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Advanced Lab Course Laser and nonlinear optics Status: 17.11.2014 Page 1 (FP13) Laser and nonlinear optics Laser-setup and operating mode, Generation and characterization of ultrashort light pulses, Generation of optical high harmonics Tutors: Wagner Martin ([email protected]) Latka Tobias ([email protected]) 1 Introduction In this advanced lab course we will present you a device that impacts our present life drastically. You will learn about the principles of lasers and their most important components. The second part will treat the principles of nonlinear optics, which describe the interaction of highly intensive light with matter. To benefit from this lab course you have to prepare yourself. ? 2 2.1 Please read the instructions to the lab course carefully and work out the asked questions. Physical Backround Interaction of radiation with matter: absorption and emission Transitions in molecules or atoms will be simplified. First we will concentrate on two energy levels |1 > and |2 > with energy gap E2 − E1 = ~ω (ω = frequency of the resonant radiation field; energy conservation!). |2> E hw |1> Interaction with the radiation field is induced by electric dipole coupling (in most cases). Advanced Lab Course Laser and nonlinear optics Status: 17.11.2014 Page 2 (FP13) The following situations occur: (a) (b) |2> |1> |1> (c) |2> (d) |2> |1> |2> |1> (a) Ground state: An isolated atom in the ground state does not change its state. (b) Absorption: Atoms or molecules from state |1 > (population density n1 ) go over to state |2 > absorbing photons (density nq ) of appropriate frequency. ∂nq = −σn1 nq ; ∂x n1 , nq population density in (cm−3 ) σ = absorption cross-section (cm2 ) ∂nq = −σcn1 nq ∂t (1) x = propagation direction c = speed of light (c) Spontaneous emission: Molecules or atoms in state |2 > (population density n2 ) go over to state |1 > emitting a photon of frequency ω. ∂n2 n2 =− ∂t τ (2) where τ = lifetime of state |2 >. (d) Induced emission: Molecules or atoms in state |2 > can go over to state |1 > by photon radiation of frequency ω. The energy difference will be emitted as a photon of the same frequency, direction and phase as the incoming photon. ∂n1 = σ 0 cn2 nq ; ∂t ∂nq = σ 0 cn2 nq ∂t where σ 0 = cross-section for stimulated emission (cm2 ) (3) Advanced Lab Course Laser and nonlinear optics (FP13) Status: 17.11.2014 Page 3 σ 0 = σ. (4) For a two level system: Processes b) and d) can be summarized for a two level system as follows: ∂nq = σnq c · (n2 − n1 ) = σnq c∆n ∂t (5) where ∆n = population inversion ∂n1 2 Applying ∂n ∂t + ∂t = 0 and taking into account the losses per unit of time due to scattering, absorption of impurities or other systematic losses, we can write: nq (t) = nq (t = 0) · e(σ(n2 −n1 )t−δt)·c (6) Amplification of the radiation field occurs if σ(n2 −n1 ) > δ. To fulfill this condition, level 2 must be higher populated than level 1 (population inversion). According to the derivation above the spectrum of the laser transition should be infinitely sharp. However, due to the following two broadening mechanisms, it has a characteristic linewidth: • homogeneous broadening • inhomogeneous broadening The dominant broadening mechanism depends on the laser medium and the operating conditions (e.g. pressure and temperature for gas lasers). According to this discussion, σ = σ(ν) is a function of frequency. ? Question 1: Compare induced and spontaneous emission with respect to a) spatial radiation characteristics and b) temporal radiation characteristics. ? Question 2: Give examples for homogeneous and inhomogeneous broadening mechanisms of absorption lines. Advanced Lab Course Laser and nonlinear optics 2.2 Status: 17.11.2014 Page 4 (FP13) Principles of lasers A neodymium ion embedded in a YLF-crystal (Yttriumlithiumfluoride) can be described by the simplified energy level scheme in figure 1: |3> |2> E hw Figure 1: Energy level scheme for a four level laser. |1> |0> Transitions from the pump level |3 > to the upper laser level |2 > and from the lower laser level |1 > to the ground state |0 > occur very rapidly (approx. 10−12 s) and without radiation (=heat), whereas state |2 > has a noticeably longer lifetime (approx. 3·10−4 s). Pump radiation can be absorbed by the lower levels and lead to a high population of the upper laser levels. Therefore, light of the frequency ω will be amplified similar to that seen in equation (6). In order to use the population inversion created by the pump several times, the active medium is placed between two mirrors with reactivities R1 and R2 (figure 2). These mirrors form the optical resonator of the laser. R1 Nd:YLF AOM R2 Figure 2: Principles of a laser resonator Lasing is triggered by a spontaneously emitted photon, that is incidentally emitted perpendicular to the mirrors. This process stops when the population inversion vanishes (e.g. stop of pumping). A fraction of the laser radiation is coupled out of the resonator by the partially transparent mirror 2. In extension of equation (6), the light intensity in the laser after one round-trip in the resonator containing a laser medium with length l is: I1 = I0 R1 R2 · e2l(α−δ) , I0 : initial intensity δ: losses per unit length (7) α: amplification coefficient (α = σ(n2 − n1 )) R1 , R2 : reflectivity of the mirrors Lasing is observed if for one round-trip in the resonator: I1 > I0 ↔ α > δ − ln(R1 R2 ) , 2l (8) Advanced Lab Course Laser and nonlinear optics Status: 17.11.2014 Page 5 (FP13) This leads to: δ ln(R1 R2 ) − . (9) σ 2l This equation is called the threshold or the oscillator condition. Therefore the oscillation threshold of a laser medium also depends on the reflectivity of the mirrors and the power coupled out of the resonator (available laser power). n2 − n1 > ? Question 3: Nd:YLF lasers are four-level lasers“. Discuss the setup of a three-level ” laser. Is it physically and technically possible to achieve a two-level laser? ? Question 4: What limits the maximum pump power of a laser? What is the role of the radiation-less process in this context? 2.2.1 The laser resonator To feed back the laser light in the resonator it must be stable. This is described by: 0 < g1 · g2 < 1 (10) where gi = 1 − rLi is the stability parameter, L is the length of the resonator, and ri is the focal length of the final mirror r1 r2 L r1 r2 L Figure 4: Possible stable resonator configurations. Figure 3: Stability diagram. With every round-trip, the field distribution in the resonator is reproduced and even constricted without losses. From electrodynamics we know that every resonator has longitudinal and transverse eigenmodes due to its boundary conditions, which has a large impact on the properties of the laser radiation. Advanced Lab Course Laser and nonlinear optics 2.2.2 (FP13) Status: 17.11.2014 Page 6 Longitudinal modes A Fabry-Perot-Interferometer consists of two plain, parallel mirrors at a distance L from each other. The possible values for the angular frequency obey the condition of a standing wave: c c · 2π 2L = nλ = n · = n · ν ω n·c·π → wn = L with n ∈ IN, n=103 . . . 107 Therefore the distance between the angular frequencies of the longitudinal eigenmodes is ∆ω = c · π/L. In order to generate ultrashort light pulses in a resonator, many of these longitudinal eigenmodes have to be amplified. This requires a minimum resonator length. Only the eigenmodes of the resonator with a gain per round-trip >= 1 contribute to the laser activity. This is schematically shown in figure 5 whereby only 5 longitudinal eigenmodes situated around the maximum of the amplification curve contribute to the laser activity. Figure 5: Eigenmodes and amplification curve ? Question 5: Determine the mode spacing ∆λ for a resonator with L = 1 m. Compare the value of ∆λ and the characteristic laser wavelength of 1047 nm. Advanced Lab Course Laser and nonlinear optics 2.2.3 (FP13) Status: 17.11.2014 Page 7 Transverse modes The transverse field distribution depends on the geometry of the laser resonator. The geometry is defined by the distance L and the radius of curvature of both mirrors r1 , r2 as well as their diameter and shape (e.g. round or rectangular). Knowing these parameters we can determine the stationary spatial distribution of the electromagnetic field in the resonator (→ assuming a stable resonator). Figure 6: Transverse laser modes Usually, the resonator is set to a TEM00 -mode (ground mode) which is best for round resonator components. Using basically round iris apertures (mode apertures) this can also be achieved in the resonator if for example the laser is over-pumped beyond the oscillation threshold. This is advantageous because there are minimum diffraction losses for TEM00 . However, non-optimal adjustment can lead to higher TEMmn -modes (m, n ∈ IN) in the resonator with spatial intensity distributions as shown in figure 6. Figure 6a shows the TEM-modes of a cylindrical coordinate system whereas figure 6b illustrates the TEM-modes of cartesian coordinates. 2.2.4 Passive mode-locking Ultrashort light pulses (10−10 s and shorter) can be achieved using mode-locking techniques. For this, the distinct longitudinal eigenmodes of the laser emission are Advanced Lab Course Laser and nonlinear optics Status: 17.11.2014 Page 8 (FP13) phase-locked (φn = const) to generate a temporally short light pulse via superposition of the field components. For passive mode-locking, a nonlinear “ switch“ is placed within the resonator in order to suppress temporally long pulse components and support short pulses (high intensities). This “ switch“ adjusts the transmission independently of external control depending on the field amplitude of the radiation. Examples are saturable absorbers or the Kerr-lens mode-locking. 2.2.5 Active mode-locking In this lab course the transmission is not controlled by the electric field itself but by an external acousto-optical modulator (AOM). The AOM consists of fused silica glass fitted with a piezoelectric transducer. The AOM is driven by an RF-source which produces an electrical signal of ' 10 V and Ω/2π ∼ 80 MHz. The driving signal induces a standing acoustical wave in the glass and, therefore, a refractive index lattice with a correlating periodicity (figure 7). Bragg: λ l > ∆2 I1 2Θ I0 , λ 2 Θ’ I0 Θ ∆ l Figure 7: Bragg-AOM At this temporally modulated induced lattice, a fraction of the light ∆T is diffracted out of the resonator by an angle of 2θ0 , which results in a periodically changing transmission. Applying a sinusoidal voltage, the transmission becomes: T (t) = T0 + ∆T cos(Ωt) ∆T iΩt = T0 + (e + e−iΩt ) 2 Therefore the components of the electric field En in the frequency domain are: En (t) = E0,n (t) · T (t) = En · T0 · eiωn t+φn + En · ∆T i(ωn −Ω)t+φn ∆T i(ωn +Ω)t+φn e + En · e 2 2 The AOM generates side bands of every single mode at ωn ± Ω (figure 8). If the modulation frequency of the AOM is set to the condition Ω = 2πc/2L, neighboring modes have the same frequency as the generated side bands. Advanced Lab Course Laser and nonlinear optics (FP13) Status: 17.11.2014 Page 9 Figure 8: Principle of active mode-locking ωn ± Ω = ωn±1 (11) Due to this overlapping, neighboring modes influence each other, and a fixed phase correlation between neighboring modes in the resonator is achieved. Therefore a temporally short pulse is generated (figure 9). Increasing the number of locked modes leads to shorter pulse durations. In non-mode-locked operation (see figure 9c), the output is only “white noise“. Figure 9: Mode-locking In the time domain, the frequency Ω of the transmission variation of the AOM corresponds to the duration of one round-trip in the resonator: Ω = 2πc/2L. Metaphorically speaking we mount a gate in the resonator which opens with the round-trip frequency. At other times, the radiation is diffracted out of the resonator by the refraction Advanced Lab Course Laser and nonlinear optics (FP13) Status: 17.11.2014 Page 10 lattice. Depending on the efficiency of this process and the duration of the “open gate“, light pulses of up to the order of several (10−12 s) can be generated. The timebandwidth product describes the quality of the mode-locked pulses. For Gaussian pulses: tp · ∆ν = 0.441 (12) where tp is the pulse duration and ∆ν is the full width at half maximum. Nonlinear processes in the components of the resonator due to the high peak intensities of the laser pulses or insufficient modulation depth of the AOM such as dispersion can lead to deviations from this ideal value. For mode-locked operation, the laser resonator emits a series of single pulses with the following properties: Pulse duration: 20-50 ps Pulse spacing: approx. 6 · 10−9 s Peak intensities: >200 W/cm2 Wavelength: 1047 nm (near infrared, invisible for the human eye) Question 6: How does the spectral width of the laser transition (bandwidth of the laser emission) effect the pulse duration of the generated laser pulses? What role does the number of locked modes play in this context? ? 2.3 Nonlinear optics 2.3.1 Classical optics Because in daily life light intensities are too weak to produce nonlinear effects, you have probably observed only classical, linear optical phenomenons so far. The laws √ of linear optics are based on the assumption that the refractive index n = in a dielectric medium and, therefore the electric susceptibility χ only depends on the wavelength: D = 0 · E D = 0 · E + P (λ) = 0 (1 + χ) · E = 0 (λ) · E (in vacuum) (ina dielectricmedium) where = dielectric constant , P = induced polarization, and E = electric field The wave equation in matter which describes the propagation of electromagnetic radiation in the x-direction is: Advanced Lab Course Laser and nonlinear optics Status: 17.11.2014 Page 11 (FP13) ∂2E ∂2E − =0 ∂x2 c2 ∂t2 The solution of this linear differential equation yields plane waves: E(x, t) = Eo · ei(ωt−kω x) + c.c. 2.3.2 (13) (14) Nonlinear polarization Higher light intensities on the order of 105 W/cm2 lead to deviations from the classical behavior. In this case, the electric susceptibility χ, which is simplified to a scalar (in isotropic media) depends also on the amplitude of the electric field E and can be developed as a Taylor series: χ = χ(1) + χ(2) · E + χ(3) · E · E + ... (15) In the following equations, the vector properties of E and P are omitted. Therefore, the induced polarization yields: P (E) = 0 · χ · E =0 · E · (χ(1) =0 ( − 1)E D(E) = 0 E +χ(2) · E + χ(3) · E · E + ...) +PN L +PN L So the linear differential equation is expanded by a perturbation term: ∂2E ∂2E 1 ∂ 2 PN L − = ∂x2 c2 ∂t2 c2 ∂t2 ? 2.3.3 (16) Question 7: How can we imagine nonlinear polarization on an atomic scale? Hint: displacement of the electrons. Generation of harmonics For the first harmonic (ωharmonic = 2 · ωf undamental ) the consideration of χ(2) is sufficient and the equation simplifies to: ∂2E ∂2E 0 ∂ 2 χ(2) E 2 − = ∂x2 c2 ∂t2 c2 ∂t2 (17) Due to the E 2 -term, this yields an induced polarization oscillating with an additional new frequency: PN L (x, t) = χ(2) E12 · ei(2ω1 t−2k1 x) + c.c. (18) This inharmonic polarization leads to a new field component with double the frequency 2ω and a propagation direction given by the wave vector k2ω . Advanced Lab Course Laser and nonlinear optics 2.3.4 Status: 17.11.2014 Page 12 (FP13) Phase matching The condition 2kω 6= k2ω normally holds in a dispersive medium. Therefore, the intensity of the first harmonic cannot increase quadratically with the crystal length but instead oscillates with sin2 . We can explain this behavior by looking at a harmonic generated at a position x, which in the case of ∆k = 2kω − k2ω 6= 0, is generally not in phase with a harmonic generated at position x + ∆x. Thus, for increasing ∆x a periodic constructive and destructive interference can be seen. E2ω ∝ Eω2 Z sin ∆k 2l l 2 i∆kx ·e !2 · l2 · Iω2 ∆k 2l · l · Eω2 ∆k 2l l = sinc2 ∆k · l2 · Iω2 2 dx = − 2l sin ∆k 2l (19) (20) I2w Dk=0 Dk=0 lkoh l Figure 10: Intensity of the first harmonic for ∆k = 0 and ∆k 6= 0 To achieve maximum amplification, we must make sure that ∆k = 0. In order to ensure this phase matching we normally use birefringent crystals with a certain orientation. 2.3.5 Optical birefringence Optical birefringence means that the refractive index of a medium depends on the polarization of the incoming light. It occurs in crystals with one or more distinguished directions called optical axes. In crystals with one optical axis (i.e. rotational symmetry of the refractive index) the axis and the propagation direction of the light define a plane. If the light is polarized perpendicularly to this plane, the ordinary refractive index no determines the propagation of the light. If the light is polarized in parallel, the extraordinary refractive index neo determines the propagation of the light. Advanced Lab Course Laser and nonlinear optics (FP13) Status: 17.11.2014 Page 13 Figure 11: Adaption of the refractive index in a single-axis negative birefringent crystal (neo < no ). If the setup is chosen so that so that the harmonic wave is polarized extraordinarily and the fundamental wave is polarized ordinarily, n2ω = nω can be fulfilled. Light propagating at an angle of Θ with respect to the optical axis of the birefringent crystal can be adjusted to any desired value ne combining both refractive indices. (neo ≤ ne ≤ no ) 1 2 ne (Θ) = = ? sin2 Θ cos2 Θ + (neo )2 (no )2 1 1 1 + − · cos2 Θ (neo )2 (no )2 (neo )2 (21) (22) Question 8: Why do the refractive indices of the fundamental and the harmonic have to be identical to efficiently generate the harmonic? Hint: speed of light in the medium Let’s look at a radiating dipole in a single-axis birefringent crystal. The phase of the ordinary polarized field propagates on spheres like in isotropic media. However, the extraordinary polarization propagates on ellipsoidal surfaces with constant phasing. In the case of type I phase matching, this is shown in figure 12 below (harmonic (2ω) extraordinarily, laser (ω) ordinarily polarized in a nonlinear, birefringent crystal) : Advanced Lab Course Laser and nonlinear optics Status: 17.11.2014 Page 14 (FP13) optical . axis ............................ .. 2ω ...........Θ ........... ....... .. no ..... ω . . . . . . . . . . . . . . . . . . . . . . . . . . ....... .. ........... ..... . . . . . . . ..... .... .. ........... k ... ..... . . . . . .. ... . .. ω ..... .. ... ... ....... .... no .. .. . . . . . . . . . ... ... . .................................................... ... ... ... .......... .. ... . ... .................... ... . . . ... . .... .. .. .... .......... . . . . . . . .. .. .. ..... .... ..... .. .. .. .. . . ... .. .. .. ... ... . ... . .. . ... .. .. .... .. ... ... ... . .. .. .. .. ... ... .. .... ... . .. .. ω .. . . . . . . . . . .. .. n2ω.. . .. neo ...... . . . . . eo . . . . .......................................................................................................................................................................................... ... .. ... .. . .. . . ... ... .. .. ... ... ... .. .. . . . . . . . . . . . ... ... ... .. . . .. ... ... ... .. ... ... ... ... . . . . . . . . .... ... .. . .. . .. ..... ... . .. . ........... . . ... ...... . . . . . . . . . . . . . . . . . . ... ............. .... . .. .......... .. ... ............................................ .. ... .. . . .. ... .... . . . ... .. . . .... ... ... .... .... .. ........ . . . . . . . .. ..... . ...... . ...... ..... .... .... ..................... ..... .. . Figure 12: The angle of phase-matching is found where the refractive index ellipsoid of the extraordinarily polarized beam of the harmonic (2ω) intersects with the refractive index circle of the ordinarily polarized laser beam (ω). 2.4 Autocorrelation measurement Usually the duration of ultrashort light pulses cannot be measured using electrical devices because of their insufficient bandwidth. The autocorrelation measurement technique is an elegant method to determine the pulse duration up to several fs (10−15 s): A light pulse is split into two equal parts. One part runs through a fixed delay line, while the other one goes through a tunable one. Both parts are spatially overlapped in an optical nonlinear crystal. In the crystal (here: BBO β-bariumoxide) we generate the harmonic of the laser pulse which is detected as a function of the relative delay time. As long as both parts overlap temporally, a measurement of an autocorrelation signal W2h can be made, otherwise it disappears. Mathematically the convolution of the pulse of intensity IL with itself is: Z ∞ W 2h (tD ) ∝ ILo (t − tD ) · ILo (t)dt (23) −∞ The amplitude Eω0 of the field should vary slowly compared to 2π/ω. In mode-locked operation, the Nd:YLF-laser generates almost gaussian pulses: IL (t) = I0 · e √ 2 2·t − 2 ln τ = I0 · e−At 2 (24) Advanced Lab Course Laser and nonlinear optics A= 4 ln 2 , τ2 Status: 17.11.2014 Page 15 (FP13) where τ : full width at half maximumum (FWHM) of the laser pulse. Equations (23) (23) and (24) (24) together yield: Z ∞ 2 2 2h e−A(t−tD ) · e−At dt = W (tD ) ∝ −∞ Z ∞ A 2 2 t 2 D · e−2At dt = |e−{z } I | −∞ {z } II Term I shows that the autocorrelation function depends on the delay line tD √. It describes a gaussian with FWHM τ 10 , which is τ 1 multiplied with a factor of 2. 2 2 τ0 = ? 3 /1/ /2/ /3/ /4/ /5/ √ 2·τ Question 9: Is it possible to determine the temporal shape of the laser pulse from the autocorrelation function? Compare W 2h (tD ) and W 2h (−tD ). Literature Koechner, Solid-State Laser Engineering, Springer (1996) Yariv, Quantum Electronics, John Wiley & Sons (1968) Siegman, Lasers, University Science Books (1986) Boyd, Nonlinear Oprics, Academic Press (1992) Svelto, Principles of Lasers, Plenum Press (1989) (25) Advanced Lab Course Laser and nonlinear optics 4 (FP13) Status: 17.11.2014 Page 16 Experimental setup Figure 13: Experimental setup Pol: polarizer, R1,2 : resonator mirrors, VD: variable delay line, BBO: nonlinear crystal (β-bariumoxide), DET 1/4/5: fast photodiode, DET 2/3: Si-avalanche-diode 5 5.1 Experiment and data analysis Characteristic line of the laser diode Record the characteristic line of the laser diode. For this, the output power at the polarizer (POL) has to be determined as a function of the input current. As the fiber produces totally un-polarized light, 50% of the laser radiation is coupled out at the polarizer. Increase the diode current starting with 5 A in steps of 0,5 A up to 20 A. The detector signal (DET4) is plotted on a computer. The maximum power of the laser diode is 10,5 W at a diode current of 20 A (U=1,8 V). Use appropriate filters in front of the detector. Do NOT increase the diode current to values higher than 20 A! This might cause permanent damage to the laser diode. Evaluation: Plot the characteristic line and determine the efficiency and the laser threshold of the diode. Advanced Lab Course Laser and nonlinear optics 5.2 (FP13) Status: 17.11.2014 Page 17 Transverse laser modes of the Nd:YLF-laser Switch on the Nd:YLF-laser and optimize the laser output power with a closed aperture in the resonator at a diode current of 15 A using the right mirror R2 and the power meter. Observe the occurring modes with the detector card. Which modes do you find for ideal adjustment of the resonator? Which changes must be made to the aperture diameter and the resonator adjustment to find the different modes of the laser? For this vary the pumping power, the aperture diameter and the position of the right end mirror R2 . Classify the observed modes using figure 6. 5.3 Slope efficiency“ of the Nd:YLF-laser in cw-operation ” Optimize the laser at a diode current of 15 A with an open aperture and then reduce the diameter until a TEM00 -mode has been reached. Now measure the laser power with the AOM switched off as a function of the optical pumping power of the laser diode using the power meter. The slope efficiency is the slope of the characteristic line of the Nd:YLF-laser beyond the laser threshold. Increase the diode current from 7 A up to 18 A in steps of 0,5 A. Evaluation: Plot the experimental data and determine the slope efficiency as well as the laser threshold. Note that only 50% of the output power of the laser diode is used to pump the resonator because of the polarizer. 5.4 Mode-locked operation of the Nd:YLF-laser Now switch on the frequency generator of the AOM (f = 79.838.001 Hz) and look for the starting point of mode-locking by tuning the resonator length with mirror R2 . In order to do this, observe the temporal development of the laser emission using the detector (DET 5), which is connected to an oscilloscope. Because of the insufficient speed of semiconductor diodes we cannot measure the pulse duration in that way. This is only possible if we look at the harmonic in the following section. 5.5 Autocorrelation function of the laser pulse Bring the beam to the autocorrelation setup by guiding it through both apertures infront of it. Optimize the signal of the harmonic by fine adjustment of the incoupling, the angle of the crystal and the position of (DET 2). Now you can optimize the laser operation by changing the resonator length and at the same time observing the error bar of the fundamental on the screen. Measure the autocorrelation function of the laser pulse of the Nd:YLF-laser by varying the delay with the measurement program, which records the autocorrelation signal of the detector (DET2). (DET 1) is used to normalize the signal. Advanced Lab Course Laser and nonlinear optics Status: 17.11.2014 Page 18 (FP13) Evaluation: Plot the measured autocorrelation function versus the delay time. Fit a gaussian to the measured curve using a suitable program. How can we determine the pulse duration from the FWHM of the autocorrelation? Please choose suitable units and an appropriate visualization of your results so that the results of your fit can also be verified graphically! 5.6 Angular dependency of the harmonics Determine the intensity of the first harmonic (DET 3) as a function of the crystal orientation. For this, the laser must be guided through the iris and the generated harmonic through the 50 µm pinhole, which is mounted 15 cm behind the crystal. Don’t forget to rotate the norm detector (DET 1) by about 180◦ from its position in the last experiment. Afterwards, vary the angle of the crystal with the motorized rotation table using the measurement program. Pay attention to an exact measurement of the principal maximum and the minima. The curve should at least be recorded until the fifth side maximum. The refractive indices of the BBO Crystals (Thickness: 4 mm) are: for 1047 nm: no/ω = 1.65547 neo/ω = 1.54273 for 523.5 nm: no/2ω = 1.6757 neo/2ω = 1.55606 Evaluation: a) Calculate the angle Θ needed for wave-vector matching. The crystal surface can be assumed to be orientated perpendicular to the laser beam. b) Plot the intensity of the first harmonic as a function of the rotation angle of the crystal. Determine graphically the angle where the argument of the sinc2 function from equation (20) equals π and π2 respectively. c) Calculate the effective path length through the crystal for the incidence angles from b).The refraction at the transition to the optically denser medium has to be taken into account. Hint: Snell’s law. d) Determine the wave-vector mismatch for ∆klef f 2 = π and π 2. e) Explain briefly why the curve does not vanish in the side minima. 5.7 Dependency of the harmonics on intensity Determine the intensity of the first harmonic (DET 3) as a function of the intensity of the fundamental (DET 1), which can be adjusted by hand using a λ/2-polarizercombination. The experimental setup is equal to the previous one. By rotating the Advanced Lab Course Laser and nonlinear optics (FP13) Status: 17.11.2014 Page 19 λ/2-plate, one can manually set different intensities of the fundamental. Evaluation: Plot the intensity of the first harmonic as a function of the laser intensity. Which shape do you expect and which dependence do you determine by fitting the data by a polynomial? 5.8 Measurements with the spectrometer Before the actual measurements you have to calibrate the spectrometer. Please pay attention to the following aspects: In order to increase the resolution, a finer grating was mounted (higher line density). This influences the internal calibration of the spectrometer. Therefore, the wavelength set at the device must be twice the actual wavelength. Calibration: For calibration we use a He-Ne-laser because of its well defined wavelength (λ=632,8 nm) its very narrow spectral width (≤ 10−3 nm). Set the spectrometer to λz = 1266 nm (slitwidth: 50 µm, for adjustment extend open slit to 500 µm) and record a spectrum. Afterwards change the wavelength by 20 nm and save this spectrum too. From these two measurements, you can determine the resolution of the spectrometer and the spectral spacing of the pixels. Recording the Spectra: With the same slitwidth, measure the spectra of the mode-locked Nd:YLF-laser and the laser diode for different diode currents (region of fluorescence, transition region, lasing). Evaluation: a) Plot the spectra of the He-Ne-laser (intensity over pixel number). Now you can determine the spectral spacing of two pixels ∆λ using the spacing of the two pixels ∆pixel : ∆λ = 20 nm/2 ∆pixel (26) Because of the new grid, you have to additionally consider a wavelength dependent offset. The conversion formula (from pixel to wavelength) is λ= λset λset − ∆λ · (P ixel − 256) + 0, 0189 · ( − 1064, 36 nm). 2 2 (27) To verify this formula, one can convert both He-Ne-spectra with it. After doing so, both laser lines must be identical. Advanced Lab Course Laser and nonlinear optics (FP13) Status: 17.11.2014 Page 20 b) Plot the measured spectra of the laser diode as a function of wavelength and describe them. Compare the laser threshold of the diode with the value from 5.1. In fact, the measured bandwidths δm of the laser lines are a convolution of the real spectrum (bandwidth δr ) with the resolution of the spectrometer δ: p 2 − δ2 (28) δr = δm c) Why does the resolution of the spectrometer, in the first approximation, correspond to the measured bandwidth of the He-Ne-laser? Determine δ using gaussians to fit the He-Ne-laser spectra. d) Plot the spectrum of the Nd:YLF-laser. Determine δr and calculate the timebandwith product using the pulse duration from the autocorrelation measurement. Compare your result with the theoretical minimum. Advanced Lab Course Laser and nonlinear optics (FP13) Status: 17.11.2014 Page 21 SAFETY INSTRUCTIONS A different safety standard must be observed with the laboratory equipment used in this experiment in comparison to usual household equipment. Competence in the use of the apparatus is required to minimize the risk to the experimenter. In particular, the risk in this experiment is LASER RADIATION. The radiation of the Nd:YLF Laser used in the experiment is in the near-infrared region, and therefore invisible to the human eye. The power output of both the laser and the pump source are strong enough that even reflected radiation can harm your retina. Therefore, the use of the LASER PROTECTING GLASES is obligatory every time the laser or the pumping source is running. Please observe the following rules: a) First think, then act! b) Never trust that insulation, readouts, etc. are fully functional. In case the laser is not operating, do not try to find the defect by yourself, but ask the tutor. c) The laser is shielded by a box of plastic windows. Never open the box while the laser is operating without the instruction of the tutor. d) The adjustment of the power supply units in the laboratory must not be changed by the experimenters without the instruction of the tutor. e) Never put your hands in the beam of the laser or the pumping source while it is running. Adding/Removing optical elements into/from the experiment is only permitted when the pumping source is turned off. Read these instructions carefully and ask the tutor if some details are not clear. I declare that I have taken note of the instructions above. Date: Name: Matriculation number: Signature: