot and the slit was standing at a longitudinal position of z slit =500 mm. is also shifted because of the linear dispersion after the lens. If the slit is moved away from the optical axis, the maximum in the intensity distribution becomes lower. This is caused by the fact that the incoming laser pulse is a Gaussian distribution and the setup is designed such that light with a wavelength of 480 nm is focused on the focal point of the lens. So moving away from the optical axis in the system is translated to moving away from the center of the Gaussian distribution in the spectrum Slit width Figure 4.6 shows a plot of the scaled intensity against the wavelength, for different slit widths. Figure 4.3, 4.5 and 4.6 show that the spectrum is not cut off straight, but there is a more gradual transition. This is caused by the fact that light with a certain wavelength does not have one single angle of diffraction but a whole interval of them, as was discussed in section 4.2. The distribution of the angle of diffraction is a function with one large peak in between several smaller peaks (see figure 2.4). It is these smaller peaks that cause the wiggles that can be seen in figure 4.3, 4.5 and 4.6. Moreover it is also these smaller peaks adjacent to the large one in the middle that cause the fact that for example the intensity distribution for a slit width of 0.3 mm has not a maximum at λ=480 nm as one may expect. For a slit width of 0.3 nm the smaller peaks in the intensity distribution for light with 26

34 4.5 Results of the model Scaled Intensity Wavelength (nm) Figure 4.6 A plot of the scaled intensity against the wavelength for a slit width of 1 mm, 0.5 mm, 0.3 mm and 0.1. For this plot the slit was standing at a longitudinal position of z slit =500 and a transverse position of x slit =0. a wavelength of 480 nm just fall outside the slit, whereas for wavelengths slightly smaller or larger than 480 nm, one of them falls inside the slit. So that is why the maximum in figure 4.6 are slightly displaced from 480 nm. 27

35 Chapter 4 Model 28

36 Chapter 5 Measurements After the setup, that is described in chapter 3, was built, measurements were performed to look whether the setup worked as we expected. These measurements are described in this chapter. All measurements of the spectra are done with a spectrometer 1 with a resolution of 0.03 nm. 5.1 Dispersion First of all the linear dispersion after the lens is measured. This is done by measuring the full width at half maximum from the spectra of the laser pulses for different slit widths. To do this the slits were placed just before the mirror. The full width at half maximum is now plotted against the slit width, which can be seen in figure 5.1. The slits were approximately placed at the focal point of the lens, so if the instrumental broadening is neglected each wavelength corresponds a specific point on the mirror. This means that the slope of figure 5.1 is the inverse value of the linear dispersion. The linear dispersion is therefore (1.026 ± 0.005) mm/nm. In paragraph was told that the linear dispersion for a first order Littrow configuration would be 0.99 mm/nm. This value, was a theoretical value for an angle of diffraction of 26 o 44. In reality, the angle of diffraction is larger than this value, because the angle of incidence was smaller then the blaze angle, as discussed in section So the angular dispersion is also higher according to equation 2.9. From the measured value of the linear dispersion can be calculated that the angle of diffraction was 28 o Longitudinal distance In chapter 4, a model is introduced which describes what the spectrum of the laser pulse would look like if the slit was translated over the optical 1 Avantes AvaSpec

37 Chapter 5 Measurements 6 FWHM spectrum (nm) Slope=0.975 nm/mm Slit width (mm) Figure 5.1 A Plot of the full width at half maximum from the spectra of the laser pulses against the slit width. The plot is fitted linear which resulted in a slope of (0.975 ± 0.005) nm/mm. axis. In this paragraph some measurements are shown to verify the model. Because the intensity of the femtosecond laser was not stable, it was only possible to do relative measurement of the spectrum. For this reason all the measurements shown in this paragraph and appendix A are scaled so that the maximum in the measured spectra is the same as the maximum in the modeled spectrum. Figure 5.2 shows a measurement of the spectrum of a laser pulse which went through a slit placed at a distance 497 mm from the lens. Because of practical issues this was the closest possible position to the mirror. The width of the spectrum of the model and the measurement agree quite well, but the plot of the model falls off more quickly then the measurements at the borders of the spectrum. This indicates that the effect of instrumental broadening should be larger, or in other words the value of N should be lower. The plot for the model, shown in figure 5.2 is made for a value of N of 7670, which was calculated in paragraph 4.2. Figure 5.3 shows a plot of the same measurement compared with the model for a value of N = The model and measurement agree much better with this value of N, so for the other plots the value for N is taken to be 3500 in the model. A possible reason that the value of N has to be lower as calculated in section 4.2 is that in the derivation [6] of equation (2.3) the wave was expected to be a plane wave, but in reality the intensity distribution of the wave is a Gaussian instead of constant. 30

38 5.2 Longitudinal distance Model Measurement Scaled Intensity Wavelength (nm) Figure 5.2 Measurement of the spectrum of the laser pulse which went through a slit of 0.5 mm at a distance z slit =497 mm Model Measurement Scaled Intensity Wavelength (nm) Figure 5.3 The same measurement as in figure 5.2, only with a value N =