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Active Modelocking of an Open-Cavity Helium-Neon Laser A Thesis Presented to The Division of Mathematics and Natural Sciences Reed College In Partial Fulfillment of the Requirements for the Degree Bachelor of Arts Sam W. Spencer May 2010 Approved for the Division (Physics) Advisor L. Illing Acknowledgements I want to begin by thanking Lucas Illing for his continual guidance, clarifications, arbitration and reassurance during the chaotic and humbling thesis process, and for helping me remember that not everything needs explanation. There are so many others who deserve acknowledgement: David Griffiths, for showing me that clarity of description begets clarity of understanding; Nicholas Wheeler, for teaching me that physics can be communicated eloquently; Joellyn Johnson, for teaching me how to move on; Robert Knapp, for letting me realize that an essay is not a proof; Mary James, for introducing me to wonderful truths; Chris Thoen, for helping me feel confident; and the 2006 MacNaughtonites and the 2010 physics students for endless friendliness. I want to thank my parents and sister for their conversations and support. Most importantly, I thank Elisabeth Hawks for making me happy and for unconditional love. Table of Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 1: Theory . . . . . . . . . . . . . . . . . . . . . . 1.1 Laser Resonator Stability . . . . . . . . . . . . . . . . 1.2 Helium-Neon Gain Medium . . . . . . . . . . . . . . 1.3 Acoustooptic Modulator . . . . . . . . . . . . . . . . 1.3.1 Standing vs. Traveling Acoustic Wave AOMs 1.3.2 Diffraction Regimes and Traveling Wave AOM 1.4 Time-domain modeling: Pulse Propagation . . . . . . 1.4.1 Gaussian Pulses . . . . . . . . . . . . . . . . . 1.4.2 Pulse Propagation . . . . . . . . . . . . . . . 1.5 Injection Locking . . . . . . . . . . . . . . . . . . . . 1.5.1 Locked Regime: ωD ≤ . . . . . . . . . . . . 1.5.2 Unlocked Regime: ωD > . . . . . . . . . . . 1.5.3 Consequences of Injection Locking . . . . . . . . . . . . . . . . . . . . 3 7 11 13 13 14 19 19 20 24 26 26 26 Chapter 2: Experimental Design . . . . . . . . . . . . . . . . . . . . . . . 29 Chapter 3: Results and Discussion 3.1 Unmodulated Operation . . . . 3.2 Approaching Modelocking . . . 3.3 Modelocking Behavior . . . . . . . . . 35 35 41 44 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . . Abstract In this thesis, we investigate the modelocking of a laser, which refers to the simultaneous excitation of multiple modes of light within the laser cavity, and the establishment of a constant uniform phase difference between pairs of these modes, resulting in the formation of laser pulses. We discuss the physical mechanisms that lead to modelocking, beginning with descriptions of the necessary laser components and presenting a mathematical model for propagating a Gaussian laser pulse through these components. We also discuss the nonlinear phenomenon of injection locking and the role it plays in making modelocking an experimental reality. We then experimentally demonstrate modelocking using a Helium-Neon laser tube with external cavity mirrors and an acoustooptic modulator. The laser is found to produce pulses of approximately 2 nanoseconds in duration, at a repetition rate of about 160 MHz and an average output intensity on the order of 110 nanowatts. We present results of the experiment, including light-intensity timeseries, optical spectra, and radio frequency spectra of the beat frequencies between modes, to characterize modelocking behavior and the injection locking process leading to modelocking. Introduction Generating short pulses of light from lasers has been an area of intense interest since the early 1960s, soon after the laser was invented. Short, intense laser pulses have acquired hundreds of applications in both industry and research roles. Examples in industry include laser ablation and micromachining [Clark-MXR, 2009], refractive index variation including lens aberration correction in the human eye [Ding et al., 2009], high speed fiber optic telecommunications [Wada, 2004], and the generation of terahertz radiation for security screening [Federici et al., 2005]. Short laser pulses are also essential in research projects like nonlinear optics studies [Boyd, 2008], inertial confinement fusion [Pfalzner, 2006], high-harmonic generation [Ferray et al., 1988], and the time-resolved spectroscopy of chemical reactions [Manescu, 2004]. Many different methods have been developed for extracting short pulses from a laser, including Q-switching, gain switching, and cavity dumping, each of which relies on electronic modulation to quickly regulate the output power of the laser [Siegman, 1986]. These techniques typically generate pulses on nanosecond (10−9 s) or picosecond (10−12 s) timescales and rely on the speed of variation of an electrical signal to form pulses. There is another method, called modelocking, that takes advantage of the fact that a laser cavity can naturally support multiple frequencies of light simultaneously, and these frequencies, if correctly superimposed, can beat together to form short pulses. First discussed by Lamb [1964] and first implemented by Hargrove et al. [1964], modelocked lasers have become the premier choice for the generation of short, intense laser pulses. Modelocked lasers usually generate ‘ultrafast’ pulses on the timescale of femtoseconds (10−15 s), with the current record held at pulses under 10 femtoseconds. This thesis documents an experimental attempt to construct a modelocked laser using a Helium-Neon laser tube and free-space optical components. We will begin by introducing the theory of modelocking and give theoretical details about the components with which we will assemble our laser. We will then motivate the design of the experiment itself. Finally, we will present data obtained while operating the laser, and analyze our results. Chapter 1 Theory This chapter gives an outline of the theory of laser modelocking and an introduction to the experimental apparatus, including the mathematical description of its components. Thus we will begin with a general description of lasers and the physical principles behind modelocking, and outline the two broad types of modelocking. Gain Medium Mirror Cavity Pump Process Figure 1.1: The schematic for a generic laser, including the three main components and a single standing-wave beam of light. Every laser requires three main components: a gain medium, an energy pumping process, and an optical cavity. A gain medium is a substance which is quantummechanically excited into higher energy levels. This energy is released as light, either randomly in the process of spontaneous emission, or by stimulated emission due to the passing of light through the substance. Stimulated emission generates photons that have both the same phase and same frequency as the stimulating photons. An optical cavity, also called a laser resonator, is a set of mirrors that surrounds the gain medium and reflects light back into it. If this reflected light encounters the excited portion of the substance, it can stimulate the emission of more light and generate a coherent beam. However, if the light instead encounters an unexcited portion of the substance, the photons may simply be reabsorbed. In order to ensure that more light is spontaneously emitted than absorbed, an energy pumping process is used to excite as much of the substance in the gain medium as possible. Most lasers operate in the ‘continuous wave’ (CW) regime, where operation in the 4 Chapter 1. Theory Cavity longitudinal mode structure Power (a) Frequency Power (b) Laser Gain Cavity Loss Frequency (c) Power Output Spectrum Frequency Figure 1.2: (a) The laser cavity supports an infinite Dirac delta frequency comb. (b) The gain of the laser gain medium (solid curve) has a Lorentzian or Gaussian shape, and must overcome the cavity losses (dashed curve) for lasing to occur. (c) Overlaying the region where gain is greater than loss onto the Dirac comb, we end up with a series of narrow emission bands bounded by a wider envelope. steady state implies generating a beam with constant frequency and intensity. The light emitted by a CW laser is ideally monochromatic. The frequency range over which most gain media amplify, called the gain bandwidth, is usually large enough such that multiple closely-spaced frequencies may oscillate simultaneously. Imagine a laser cavity of optical path length L, formed by two inward-facing highly reflective mirrors. This cavity will support standing light waves of certain evenly-spaced wavelengths, given by L = mλ , where m is a positive integer. The frequency corresponding to 2 this standing wave is ν = mc , implying that the difference in frequency between the 2L adjacent m and m + 1 modes is given by ∆ν = c . 2L (1.1) This forms an evenly-spaced comb of Dirac delta peaks in frequency space, representing ‘axial modes’ that will be supported by the cavity. We thus call ∆ν the ‘axial mode spacing.’ If the cavity is long enough (and thus the allowed modes close enough in frequency), then multiple axial modes will lie within the laser’s gain bandwidth. 5 Some of these modes will experience enough gain to overcome the losses that they experience when circulating in the cavity. See Fig. 1.2. In some lasers, called ‘inhomogeneously broadened’ lasers1 , these axial modes can lase simultaneously. Conversely, in ‘homogeneously broadened’ lasers2 , normal circumstances will allow only one axial mode to lase at a time, even if multiple modes fall within the region where the gain is greater than cavity losses. In either case, if multiple axial modes can be forced to lase simultaneously, and if certain phase relationships between the modes are maintained, then the modes will beat together to form laser pulses. Suppose a laser has been forced to lase in multiple axial modes simultaneously. If each neighboring pair of axial modes has the same relative phase shift, then all modes will beat together to form pulses. If the mode pairs continue to have the same relative phase shift over time, then the pulses will remain. We can call this condition ‘constant uniform mode pair phase difference,’ or we can simply say that the laser is ‘modelocked’. Maintaining a constant uniform mode pair phase difference is the primary challenge of modelocked laser design. This challenge arises because the phases of a free-running laser’s modes are typically not linked in any meaningful way. The relative phases of the modes may fluctuate randomly with respect to one another over time, a phenomenon called phase noise, which is due to quantum noise, mechanical vibrations, and temperature fluctuations. Experimentalists cannot hope to eliminate phase noise completely, but if the noise for each mode is correlated, then pulses will still form, and we can still consider this state modelocking (see Fig. 1.7) [Wollenhaupt et al., 2007]. In a practical laser, part of the power of the circulating pulses will be coupled out of the cavity. The frequency at which pulses are emitted from the cavity, which we call the ‘round-trip frequency,’ is νRT = c . 2L (1.2) Note that this is equal to the axial mode spacing, ∆ν. Thus lasers with long cavities, and therefore close axial modes, will rapidly emit pulses. Modelocked lasers fall into two primary classifications, based on the technique used to bring the axial modes into phase. Some modelocked lasers employ an electronicallydriven modulator to periodically change the cavity loss or frequency of light; this technique is called active modelocking, because the modulator actively shifts optical power between axial modes. Loss modulators achieve modelocking by varying the cavity loss at the round-trip frequency, so that the wings of the pulse experience loss while the pulse peak does not, and thus the pulse is shortened. Frequency modulators, on the other hand, the kind used in this thesis, shift part of the light passing through 1 This means that different atoms in the gain medium experience different frequency-shifting effects, so that different atoms have gain spectra that are slightly shifted relative to one another. In the HeNe and other gas lasers, this is mostly due to Doppler shifts that arise from the velocity distribution of the atoms [Siegman, 1986, Section 12.2, pp. 462-465; Section 30.3, pp. 1185-1186]. 2 Homogeneous broadening means that all atoms in the gain medium experience the same frequency-shifting effects, so the gain spectra of the atoms lie on top of one another. In homogeneously broadened lasers, like those with solid-state gain media, the lasing of one mode ‘saturates’ the gain of other modes. We will discuss gain saturation in more detail later. Chapter 1. Theory PSD Power, E 6 0 t Ν Figure 1.4: All modes in phase. Ideally, with a constant mode pair phase difference, the laser generates a Gaussian pulse train. The thick dashed line represents power, and the thin solid line represents electric field. Power, E Power, E Figure 1.3: Power spectral density (PSD) of a multimode laser. Simulated as a Dirac delta comb with a Gaussian envelope. The following figures show resulting waveform for different phase relationships. 0 0 t t Figure 1.6: Same phase noise. Although the phase of the electric field is noisy, pulses are still formed in the power waveform and we have modelocking. Power Power, E Figure 1.5: Random phases, no noise. If phases are random but fixed in time, then the waveform is deformed but periodic. Mode pair phase differences are constant but nonuniform. 0 t 0 Figure 1.7: Correlated phase noise. t Pulses will form, but the waveform will not be clean. This is the best that experimen- Figure 1.8: Independent phase noise. talists can accomplish under normal circum- Waveform will be noisy. This is the case stances. in a multifrequency continuous wave laser. 1.1. Laser Resonator Stability 7 them to a higher or lower frequency. This shifted light can excite other axial modes, and, by a process we will discuss later, can even establish the fixed mode pair phase difference necessary for modelocking. The other class of modelocked lasers, called passively modelocked lasers, involve a device that modulates light without being driven. Passive modelocking usually implies the use of a ‘saturable absorber’: a substance that will absorb light up to a certain intensity, and transmit more intense light. Like an active loss modulator, this has the effect of absorbing the wings of a pulse yet letting the pulse peak through. The advantage of the saturable absorber is that it can react very quickly to intensity variations, and thus can produce pulses of shorter duration than active loss modulators, which are limited by their driving frequency. It is passively modelocked lasers that hold the records for shortest pulse duration; Kerr-lens passive modelocking of Ti:sapphire lasers have produced pulses of under 6 femtoseconds, spanning less than two optical cycles [Sutter et al., 1999]. Despite the advantages of passive modelocking, actively modelocked lasers are arguably conceptually simpler, and, with most of the resources available on-hand in the Reed physics department, the actively modelocked laser was chosen for this experiment. In the following sections of this chapter, we will describe in turn the major components of the laser assembled in this thesis. We will begin by examining the laser cavity and the conditions necessary for the support of a laser beam between the cavity mirrors. We will then turn to our gain medium, the Helium-Neon discharge tube, and describe how the medium amplifies light by stimulated emission. With an understanding of how the gain medium generates coherent light, we will move on to the active modulator employed in the laser, an acoustooptic modulator, and explain how the device works as a frequency modulator to bring the axial modes into phase. With the theory of the components laid out, we will then model the operation of the modelocked laser by propagating a Gaussian electromagnetic pulse through each laser component. This argument will make some idealizing assumptions, including the assumptions that our frequency modulator can be tuned perfectly and that phase noise is negligible. However, we will see that because the laser system is not perfectly linear, the reality of imperfect tuning and the presence of phase noise still allow for modelocking to occur. 1.1 Laser Resonator Stability Laser resonators, or laser cavities, come in a variety of styles. The simplest case of a linear optical cavity consists of two spherical mirrors: a ‘high reflector’ (HR) that maximizes the reflected light, and an ‘output coupler’ (OC) that transmits a small fraction of the light incident on it. Such a cavity can be designed to support a beam that is Gaussian in geometry (with transverse intensity described by the Gaussian function). If a Gaussian beam can be reflected in on itself perfectly at both ends of the cavity, essentially none of the light in the beam will escape out of the sides of the 8 Chapter 1. Theory L Beam Waist R1 R2 Z1 Z=0 Z2 Figure 1.9: A linear laser cavity with two spherical mirrors, stably supporting a Gaussian beam. The cavity length, mirror radii, beam waist and distances to mirrors are all indicated. R1 R2 L < R1 L < R2 Stable L L < R1 L > R2 Unstable L > R1 L > R2 Stable L < R1+R2 Stable L = R1+R2 Unstable L > R1+R2 Figure 1.10: Various conditions under which a Gaussian beam may or may not be supported by a cavity. 1.1. Laser Resonator Stability 9 cavity. Such a cavity is said to stably support the Gaussian beam.3 Whether a cavity is stable, and thus able to support a Gaussian beam, depends on its geometry: the length of the cavity and the curvatures of the mirrors. Consider a cavity composed of two concave mirrors, with radii of curvature R1 and R2 respectively. We place the z axis along the optical axis, with z = 0 at the beam waist, the narrowest portion of the beam. The length of the cavity can be expressed as the sum of the distance from the R1 mirror to the beam waist, called z1 , and from the beam waist to the R2 mirror, called z2 : L = z2 − z1 . (1.3) Note that z1 is defined to be negative, because it runs from the beam waist inside the cavity to the mirror R1 , opposite to the direction of the optical axis [Siegman, 1986]. This causes L to always be a positive quantity. See Fig. 1.9. Suppose that we are given two mirrors, of curvature R1 and R2 , and we want to find the cavity lengths that will stably support a Gaussian beam of wavelength λ. To make sure that the beam is stable within the cavity, we need the curvature of the wavefronts to match the curvature of the mirrors, so that the mirrors reflect the wavefronts perfectly. Following Chang [2005, p.50], we first write down, for the case at hand, formulae relating the Gaussian beam curvature to z, zR2 , z1 z2 R2 = z2 + R , z2 −R1 = z1 + (1.4) (1.5) πw2 where zR = λ 0 is the Rayleigh range, and w0 is the radius of the beam waist. The minus sign in Eq. 1.4 is needed to settle a discrepancy in conventions. Gaussian wavefront curvature is taken to be negative for a converging beam, traveling in the direction of the optical axis. Mirror curvatures, on the other hand, are positive for concave mirrors (looking out from the inside of the cavity). The minus sign allows us to relate a negative wavefront curvature to a positive mirror curvature [Siegman, 1986]. We can solve Eq. 1.4 and 1.5 for z1 and z2 , q −R1 1 ± R2 − 4zR2 , (1.6) z1 = 2 2q 1 R2 1 z2 = ± R22 − 4zR2 . (1.7) 2 2 and plug them into Eq. 1.3, yielding a formula for the cavity length: q q R2 1 2 2 2 2 L= ± R2 − 4zR − R1 − 4zR . 2 2 3 (1.8) Note that so-called unstable cavities can still lase. Such laser cavities are usually designed so that light will leak out around one of the cavity mirrors. See Siegman [1986] for more details. 10 Chapter 1. Theory g1 g2 1.0 0.8 0.6 0.4 0.2 20 40 60 80 100 L Figure 1.11: Stability ranges of L. A graph of the product g1 g2 for a cavity with a 60 cm high reflector and a 45 cm output coupler. The upper stability limit is shown as a dashed line. The cavity will be stable if 0 < L < R1 or R2 < L < R1 + R2 , so in this case, 0 < L < 45 cm or 60 cm < L < 105 cm. Some algebra and squaring leads to the following expression for zR : zR2 = L (L − R1 ) (L − R2 ) (R1 + R2 − L) . (−2L + R1 + R2 )2 (1.9) The Rayleigh range is positive real by definition, so the right hand side of Eq. 1.9 must be positive as well. The denominator is always positive, by the reality of the variables involved. The numerator must also be positive, which places restrictions on the values of R1 and R2 . Either two of the parenthesized terms in the numerator are negative, or none of them are. This implies two possible inequalities, which represent two separate cases: 1. R1 + R2 ≥ L ≥ max (R1 , R2 ), 2. min (R1 , R2 ) ≥ L > 0. Physically, this means that a cavity will be stable in two cases: either the centers of curvature of both mirrors lie within the cavity, or both centers lie beyond the cavity.4 See Fig. 1.10 for an illustration of these conditions. A single, useful inequality thatapplies for both cases can be derived algebraically. L Define g1 = 1 − R1 and g2 = 1 − RL2 . In case 1, the quantities g1 and g2 are both negative, so their product must be positive. Likewise, in case 2, g1 and g2 are both positive, and the product must again be positive: g1 g2 > 0. (1.10) Also, note that the length L and curvatures R are always positive, so g1 and g2 must each be less than 1, and thus so must their product: 1 > g1 g2 . 4 (1.11) In the case that the curvatures are equal, it is possible for R1 = R2 = L, so both cases are satisfied simultaneously. 1.2. Helium-Neon Gain Medium 11 Together, Eqs. 1.10 and 1.11 form the ‘stability equation’ for laser resonators: 1 > g1 g2 > 0. (1.12) With the mirror curvatures R1 and R2 in hand, we can use Eq. 1.12 to place boundaries on the length of the cavity. Graphing g1 g2 as a function of L shows that Eq. 1.12 is equivalent to cases 1 and 2 above: the cavity is stable up to the radius of curvature of the first mirror, and again from the raius of the second mirror to the sum of the radii. See Fig. 1.11 for a graph with the mirror curvatures used in this thesis. 1.2 Helium-Neon Gain Medium This thesis utilizes the common helium-neon gas discharge tube as a gain medium. As explained in Siegman [1986], the tube is filled with a low-pressure mixture of helium and neon gas, with about ten times as much helium as neon. The potential difference between anode and cathode at opposite ends of the tube is over 1000 volts. This voltage accelerates electrons that collide with the helium atoms, exciting them to the long-lived 21 S state. If an excited helium atom collides inelastically with a ground state neon atom, the neon atom can be excited to the 3s state, which has a 632.8 nm laser transition to a 2p state. Photons of this frequency that interact with excited neon atoms can stimulate the emission of additional photons of identical frequency and phase. After many round-trips through the cavity, the photons begin to form a coherent beam. Consider the light circulating in a laser cavity as it passes by the gain medium. The amount by which light is amplified depends critically on frequency, so we express gain mathematically in terms of frequency. Transforming the electric field of the light, Ẽ(t),to the frequency domain, Ẽ(ω), we can approximate the effect of the gain medium by multiplying Ẽ(ω) by the transfer function g(ω):5 αp . g(ω) = exp 2 + 4i(ω − ω0 )/∆ω0 (1.13) Here α is the laser gain coefficient, which is less than 1, p is the gain medium length, ω0 is the peak laser transition frequency, and ∆ω0 is the full-width halfmaximum (FWHM) gain bandwidth. For the 632.8 nm HeNe transition, values are 0.02 cm−1 < α < 0.1 cm−1 (typically), ω0 ≈ 4.737 × 1014 Hz, and ∆ω0 ≈ 1.56 × 109 Hz (as found by Niebauer et al. [1988]). The real part of Eq. 1.13 is pα∆ω02 Re(g(ω)) = exp , 2 (∆ω02 + 4 (ω − ω0 ) 2 ) 5 (1.14) This expression reproduced from Siegman [1986, p.1062], though a factor of 1/2 is introduced as Siegman’s expression describes an entire round trip, and thus propagation through twice the cavity length L. Note that atomic collisions and Doppler broadening are ignored in this formula for the sake of simplicity; these phenomena cause broadening of the gain lineshape. 12 Chapter 1. Theory ReHgHΩLL 1 0 Ω - Ω0 Figure 1.12: Small-signal gain vs. detuning from resonance. Dependence of laser gain on the detuning of the incident field from a resonance of the gain medium. Ignoring line broadening, the lineshape is Lorentzian. representing the small-signal gain, or the maximum gain of light passing through the gain medium at frequency ω. A representative plot of this function for an arbitrary gain medium can be seen in Fig. 1.12. This shows that only a narrow band of frequencies are amplified by the medium. The gain values given by Re(g(ω)) are those experienced only by very small signals, and in general, the gain that any particular laser mode experiences is not constant; otherwise, the mode would continue to gain power forever. Instead, as the mode increases in intensity, the gain ‘saturates,’ decreasing until it equals the cavity losses, at which point the mode intensity equals its steady-state value. This saturation is due to the fact that the atoms in the gain medium need a finite amount of time to become re-excited. Consider a single mode of light at an intensity greater than the steady-state intensity. The presence of this light stimulates emission from the gain atoms more quickly than the atoms can become re-excited by the pump process. Meanwhile, losses continue at a constant rate and the mode intensity decreases, so at this instant the gain is less than the losses. As the mode loses intensity, the gain will increase until it exactly equals the cavity losses. This shows that lasing in the steady state is an equilibrium point in the dynamics of a laser. The imaginary part of Eq. 1.13 is pα∆ω0 (ω − ω0 ) , (1.15) Im(g(ω)) = exp − ∆ω02 + 4 (ω − ω0 ) 2 representing the phase shift introduced to each frequency. The gain medium is thus dispersive, and we will see in a later section that a light pulse will be broadened upon interacting with the gain medium. A representative plot is given in Fig. 1.13. The bandwidth of the gain medium is relevant to modelocked laser design, since the bandwidth limits the minimum duration of pulses produced by the laser. The minimum pulse duration is approximately equal to the inverse gain bandwidth. For the HeNe gain medium, with a gain bandwidth of ∆ω0 = 1.56 GHz, this estimation yields a minimum pulse duration of about 640 picoseconds. The current standard for research-grade ultrafast pulse generation is the titanium-doped sapphire crystal. 1.3. Acoustooptic Modulator 13 ImHgHΩLL 1 0 Ω - Ω0 Figure 1.13: Phase shift vs. detuning from resonance. Phase shift induced by a gain medium as a function of the detuning of the incident field from a resonance of the gain medium. With a gain bandwidth of about 128 GHz, it can generate pulses lasting under 10 femtoseconds. Comparatively, the HeNe is a relatively mediocre choice of gain medium, but it is much cheaper and easier to set up. 1.3 Acoustooptic Modulator This section is intended to describe the variety of acoustooptic modulators (AOMs) used in active modelocking, and to present a theory of how the AOM in this thesis interacts with laser light. An acoustooptic modulator is a mechanical device that consists of a piezoelectric transducer attached to a transparent crystal. The transducer sends radio-frequency acoustic vibrations into the crystal, and the compression waves induce changes in index of refraction to form a ‘phase grating’ off of which laser light can diffract. Depending on various operational and fabrication parameters, this device can act either like a standard diffraction grating that produces many diffracted beams, or as a Bragg scatterer that produces a single diffracted beam at a specified angle, or the behavior can lie somewhere in between. The distinction between these regimes will be discussed in a following subsection. In either case, light from the principle beam is diffracted into one or more higher-order beams. A second and arguably more important distinction between classes of AOMs is whether the acoustic wave is a standing wave or a traveling wave. 1.3.1 Standing vs. Traveling Acoustic Wave AOMs The standing wave AOM creates a stationary phase grating that oscillates in intensity. Sending a laser through this type of AOM causes the beam to undergo diffraction periodic in time. Such an AOM can be placed inside the laser cavity and used as a loss modulator. At zero amplitude of the stationary wave, the beam passes through unimpeded, but, as the amplitude increases, some of the incident optical power is 14 Chapter 1. Theory redirected into the diffracted beams, which escape out of the sides of the cavity, resulting in a loss. If this periodic loss modulation occurs at the round-trip frequency, a short pulse can form that coincides with low-loss periods. This is because every time the pulse reaches the AOM, the pulse wings are subject to more loss than the pulse peak, shortening the pulse. This technique is called modelocking by amplitude modulation. The traveling wave AOM, the type employed in the experimental section of this thesis, creates a moving phase grating. A laser beam passing through this AOM will constantly be diffracted by the grating, but the traveling nature of the grating causes higher and lower diffraction orders to become frequency-shifted by various amounts. This can be thought of, as will be explained in a following subsection, as the addition (or subtraction) of the energy and momentum of the acoustic waves to the optical wave. The frequency of each successive diffraction order is shifted by another multiple of the acoustic wave frequency. The upshot is that these frequency-shifted beams are generated in phase with the incident beam. If the acoustic frequency is set to equal the round-trip frequency, then diffracted beams that are reflected back into the laser cavity will resonate within it as axial modes, and proper alignment can ensure that the modes stay in phase as light circulates within the cavity. Thus, a traveling wave AOM can be used to bring the axial modes of a laser into phase. This technique is called modelocking by frequency modulation. With this application in mind, we will now turn to focus on the theory of operation of a traveling wave AOM and the use of this device for shifting the frequency of light. 1.3.2 Diffraction Regimes and Traveling Wave AOM Theory Based on how an AOM is constructed and employed, the device can operate in two diffraction regimes: the Bragg regime and the Raman-Nath regime. In the Bragg regime, light passing through a thick AOM crystal interacts with many acoustic wavefronts to form a single, intense diffracted beam that only appears at certain angles of incidence. This is analogous to the diffraction of X-rays off of crystal layers; thus the reference to Bragg. In the Raman-Nath regime, a relatively thin AOM acts as a standard diffraction grating, generating many higher-order beams regardless of incident angle. Although the AOM used in this thesis operates somewhere between the regimes, it operates closer to the Raman-Nath (diffraction grating) regime, so the theory of that regime will be discussed in detail.6 Under most circumstances, the regime in which an AOM operates is determined by geometrical factors7 . For Λλd2 1, the AOM is in the Bragg regime, and for Λλd2 < 1, we are in the Raman-Nath regime. Here λ is the optical wavelength of incident light, Λ is the wavelength of the acoustic wave, and d is the width of the crystal. To understand the origin of this condition, consider first the Bragg diffraction 6 This theory follows Boyd [2008, Section 8.4], who gives detailed descriptions of both diffraction regimes. The Bragg regime is described in Section 8.4.1, while the Raman-Nath regime is discussed in Section 8.4.2. 7 For a more accurate method for distinguishing between regimes, the degree of modulation of the index of refraction must be taken into account. See Moharam and Young [1978]. 1.3. Acoustooptic Modulator 15 θ L L Λ Figure 1.14: Bragg diffraction. A plane wave scatters off of the layers in a crystal. If the incident angle is correct for constructive interference, the ray diffracted by the lower crystal face will travel an extra distance 2L = 2Λ sin θ = mλ, where m is an integer, so that the rays scattered by the upper and lower planes are in phase. regime, shown in Fig. 1.14: interference between reflections from multiple acoustic waves generates a diffracted beam, the angle of which is specified by the relative wavelengths of the optical and acoustic waves: sin θ = mλ , 2Λ (1.16) where θ is the half-angle of the diffracted beam away from the acoustic plane, and m is the diffraction order. Note that the efficiency of diffraction should increase if the laser beam encounters more acoustic wavefronts before exiting the AOM. Thus the intensity of the diffracted beam will be greatest if the transverse distance covered by the beam inside the AOM is much larger than the acoustic wavelength. The transverse distance, z is found geometrically by examining Fig. 1.15: z = d tan θ. (1.17) Encountering multiple wavefronts requires that z Λ. Here d is the width of the AOM. In most cases, θ is small, so we can say that sin θ ≈ tan θ ≈ θ. In this approximation we can combine Eqs. 1.16 and 1.17 and simplify the result to find the condition that must hold for Bragg scattering to occur: λd 1. Λ2 (1.18) If instead the AOM is thin enough that the beam only encounters one acoustic wavefront within the AOM, we will have Raman-Nath scattering: z = d tan θ < Λ. This is shown in Fig. 1.15. (1.19) 16 Chapter 1. Theory d z = d tan θ θ Λ Figure 1.15: Requirements for Raman-Nath diffraction in a traveling-wave AOM. The wavelength of the acoustic waves, Λ, must be greater than the transverse distance covered by the light beam through the AOM, d tan θ. This ensures that the beam does not diffract off of multiple wavefronts. It will be shown that Raman-Nath scattering generates diffracted beams at angles similar to a diffraction grating. Assuming for now that this is the case, we can combine the above inequality with the diffraction grating equation, sin θ = mλ , Λ (1.20) to obtain, for the first order diffracted beam (m = 1), again in the small-angle approximation, λd < 1. Λ2 (1.21) Thus the relative dimensions of the AOM and the wavelengths involved give an indication as to the type of scattering the acoustooptic modulation will produce. The AOM used in this thesis consists of a tellurium dioxide crystal, for which the ratio λd/Λ2 ≈ 5, placing the AOM between regimes. For the sake of brevity, only the Raman-Nath theory is presented here; this theory has the advantage of being more closely related to expressions about general frequency modulators than the Bragg theory. We will now present the theory of Raman-Nath scattering. We first note that the piezoelectric transducer induces a sinusoidal density variation ∆ρ̃ = ∆ρ sin(qz − Ωt). (1.22) This will generate a refractive index variation ∆ñ = ∆n sin(qz − Ωt). (1.23) To find the amplitude ∆n of the refractive index variation in terms of the known ∆ρ, we can employ the relationship between relative permittivity change and density 1.3. Acoustooptic Modulator 17 change, which we assume is small enough as to be linear: ∆˜ = ∂ ∆ρ̃ = γe ∆ρ̃/ρ0 , ∂ρ (1.24) where γe is the electrostrictive constant and ρ0 is the bulk density, both of which are measurable. Now note that the total refractive index, n = n0 + ∆ñ, is p n = 0 + ∆˜, (1.25) as long as we assume that the relative permeability is 1 (a good approximation for most dielectrics). Since the relative permittivity variation ∆˜ is small, we can make √ √ √ the approximation n ≈ 0 + ∆˜/(2 0 ), which results in the relation n0 = 0 , and ∆ñ ≈ ∆˜ . 2n0 (1.26) Using Eq. 1.24 we can finally relate the variation of the refractive index to density variations of the AOM crystal: ∆n = γe ∆ρ . 2n0 ρ0 (1.27) Now we can figure out what happens to an incident light wave as it passes through the AOM. Sinusoidal voltage signals applied to the AOM-transducer result in density variations of the crystal, which lead to variation of the refractive index. This, in turn, will cause a phase shift. The exact shift depends on both position and time. To calculate the relative phase shift between different parts of the light wave after traveling across the AOM, we can use a varying wavenumber, call it ∆κ: ω ω ∆nd sin(qz − Ωt) (1.28) φ = ∆κd = ∆ñd = c c ≡ δ sin(qz − Ωt), (1.29) which serves to define δ. If we write the incident field in complex form as Ẽ(x, t), then the transmitted field Ẽ 0 (x, t) is simply the incident field multiplied by a complex transmission function eiφ : Ẽ 0 (x, t) = Ẽ(x, t)eiδ sin(qz−Ωt) . (1.30) We can put this result into a more revealing form by using a variation of the JacobiAnger expansion: eiδ sin θ = ∞ X Jm (δ)eimθ , (1.31) m=−∞ which employs Bessel functions of order m acting on δ. Applying this expansion to the transmitted field gives 0 Ẽ (x, t) = Ẽ(x, t) ∞ X m=−∞ Jm (δ)eim(qz−Ωt) . (1.32) 18 Chapter 1. Theory Consider the effect that the modulation has on a plane wave of frequency ω and wave vector ~k = kx̂. We can assume that the beam covers a large enough transverse (ẑ-direction) distance to interact with multiple wavelengths of the acoustic wave. Representing the plane wave in complex notation, we get 0 i(kx−ωt) Ẽ (x, t) = A[e −i(kx−ωt) +e ] ∞ X Jm (δ)eim(qz−Ωt) (1.33) m=−∞ = A = A ∞ X m=−∞ ∞ X Jm (δ)[ei[(kx+mqz)−(ω+mΩ)t] + e−i[(kx+mqz)−(ω+mΩ)t] ] (1.34) Jm (δ) sin[(kx + mqz) − (ω + mΩ)t]. (1.35) m=−∞ This reveals that the transmitted field is in fact a superposition of plane waves of varying magnitude, direction and frequency. The diffracted waves are essentially formed by adding or subtracting the momentum and energy of the acoustic wave to the incident optical wave. Higher-order diffracted beams vary in frequency in increments of Ω, the acoustic wave frequency, and vary in wavevector in increments of ~q, the acoustic wavevector. Thus varying the AOM-transducer frequency provides control over the frequency of diffracted beams. Since the speed of sound in the crystal is independent of frequency, changing the transducer frequency also changes the wave number and thus the deflection angle. (Traveling wave AOMs are often used as beam deflectors that can precisely alter the direction of diffracted orders without much dependence on incident beam angle). Imagine the wavevector of the m-th order output from the AOM as the hypotenuse of a right triangle, with the optical wavevector ~k as one leg, and an integer number of acoustic wavevectors m~q as the other leg. Then the angle of the diffracted mode away from the optical wavevector ~k is given by the trigonometry of the triangle, tan θm = mλ mq = , k Λ (1.36) which at small angles is identical to the diffraction grating equation. When using a traveling wave AOM for a modelocking experiment, one would, ideally, capture all diffracted beams and redirect them so that they recirculate inside the laser cavity. Doing so would eliminate the z-dependence of the phase shift, so the total phase shift equation would read φ = δ sin(Ωt). (1.37) This idealization will be useful shortly when modeling the behavior of a modelocked laser. In practice, it is possible to capture just one diffracted beam, typically the first-order beam, and redirect it so that it circulates in the laser cavity, allowing a one-directional transfer of power between frequencies in the cavity. 1.4. Time-domain modeling: Pulse Propagation 1.4 19 Time-domain modeling: Pulse Propagation Having described the components of the modelocked laser and described how each affects passing light, we can introduce a theory of optical pulses and propagate them through each component. Following the argument of Siegman [1986, Section 27.3], this section is intended to demonstrate how an optical pulse can maintain a Gaussian shape throughout a round-trip through the laser. Unfortunately, some of the approximations used are invalid in the experimental setup described later in this thesis, but the idealized argument presented here is nonetheless useful for understanding the generation of Gaussian pulses in a general modelocked laser system. We will begin by developing the mathematics for the Gaussian pulse, move on to find the effect each optical element has on the pulse, and finally find a stable round-trip pulse shape. 1.4.1 Gaussian Pulses We can approximate the pulses that circulate in the laser as having Gaussian intensity profiles. In the time domain, we can describe a Gaussian pulse in the form 2 2 Ẽ(t) = e−at ei(ωc t+bt ) = exp(−Γt2 + iωc t), (1.38) where a is the envelope parameter, ωc is the carrier frequency, and Γ ≡ a − ib is the ‘Gaussian pulse parameter’. If b is nonzero, then the quadratic dependence of optical phase on time implies that the instantaneous frequency varies linearly along the pulse. This linear frequency variation is called ‘chirp,’ and is a result of the shifting of parts of the pulse spectrum. The instantaneous intensity of the pulse is 2 2 I(t) = e−2at = e−(4 log 2)(t/τ ) , (1.39) p where τ = 2 log 2/a is the full width at half-maximum (FWHM) of the intensity profile. Thus a large a parameter implies a short pulse, while a small a implies a long pulse. Propagation of the pulse through most optical components is more easily accomplished in the frequency domain than in the time domain. Taking the Fourier transform of the field results in Z ∞ Ẽ(ω) = Ẽ(t)e−iωt dt (1.40) −∞ Z ∞ 1 2 = √ e−Γt eiωc t e−iωt dt (1.41) 2π −∞ 1 (ω − ωc ) 2 = √ exp − . (1.42) 4Γ 2Γ With this result in hand, we can turn to propagating the pulse through the laser. 20 Chapter 1. Theory 1.4.2 Pulse Propagation We can propagate our pulse through a simple laser setup by multiplying the field by either a transmission function in the time domain or a transfer function in the frequency domain. Consider a linear laser consisting of generic laser gain medium (with a Lorentzian lineshape) and a generalized frequency modulator within a twomirror cavity. Suppose the cavity has length L. We begin with the standard Gaussian pulse, in frequency representation, and assume that the carrier frequency is equal to the peak frequency in the gain medium: (ω − ω0 ) 2 1 exp − . (1.43) Ẽ0 (ω) = √ 4Γ0 2Γ0 First let us consider the effect the gain medium has on the pulse during one roundtrip. We will multiply the field by the transfer function g(ω), from the section on gain media, but first we can make an approximation that will be useful in analysis later8 : αp (1.44) g(ω) = exp 2 + 4i(ω − ω0 )/∆ω0 ω − ω0 8 2 ≈ exp αp 2 − 4i (ω − ω0 ) . (1.45) − ∆ω0 ∆ω02 Here we have made a Taylor expansion approximation based on the assumption that the spectrum of Ẽ(ω) is much narrower than the total gain bandwidth of the laser transition.9 Multiplying the Gaussian field by g(ω) yields a new field (ω − ω0 ) 1 8αp 1 2 0 exp 2αp − 4iαp − + (ω − ω0 ) . (1.46) Ẽ (ω) = √ ∆ω0 4Γ0 ∆ω02 2Γ0 The three terms in the exponent represent different physical effects that the gain medium has on the field. The first term represents the amplification afforded to each frequency, which is independent of frequency (according to the approximation we made above, that the bandwidth of the pulse is much narrower than the FWHM of the gain medium). The second term is imaginary, and thus represents a phase shift in frequency space, corresponding to a delay of the field in time. We will consider this term later. The last term is real and quadratic in ω, and corresponds to the new waveform of the field. This term is, like the exponent of Eq. 1.42, proportional to −(ω − ω0 )2 , so it is again a Gaussian waveform. We can describe this pulse with a new Gaussian pulse parameter, Γ0 : 1 8αp 1 = + 0 4Γ 4Γ0 ∆ω02 Γ0 ∆ω02 =⇒ Γ0 = . 32αpΓ0 + ∆ω02 8 (1.47) (1.48) Note that we are passing through the gain medium twice during the round-trip. This assumption is invalid for the experimental setup of this thesis, because at practical cavity lengths, the HeNe’s narrow gain bandwidth covers only a few intermode frequency spacings. 9 1.4. Time-domain modeling: Pulse Propagation 21 Decomposing this result into real and imaginary parts, Γ0 = a0 − ib0 , we have a0 = b0 = ∆ω02 (32 (a2 + b2 ) pα + a∆ω02 ) 1024b2 p2 α2 + (32apα + ∆ω02 ) b∆ω04 2, (1.49) 2. (1.50) 1024b2 p2 α2 + (32apα + ∆ω02 ) These are complicated expressions, and the behavior of a and b over many round trips depends on the initial values of the parameters. However, we can run a simple simulation to show that a pulse circulating in a cavity containing only a gain medium will broaden over time, returning to a sine wave. This is done by choosing values for the constants and initial values a0 and b0 , and then iterating Equations 1.49 and 1.50. A representative plot of the evolution of a and b after each round-trip can be seen in Fig. 1.16. Both a and b decrease slowly over time, meaning that the pulse width increases to infinity and the chirp decreases to zero, approaching a sine wave. However, if the initial pulse has a high chirp (large b), then there will be a transient effect where the pulse width decreases before the long-term increase begins. This suggests that a chirped pulse, and thus a pulse with phase-shifted modes, will be narrowed in time when passing through the gain medium. However, without modulation, the gain medium will eventually cause the pulse to broaden, and the chirp to decrease, due to a narrowing of the pulse spectrum. a, b Round trips Figure 1.16: Evolution of pulse parameters in a cavity with gain. Calculated iteratively over many round trips. The thick line represents a, and the thin line represents b. Both a and b decrease slowly over time. The dashed line shows the initial value of a. Let us now consider the phase shift component of the exponential in Eq. 1.46, 4iαp (ω − ω0 ) /∆ω0 . Taking the inverse Fourier transform of this part of the pulse spectrum reveals that the constant term, −4iαpω0 /∆ω0 , simply shifts the phase of the pulse in the time-domain. The frequency-dependent term, 4iαpω/∆ω0 , on the other hand, actually translates the pulse in the time domain, so that the pulse is delayed. This small delay, due to the gain medium, means that the round-trip time is not exactly 2L/c, and the axial mode spacing is not exactly c/2L. However, the product αp and the difference ω − ω0 are usually quite small compared to ∆ω0 . 22 Chapter 1. Theory While the perturbation is theoretically nontrivial, its magnitude is usually within the uncertainty of an experimentalist’s tools for measuring the cavity length or roundtrip frequency. Thus experimentalists usually ignore this perturbation and estimate the axial mode spacing to be c/2L, as they can tune cavity length and modulation frequency empirically for best modelocking. Finally, we consider the effect of the frequency modulator on the pulse. We developed the modulator theory in the time domain, so we use the time-domain Gaussian pulse. The gain and phase-shift terms induced by the gain medium are inconsequential in the steady state10 , so we focus on the term involving the pulse parameter: Ẽ(t) ∝ exp[−Γ0 t2 ], (1.51) The transmission function from Eq. 1.37, this time replacing the sine function with a cosine function, is given by T (t) = eiδ cos(Ωt) . We can make a simplifying approximation of the cosine function by Taylor expansion, but we first need to figure out at which point in time the pulse will cross the modulator. Suppose the pulse crosses the modulator at t = nπ/2 for odd integer values of n; that is, when cosine sweeps through zero and the phase changes most rapidly. This would yield a transmission function approximately proportional to exp[iδΩt]. This would change the instantaneous frequency of the pulse Ẽ(t), bringing the central pulse frequency outside of the bandwidth of the gain medium after many round-trips. Thus this pulse will not be amplified and will quickly die off due to cavity losses. To form pulses that survive in the steady-state, then, we want the pulse to avoid periods of rapidly-changing phase. If the pulse passes through the modulator when the phase is nearly constant (at the crest or trough of the cosine function), then we can avoid terms in the transfer function that change linearly with time, and thus avoid shifting the central frequency of the pulse. If we assume the pulse duration to be much shorter than the modulation period11 , we can make the following quadratic approximation for the transmission function: T (t) ≈ exp iδ 1 − Ω2 t2 /2 . (1.52) After passing through the modulator, then, the field is Ẽ 0 (t) ∝ exp −Γ0 t2 + iδ 1 − Ω2 t2 /2 = exp iδ − Γ0 + iδΩ2 /2 t2 . (1.53) (1.54) The iδ term is a small constant phase shift. The coefficient of t2 is the new Gaussian pulse parameter: Γ00 = Γ0 +iδΩ2 /2. This shows that |b00 | > |b0 |, so the chirp of the pulse is increased during modulation, but a00 = a0 , so modulation does not directly compress 10 The gain per round trip depends on the degree to which the gain medium is saturated, and in the steady state, it is exactly cancelled by the cavity losses. The phase shift accumulates over every round trip, so emitted pulses generally differ in optical phase. 11 This is another approximation that fails in our experimental case, but is true for modelocked lasers employing broadband gain media. 1.4. Time-domain modeling: Pulse Propagation 23 the pulse. However, increasing the chirp of the pulse does broaden the spectrum, and looking back at Eq. 1.49, we see that the when the pulse passes through the gain medium again, the parameter a increases if the magnitude of b had increased during the previous round-trip. Thus upon exiting the gain medium, the pulse is narrowed slightly. This indirect pulse narrowing causes the spectrum to broaden and the pulse duration to decrease over many round trips, until a steady state is reached. When we include the modulation in our simulation, as in Fig. 1.17, we see that both a and b stabilize to reach approximately the same value. This stable value is independent of the initial pulse parameters; the stable value is reached even when a0 = b0 = 0. a, b Round trips Figure 1.17: Evolution of pulse parameters with both gain and modulation. The thick line represents a, and the thin line represents b. Whenever the lines cross, one of the values reaches a maximum or a minimum. The dashed and dotted lines represent the theoretical steady-state values of a and b respectively, as obtained from Eq. 1.58. Finally, we can solve our formulae analytically for the steady-state pulse parameter. In the steady state, Γ00 = Γ0 p 2ipαδΩ2 + −4p2 α2 δ 2 Ω4 + ipαδΩ2 ∆ω02 =⇒ Γ = 8pα (1.55) (1.56) (we choose the root such that a is positive). Expanded into real and imaginary parts, we have √ iδΩ2 δΩ (16p2 α2 δ 2 Ω4 + ∆ω04 ) 1/4 Γ= + (Cos[ψ] + iSin[ψ]), (1.57) √ 4 8 pα 1 with ψ = argument pαδΩ2 −4pαδΩ2 + i∆ω02 . (1.58) 2 This result is complicated. Numerically, we find that b is slightly higher than a for the values used in the iterative simulation, as seen in the Fig. 1.17. In this section, we have shown that a steady-state Gaussian pulse solution exists, where an initial Gaussian pulse maps back onto itself after propagating through the 24 Chapter 1. Theory gain medium and intracavtiy frequency modulator. We have also provided evidence that this solution is stable in the sense that any initial Gaussian evolves toward this solution. 1.5 Injection Locking There are two practical problems with modelocked laser experiments that we implicitly ignored in the preceding theory. The first is the fact that we cannot hope to perfectly tune the modulation frequency Ω to line up with the axial mode spacing ∆ν. The frequency-shifted light will, in every practical case, be detuned slightly from the axial mode frequencies that the laser cavity supports. We expect that the spectra of both the allowed cavity modes and the frequency-shifted light will be broadened by noise, but the overlap of these broadened peaks will not be complete, resulting in low gain of the frequency-shifted light. Thus there would be little power in any of the modes except the principle mode, forming broad output pulses at best. The second, and perhaps more important, problem we ignored was that of phase noise. When we began our discussion of modelocking, we emphasized that a goal of the technique is to minimize phase noise differences between mode pairs. The pulsepropagation modeling we presented in the previous section showed that the spectrum of a pulse will be broadened inside our laser (spectral broadening being the other main goal of modelocking), but it assumed that the pulse was noiseless, and that the optical components did not contribute any noise. In reality, every component contributes noise to the system, whether it be from spurious vibrations of the mirrors or AOM, thermal deformations of those components, or variations of index of refraction in the air between components. Thankfully, it turns out that modelocked lasers can operate effectively in the presence of both detuning of the modulation frequency and standard experimental noise. This is due to the nonlinearity of the system. Suppose two axial modes, at ω1 and ω2 = ω1 + ∆ν, are oscillating simultaneously in a laser that includes a frequency modulator that is currently turned off. The two modes have different phase noise. Now suppose that the frequency modulator is turned on. All we want this modulator to do is take part of the incident light and increase its frequency by Ω, where Ω is close but not equal to the axial mode spacing ∆ν. Considering only the ω1 mode for now, the modulator will therefore output two frequencies of light, ω1 and ω∗ = ω1 +Ω, where the ω∗ light has some phase relationship with the ω1 signal. We say that the light at ω∗, which does not lie precisely at any axial mode, has been ‘injected’ into the cavity. If the laser were a completely linear system, we would have three frequencies of light superimposed in the cavity: ω1 , ω2 and ω∗. However, because of nonlinear effects, such as cross-gain saturation, the presence of light at ω∗ affects the generation of light at ω2 . In fact, if the frequency ω∗ is close enough to ω2 , then the frequency of the ω2 mode will be ‘pulled’ to the frequency ω∗, and the phase of this pulled mode will follow the phase of the ω∗ signal. Thus by turning on the modulator, we have effectively replaced the mode at ω2 with a mode at ω∗, and because the ω∗ mode was generated from ω1 , the two modes in the cavity have a constant phase relationship. 1.5. Injection Locking 25 This synchronization phenomenon, called ‘injection locking,’ occurs in almost any nonlinear oscillatory system in which a periodic signal is injected near resonance. It was described mathematically in a paper by Adler [1973], and this description was applied to laser oscillators by Kurokawa [1973]. To show how injection locking works, we will consider the behavior of a general selfsustained oscillator when injected with a periodic signal (a more in-depth treatment of the injection locking of lasers is given in Siegman [1986, Chapter 29]). The system is described by the so-called Adler’s equation12 , dψ = −ωD + sin ψ, dt (1.59) where ψ is the phase difference between the oscillator and the injected signal, ωD is the detuning of the injected signal’s frequency from the natural frequency of the oscillator, and will be shown to be the injection locking range, which has the units of frequency. We can solve this equation for ψ(t), which tells us how the phase of the oscillator relates to the phase of the injected signal over time: " − ψ(t) = 2 arctan ωD "p ## p 2 2 ωD − 2 ωD − 2 tan t ωD 2 (1.60) There are, then, two general regimes of solutions, which depend critically on the relative magnitudes of ωD and . ΨHtL Π Ψf Ψ0 0 T 2T 3T t Figure 1.18: Evolution of phase difference, ωD < . Solution to Adler’s equation for small detuning. The phase of the oscillator approaches a fixed phase difference, ψf , with respect to the injected signal, so the frequencies are identical and injection locking occurs. The dashed line shows the final phase difference. 12 The equation in this form was adapted from Pikovsky et al. [2001], which derives Adler’s equation and provides a generalization that can be applied to many situations. 26 1.5.1 Chapter 1. Theory Locked Regime: ωD ≤ If the the detuning ωD is less than the parameter , then the radical factor becomes imaginary: q 2 ωD − 2 = ih, (1.61) which serves to define h. The solution (1.60) can now be written ih ih ψ(t) = 2 arctan − tan t . ωD ωD 2 (1.62) We can solve the problem of the tangent function taking an imaginary argument by making the replacement tan(iθ) = i tanh(θ): h h ψ(t) = 2 arctan t . (1.63) + tanh ωD ωD 2 We plot this function in Fig. 1.18. We can see that the phase difference moves from √ 2 + 2 −ωD the initial value ψ0 to a final constant value ψf = 2 arctan , indicating ωD that the oscillator is oscillating at the same frequency as the injected signal but with a shifted phase. Here T is the natural period of the oscillator when undisturbed. This is the injection locked regime. Note that if we had set ωD = , then the solution would be ψ(t) = 2 arctan(1) = π/2, so the phase difference would still be constant and we can still consider this situation injection locking. 1.5.2 Unlocked Regime: ωD > If the detuning ωD is greater than the parameter , then the radical factor is real, and we can call Eq. 1.60 our solution. This is plotted in Fig. 1.19 for ωD slightly larger than (1.2:1). What is happening here is that, as long as ψ evolves slowly, the oscillation is basically at the injected frequency, but the phase periodically ‘slips,’ changing the instantaneous frequency of oscillation. If we increase ωD to be much greater than (2:1), shown in Fig. 1.20, then the phase slips become more frequent, and the period of the phase slips approaches the period of the undisturbed oscillator. Thus, for large detunings, the oscillator essentially oscillates at the undisturbed frequency. 1.5.3 Consequences of Injection Locking In the context of our laser, injection locking helps overcome the two difficulties presented in the beginning of this section: modulator detuning and phase noise. As long as the detuning of the modulator is small enough, injected frequency-shifted modes will replace the naturally occurring axial modes. Also, the phase noise of the injected mode replaces the noise of the natural axial mode, essentially coupling the noise of 1.5. Injection Locking 27 ΨHtL Π Ψ0 T 2T 3T t -Π Figure 1.19: Evolution of phase difference, ωD & . Solution to Adler’s equation for detuning slightly larger than . The frequency of the oscillator is usually close to that of the injected signal, but the phase ‘slips’ periodically, though less frequenty than the undisturbed period T . The dashed line shows the initial phase, to give a sense of the phase slip period. ΨHtL Π Ψ0 T 2T 3T t -Π Figure 1.20: Evolution of phase difference, ωD . Solution to Adler’s equation for detuning far larger than . The period of phase slips approach the period of the undisturbed oscillator, so the oscillator usually oscillates near the undisturbed frequency. each mode in the cavity. But how small does the detuning have to be? That is, what can we expect to be, and how can we increase it? Siegman [1986, Eq. 29.30] gives r Iinj =γ , (1.64) I0 where Iinj is the intensity of the injected signal, I0 is the intensity of the undisturbed mode, both measured inside the cavity, and γ is the ‘cavity decay rate,’ which is δ , where δ is the percentage of light that is emitted from approximately equal to TRT the output coupler mirror, and TRT is the round-trip time through the cavity, which is just 1/νRT . What all this means is that we can improve our chances of modelocking by increasing the intensity of the light we inject into the laser, by increasing the axial mode spacing (making our cavity shorter), and increasing the amount of light transmitted through our output coupler mirror. Thus the laser will be more difficult 28 Chapter 1. Theory to injection-lock if we use a long cavity to reduce the duration of our pulses, so injecting as much intensity into the cavity is a priority. In this section, we have seen that the two problems we feared would ruin our experimental attempts at modelocking, being modulator detuning and phase noise, are mitigated by injection locking, which results from the inherent nonlinearities of the laser. On this positive note, we make the assumption that the theory presented in this chapter will be sufficient for guiding us through the construction and analysis of a modelocked laser, and turn now to the experimental design of our laser. Chapter 2 Experimental Design In this chapter, we will describe and motivate the experimental setup used to achieve modelocking. First we will consider a simple design that employs a traveling-wave AOM for modelocking. We will then describe the practical problems with this design and introduce the actual experimental setup. HR L0 er AOM HR t 1s d or 0th order HeNe tube OC L0 Figure 2.1: Theoretically simple split cavity for modelocking. Each leg of the cavity is the same length, so a circulating pulse that splits at the AOM will be recombined at the AOM with no relative delay between the legs. There are two high-reflecting (HR) mirrors and one output-coupling mirror (OC). In this configuration, each mirror can ideally have the same radius, equal to L. The design of our modelocked laser is centered around the modulator available, a traveling-wave acoustooptic modulator (AOM). As explained in Chapter 1, the traveling-wave AOM diffracts incident light into multiple beams, each of which is frequency-shifted by a certain amount. A modelocked laser can be constructed by capturing both the 0th order beam, which is not frequency shifted, along with one of the higher-order frequency shifted beams, such as the 1st order beam, and allowing both to beat together inside of an optical cavity. A theoretically simple setup that will contain both 0th and 1st order beams within an optical cavity involves three 30 Chapter 2. Experimental Design AOM 0,1 AOM 1,1 Reflected 1st order 0,0 Reflected 0th order 0,-1 Figure 2.2: After reflecting off of the output coupler, the 0th order is reflected exactly back in on itself and diffracted once again in the AOM. The 0th order of this second diffraction, labelled (0,0), is contained by the cavity, while the other orders are lost. 1,0 1,-1 Figure 2.3: Like the 0th order, the 1st order is reflected exactly back in on itself and diffracted once again in the AOM. This time the 1st order of this second diffraction, labelled (1,1), continues to the HeNe tube and is contained by the cavity, while the other orders are lost. mirrors and is shown in Fig. 2.1. In this setup, an AOM and a third mirror have essentially been placed inside of a linear two-mirror laser cavity. The third mirror reflects the 1st order back into the AOM, so both 0th and 1st orders are reflected back into the AOM. These reflected beams also diffract. Examining the diffraction of the reflected 0th order beam in Fig. 2.2, we see that only the unshifted part of this diffraction pattern (beam 0,0) remains inside the cavity. The diffraction of the reflected 1st order beam is seen in Fig. 2.3; in this case the first shifted beam of this diffraction pattern (beam 1,1) remains inside the cavity. Thus the 1,1 beam is frequency-shifted by the AOM twice: once propagating left and once propagating right, so we set the AOM modulation frequency to half of the round-trip frequency. This twice-shifted beam injection locks the modes circulating between the HeNe tube and the OC, modelocking the laser. It is important in this configuration that the distance from the AOM to each mirror on the right-hand side of the the setup is the same, so that the 1st and 0th order beams traverse the same distance before recombining. If we think of a pulse propagating rightward in the time domain through this laser, we can see that it will split at the AOM, the split pulses will traverse each leg separately, and then recombine again at the AOM. To make sure that the recombined pulses constructively interfere, each leg of the cavity must be the same length. This length is labeled as L0 on Fig. 2.1. This split cavity is theoretically sound, but difficult to construct for a number of practical reasons. Most importantly, the losses that the beam incurs through the AOM are much greater than the gain from the HeNe tube during each round trip, so there is no hope that enough photons will circulate in the cavity simultaneously for lasing to occur. In the standard two-mirror continuous wave configuration, the HeNe tube used in this thesis has such a low gain that even a glass coverslip, if introduced into the cavity as a beamsplitter, presents too much loss. If a thin coverslip cannot be placed into the cavity, then a thicker crystal will definitely not be supported. The solution to the problem of too much loss is to place the AOM outside of 31 the laser cavity, allowing unimpeded amplification inside the cavity, and then couple the frequency-shifted light back into this cavity. Figure 2.4 shows the experimental setup used in this thesis. The setup consists of a ‘primary cavity’ configured like a standard continuous-wave laser cavity, and a ‘secondary cavity’ formed by using a third spherical mirror to couple light back into the primary cavity. Modelocking with an external modulator was first accomplished by Foster et al. [1965], using a tank of water instead of an acoustooptic device. The setup used in this lab follows the layout in the graduate lab of Jones [2009] at the College of Optical Sciences of the University of Arizona. L L R = 60 cm HR R = 45 cm OC AOM R=1m HR y r vit e a d r Edge yC to ar 1s d Mirror n co e S 0th order Primary Cavity Scope ESA FPI Fast Photodiode Figure 2.4: Experimental setup. The 0th order diffracted from the AOM is reflected by an edge mirror into either a Fabry-Perot interferometer or a fast photodiode. The entirety of the optical setup was built on a Newport RS 300 floating optical table. The primary cavity consists of a HeNe laser tube and an output coupler (OC) mirror through which the laser beam is emitted. The HeNe tube is a Melles Griot 05LHB-570 powered by the power supply of a Spectra Physics 132 laser. The right end of the HeNe tube is terminated in a glass window that lies perpendicular to the Brewster angle of 632.8 nm as it transitions from air to glass. This setup, called a Brewster window, minimizes the reflectivity of the glass-air interface for light polarized parallel to the window. The Brewster window thus helps polarize the laser beam. The other end of the HeNe tube is terminated in a high-reflecting (HR) mirror with a radius of 60 cm. The spherical output coupler mirror, on the right beyond the Brewster window in Fig. 2.4, has a radius of curvature of 45 cm, but the manufacturer of the 32 Chapter 2. Experimental Design mirror is unknown. The output coupler is mounted on a micrometer stage for fine adjustment of the cavity length. According to Section 1.1 of the Theory chapter, this pair of mirrors can theoretically support Gaussian beams at cavity lengths from 0 to 45 cm, and from 60 to 105 cm. Maximizing the cavity length is desirable, because this decreases the axial mode spacing, which increases the number of modes that can lase simultaneously and thus minimize the duration of our pulses. At 105 cm, the round-trip frequency is νRT = 157.5 MHz, and the round-trip time is approximately 7 ns. The longest cavity achieved during experimentation was approximately 94 cm long (νRT ≈ 141 MHz; t ≈ 6.3 ns), at which point the intensity began to fluctuate and further alignment became difficult. Light emitted from the output coupler travels a short distance before encountering a NEOS 46110-1-LTD acoustooptic beam deflector, driven in the range of 80 MHz to 130 MHz by a NEOS 21110-1ASVCO AOM driver. The drive frequency is controlled by an external voltage, supplied by a Tektronix PS280 power supply. The traveling acoustic wave in the AOM causes the incident light to diffract into a number of beams; only the 0th and 1st order beams are of interest to us. The 0th order beam is not frequency-shifted, so it can be measured for an accurate description of the field inside the primary cavity. The 0th order beam is deflected by an edge mirror towards either a photodiode or Fabry-Perot interferometer. A Thorlabs DET10A fast photodiode is used for measurement of the intensity timeseries and the radio-frequency beats between laser modes. This detector was chosen for its risetime of 1ns and reasonable responsivity. The detector is connected to either a Tektronix TDS 620B oscilloscope or an Agilent E4411B electronic spectrum analyzer. Alternatively, the 0th order beam can be deflected into a Burleigh SAPlus Laser Spectrum Analyzer1 . This is a Fabry-Perot interferometer (FPI), which consists of an optical cavity of two partially-transmitting confocal mirrors in front of a photodiode. Like the laser cavity, this Fabry-Perot cavity only supports distinct, evenly-spaced frequencies. Supported frequencies are transmitted through the cavity to the photodiode2 , allowing the measurement of the intensity of a very narrow portion of the optical spectrum. The distance between mirrors is varied by a piezoelectric transducer over time, allowing the measurement of a range of optical frequencies. By connecting the photodiode in the interferometer to our scope, we can measure the relative intensities of different optical frequencies in our laser signal. The mirror set used in the cavity reflects light in the range 450-700 nm. An important property of the FPI cavity is the axial mode spacing3 , also called the ‘free spectral range’ in the case of interferometers. This is important because it limits the frequency range over which the interferometer can measure. Suppose light at a frequency of ν, and thus wavelength λ = c/ν, enters the FPI as the cavity 1 The interferometer is equipped with a SA-98-B1 mirror set and driven by a Burleigh RG-91 ramp generator. 2 The photodiode signal runs through a Burleigh DA-100 amplifier. 3 Because of the confocal geometry of the Fabry-Perot interferometer used, it is possible for light to be injected into the cavity off of the optical axis. In the paraxial approximation, light then travels f our cavity lengths per round-trip, so the formula for the axial mode spacing is νRT = c/4L, except in the case that the input beam is aligned exactly on the optical axis, in which case νRT = c/2L. 33 length sweeps from L1 = nλ/2 to L2 = (n + 1)λ/2. Then the light will resonate in the cavity at both L1 and L2 , so light will be transmitted to the photodiode at both lengths, and the monochromatic light will cause the FPI to measure a signal at what appear to be two different frequencies, when in actuality the spectrum is just repeated. The apparent frequency difference between these transmission peaks is the free spectral range. The Fabry-Perot cavity we use in this experiment is very short, with a free spectral range of 8 GHz4 , so the spectrum from a narrow-bandwidth laser like the HeNe is sure to lie within this range, preventing any part of the spectrum from overlapping with other parts. While the 0th order beam emanating from the AOM is used for measurement, the 1st order beam is used to modelock the laser. This is done by retroreflecting the 1st order beam back into the AOM with a third spherical mirror, a Newport 10DC1000ER enhanced aluminum mirror with a radius of 1 m. As in the split-cavity configuration, the 1,1 beam travels along the same path as the light incident on the AOM, but in the opposite direction. This 1,1 beam is coupled into the primary cavity. As in the split cavity case, the AOM drive frequency is half of the roundtrip frequency of the primary cavity, so that the twice-shifted beam that is coupled back into the primary cavity has the same frequency as one of the primary cavity’s axial modes. The shifted beam then injection-locks the modes inside of the primary cavity, modelocking the laser. As labeled in Fig. 2.4, the distance between the output coupler and the third mirror should be as close as possible to the length of the primary cavity. This is because a pulse circulating in the laser is split at the output coupler; one part traveling out to the secondary cavity, and the other returning to the primary cavity. After one round-trip time, these pulses will once again meet at the output coupler, and if there is any delay between them, the two split pulses will not maximally constructively interfere. This is similar to the requirement that each of the legs of the split cavity be the same length, but in this case the splitting at the output coupler is important, while the splitting at the AOM is not (the 0th order beam is measured, not reflected back into the cavity). Thus there are three variables that must be in relatively good agreement for modelocking to occur: the round-trip frequency of the primary cavity, the round-trip frequency of the secondary cavity, and the doubled modulation frequency. Thankfully, due to the miracle of injection locking, these three frequencies do not have to be precisely the same for modelocking to occur, as we will see in the Results chapter. Note that, although the AOM operates closer to Raman-Nath regime than the Bragg regime, generating multiple diffracted beams, the angle of the AOM still affects the relative intensities of the beams. Specifically, the intensity of a given diffraction order is maximized when the angle between the acoustic waves and the incident light is at one of the Bragg angles. For this experiment, the AOM was rotated so that the intensity of the first diffracted order was maximized, to ensure that sufficient power was fed back into the laser for modelocking to occur (recall that increasing the intensity of the injected light can increase the range of modulation frequency 4 The range depends on the cavity length, which changes over time, but we assume that these changes are very small compared to the total length. 34 Chapter 2. Experimental Design detunings over which injection locking can occur). While this necessarily decreased the intensity of the zeroth order beam, and thus decreased the percentage of optical power redirected to the photodiode, a signal was still resolvable above noise when analyzed by our measuring equipment. Chapter 3 Results and Discussion Having provided a description of the experimental setup in the previous chapter, we can turn to the results of the experiment and attempt to analyze them. First we will describe the modelocking procedure and examine the behavior of the laser during this process. The first, and most time-consuming, step in the modelocking procedure was to align the primary cavity for lasing. This was increasingly difficult at longer cavity lengths. The laser was aligned at a cavity length of about 91 cm, yielding a theoretical axial mode spacing of about ∆ν ≈ 165 MHz. (A small axial mode spacing was necessary to maximize the number of modes within the modelocked frequency spectrum.) Once the primary cavity was lasing, the AOM was aligned such that the 1st order mode was most intense. The third mirror, reflecting the 1st order mode back into the AOM, was then set up at a distance L − 3.5 cm away from the output coupler, where the correction is necessary because of the high index of refraction of the AOM crystal. 3.1 Unmodulated Operation With the components laid out, data collection began. The unmodulated output of the laser, with the AOM turned off, was measured first. The laser was found to operate in only one axial mode for most of the time, whereas two-mode operation was observed only once in a while. We will call these modes ‘primary modes,’ to distinguish them from the ‘shifted modes’ that will arise later due to modulation. Fig. 3.1 shows timeseries data of the intensity waveform for one-mode operation, taken with the fast photodiode. This is essentially continuous-wave output. The mean voltage was approximately 0.425 mV, while the RMS voltage was 0.450 mV. An optical power meter, set up just outside of the output coupler mirror, measured the average power to be about 110 nW, far weaker than the average laser pointer (1-5 mW). We believe that the noise in the output was a combination of electronic noise (measured to have an RMS voltage of 0.220 mV) as well as fluctuations in laser power due to changing cavity losses. Evidence that this output was due to single-mode operation comes from Figs. 3.2 36 Chapter 3. Results and Discussion 0.8 Intensity HmVL 0.6 0.4 0.2 0.0 -20 -10 0 Time HnsL 10 20 Figure 3.1: Timeseries waveform: AOM off, one mode. Single-mode output at 91 cm. Mean voltage is 0.425 mV, RMS 0.450 mV. -1 0 1 Relative frequency HMHzL 2 3 4 5 6 7 8 9 Optical spectrum HmVL 20 15 10 5 0 0 5 10 Scan time HmsL 15 Figure 3.2: Free spectral range: AOM off, one mode. Power spectral density data of single-mode laser operation from the FPI, covering the free spectral range (8 GHz) of the device. The upper horizontal axis gives relative frequency spacing, while the lower axis gives sweep time. The sweep rate is calculated to be 523.56 MHz per ms. 3.1. Unmodulated Operation -1000 37 Relative frequency HMHzL -500 0 500 1000 Optical spectrum HmVL 40 30 20 10 0 -2 -1 0 Scan time HmsL 1 2 Figure 3.3: FPI spectrum: AOM off, one mode. Detail of single-mode laser operation. The sweep rate is the same as for Fig. 3.2. The peak appears to have a FWHM of about 100 MHz. 1.2 Intensity HmVL 1.0 0.8 0.6 0.4 0.2 0.0 -20 -10 0 Time HnsL 10 20 Figure 3.4: Timeseries waveform: AOM off, two modes. The period of the beats is about 6 ns, close to the expected period. 38 Chapter 3. Results and Discussion and 3.3, which show a single mode in the optical spectrum taken from the FabryPerot Interferometer (FPI). Figure 3.2 shows the output of the FPI when the range of frequencies swept out by the interferometer’s cavity is greater than the free spectral range of 8 GHz. As a result, the spectrum repeats after 8 GHz. The mirrors in the FPI oscillate to sweep across a frequency range, so the output of the FPI is actually a timeseries. We used the free spectral range sweep presented in Fig. 3.2 to find the ratio between sweep time and frequency spacing. All of the FPI data presented in this thesis have the same sweep rate as in Fig. 3.2, which is ∼523.56 MHz per ms. Knowing the sweep rate allows us then to zoom in on small frequency ranges for a better spectral resolution. Looking in detail at the single mode output by the FPI, as in Fig. 3.3, we see that the optical mode had a FWHM of about 100 MHz, although this may be artificially broadened by the combination of the limited risetime of the FPI’s photodiode and the FPI’s fast sweep rate. The FPI cannot directly measure absolute frequency, but we can assume that the single lasing mode is close to the standard HeNe 632.8 nm transition (at a frequency of 474.1 THz). It is also possible for the laser to switch briefly to two-mode operation. This was generally an unstable state, and would survive for only a few seconds at most. A sample timeseries waveform can be seen in Fig. 3.4. The corresponding radiofrequency beat signal on the electronic spectrum analyzer (ESA) is shown in Fig. 3.5, and a typical FPI spectrum is shown in Fig. 3.6. In Fig. 3.4 we can see the rather clear beating of the modes, with a periodicity of about 6 ns, which is close to the round-trip frequency associated with a cavity of 91 cm, T = 6.25 ns. The beat frequency is also visible on the ESA, as in Fig. 3.5, and gives a far more accurate measure of cavity length than can be achieved with a ruler. In this case, the cavity appears to have a length of 91.64 cm. The broadened nature of this peak is due to noise. The apparent width of the peak is deceptive because Fig. 3.5 is a semi-logarithmic plot; plotting the same data linearly reveals that the FWHM of the peak is only about 2.42 kHz. The clearest evidence that two modes can coexist in the cavity is seen in the FPI optical spectrum, as in Fig. 3.6. The frequency labeled “∆ν” in the upper horizontal axis marks the axial mode spacing corresponding to the beat frequency that we see on the ESA in Fig. 3.5, at about 163.684 MHz. The distribution of power between the modes was seen to fluctuate over time, but the mode spacing remained constant, until the spectrum collapsed into a single mode. Note that the vertical scale is this case is the same as in the plot of one-mode operation (Fig. 3.3), so the total power in the spectrum is roughly the same in both cases. When we discussed laser spectra in the Theory chapter, we predicted that, if the axial mode spacing was small enough, multiple axial modes would fall within the laser’s gain bandwidth, and these modes could lase simultaneously, even without the modulation necessary for modelocking. Why, then, is the presence of multiple modes so rare? We expect that there is simply too much loss in the cavity for more than one mode to stably lase at once. That is, only the highest portion of the gain curve is greater than the cavity losses, and only the modes that lie within this region have a chance to lase. See Fig. 3.7. A possible explanation for the relatively instability of multi-mode operation is a 3.1. Unmodulated Operation 39 -6 RF spectrum HdBL -7 -8 -9 -10 -11 -12 163.675 163.680 163.685 Frequency HMHzL 163.690 Figure 3.5: ESA spectrum: AOM off, two modes. Radio-frequency beat signal visible on ESA, indicating the presence of two modes in the cavity. -1000 Relative frequency HMHzL -500 0 DΝ 500 1000 Optical spectrum HmVL 40 30 20 10 0 -2 -1 0 Scan time HmsL 1 2 Figure 3.6: PSI spectrum: AOM off, two modes. Detail of two-mode laser operation. The sweep rate is the same as for Fig. 3.2. The mode spacing ∆ν was found from the ESA plot in Fig. 3.5. Chapter 3. Results and Discussion Power 40 Cavity Loss Only 1-2 modes lase Laser Gain Frequency Figure 3.7: Possible relative position of gain curve to axial modes, with large cavity losses preventing lasing in more than two modes. Relative frequency HMHzL -1000 -500 0 500 1000 1500 Optical spectrum HmVL 40 30 20 10 0 -2 -1 0 1 Scan time HmsL 2 3 Figure 3.8: FPI spectrum: AOM on, significant detuning. Typical spectrum arising when AOM is turned on and aligned correctly. Each shifted mode is shifted by the same frequency. The shifted modes are weak because they are not supported by the primary cavity. 3.2. Approaching Modelocking 41 nonlinear phenomenon called ‘cross-saturation.’ In the Theory chapter we discussed the general phenomenon of gain saturation, where the gain experienced by any mode decreases to the level of cavity losses as the mode intensity increases to its steadystate value. This prevents mode intensities from increasing indefinitely. In some gain media, stimulated emission by one mode not only saturates the gain of that mode, but also contributes to the saturation of other modes nearby. This effect, called crosssaturation, can cause one particularly intense mode to reduce the gain of nearby modes below the cavity loss line, so that the other modes die out. Cross-saturation is usually not stated to be a significant cause for concern in gas lasers like the HeNe [Siegman, 1986, Section 12.2, pp. 462-465]. However, because our laser has such a small axial mode spacing of 160 MHz, and because such a small portion of the gain curve seems to be available above the cavity loss line, we suspect that cross-saturation could be responsible for the domination of one mode in the spectrum, as well as the instability observed during two-mode operation. 3.2 Approaching Modelocking With the laser operational and the third mirror correctly aligned, the final step in the modelocking procedure was the tuning of the AOM modulation frequency Ω to coincide with half of the axial mode spacing ∆ν. In this section, we will describe the effect of injecting frequency-shifted light into the primary cavity for a range of detunings. When the AOM was turned on, part of the light emitted by the primary cavity as primary modes was frequency-shifted by the modulation, becoming shifted modes. These shifted modes were visible not only in the optical spectrum as seen by the FPI, but also created beat frequencies by interfering with the primary modes. These beat frequencies were visible on the ESA. When the modulation frequency was significantly detuned, shifted modes became visible on the FPI spectrum (Fig. 3.8). The intense mode at 0 MHz is the primary mode, and the weaker modes to the right are shifted modes. The shifted modes were weak because they did not line up with axial mode frequencies supported by the primary cavity. The beating of these shifted modes and the single cavity mode, shown in Fig. 3.9, was visible on the ESA. The central beat frequency was at twice the modulation frequency, 2Ω. The sidebands at approximately ±60 kHz are of unknown origin; the spacing and magnitude of these sidebands appeared to be independent of Ω and cavity length, so it is conceivable that the AOM itself has an unintended resonance at this frequency. Fortunately, the sidebands had little effect on laser operation, since they were about 15 decibels less intense than the central beat frequency. The entirety of the beat structure could also be seen at higher multiples of 2Ω, corresponding to the beating of the higher-order shifted modes with the primary mode and with one another. These higher-frequency beat signals we much weaker than the structure at 2Ω, because the higher-order shifted modes themselves were much weaker than the primary mode. Important to modulation was the structure of the central peak of the beat signal. Unfortunately, the magnitude of the peak was quite noisy. We expect that this noise 42 Chapter 3. Results and Discussion RF spectrum HdBL -90 -100 -110 -120 -0.05 0.00 0.05 Relative frequency HMHzL Figure 3.9: ESA spectrum: AOM-shifted signal. Beat frequencies arising when AOM is turned on and aligned correctly. The central peak lies near twice the modulation frequency (2Ω ≈ 180 MHz in this case). Relative frequencies are given since the structure is virtually independent of Ω. RF spectrum HmVL 12 10 8 6 4 2 0 175.175 175.180 175.185 175.190 Frequency HMHzL 175.195 175.200 Figure 3.10: ESA spectrum: Detail of AOM-shifted signal. Plot averaged over eight acquisitions. The linear scale helps emphasize the importance of the central peak compared to the rest of the beat spectrum generated by the AOM. 3.2. Approaching Modelocking 43 RF spectrum !dB" !60 a) Primary !70 !80 Second Primary & Shifted !90 !100 !110 163.0 163.5 164.0 164.5 165.0 Frequency !MHz" Primary !70 !80 !90 !100 !110 165.5 RF spectrum !dB" RF spectrum !dB" !60 Primary & Shifted (2Ω) b) 166.0 163.4 163.5 163.6 163.7 163.8 163.9 164.0 !60 !70 !80 c) Primary & Shifted (2Ω) !90 !100 !110 !120 RF spectrum !dB" !60 !70 163.4 163.5 163.6 163.7 163.8 163.9 164.0 d) !80 !90 !100 !110 163.4 163.5 163.6 163.7 163.8 163.9 164.0 RF spectrum !dB" !60 e) !70 !80 !90 !100 !110 163.4 163.5 163.6 163.7 163.8 163.9 164.0 Frequency !MHz" Figure 3.11: Beat frequencies visible on the ESA as the detuning of the modulation frequency from the axial mode spacing is reduced. (a) Beat signals when two primary modes lase simultaneously. Each primary mode beats with the first shifted mode. (b) Detail of beat between primary modes alone. Dashed line used to indicate ∆ν in following figures. (c) Beat signal as shifted modes approach axial mode spacings. Dotted line indicates 2Ω. (d) Noise due to rapid switching in and out of modelocking. (e) Shifted modes within modelocking region (2Ω ≈ ∆ν). 44 Chapter 3. Results and Discussion was from the vibrations of the AOM itself. Fig. 3.10 shows the average of eight plots of the structure of the central peak, taken over about 30 seconds. Note that this figure is vertically linear, whereas the other ESA plots have been logarithmic, so we are focusing on only the most intense beat frequencies. The upshot of averaging the noisy signal over time is that it allows us to estimate an upper bound on the FHWM of the central AOM beat frequency. In this case, the FWHM is about 3.2 kHz, which was on the same order of magnitude as the FWHM of the primary-primary beat. This suggests that the noise in the modulation would not a have a huge impact on the modelocking of the laser. With the AOM turned on and detuned, it was still possible (although uncommon) for two primary modes to lase simultaneously, just as we saw in the previous section on unmodulated operation. In this case, there were three significant beat frequencies: one between the two primary modes, one between the first shifted mode and the first primary mode, and one between the second primary mode and first shifted mode. The occurrence of three beat signals can be seen in Fig. 3.11 (a). This gave us an indication about what the axial mode spacing (the primary-primary beat frequency, Fig. 3.11 (b)) was at that point, and how far we would have to tune the modulation frequency so that the shifted modes lined up with axial modes. Fig. 3.11 (c) through (e) show the structure of the beat signals as the modulation frequency Ω was decreased. The dashed line represents the axial mode spacing frequency ∆ν and the dotted line represents 2Ω. Part (c) shows the beat structure that arose at small detuning. The entirety of the shifted beat signal was amplified, including the ±60 kHz sidebands flanking the peak. Part (d) shows the behavior on the edge of the modelocking region, with detuning around 40 kHz: significant noise appears near 2Ω as the laser rapidly switches in and out of modelocking. In part (e), the detuning is small enough (∼20 kHz) that the laser is modelocked. The noise is reduced and the peak of the beat signal is amplified significantly. 3.3 Modelocking Behavior When the beat signal was tuned to within about 20 kHz of the axial mode spacing, modelocking occurred. Fig. 3.12 shows the modelocked intensity waveform, with pulses separated by about 6 ns, with a FWHM of about 2 ns. The laser maintained this output as long as the modulation frequency did not drift outside of the locking range. The average output signal measured by the photodiode during this acquisition was 0.689 mV, which was higher than the unmodulated case of 0.425 mV, probably because modelocking fills more of the gain medium’s bandwidth. For reference, Fig. 3.13 shows the beat signal for this modelocked data, showing a structure very similar to Fig. 3.9. Figure 3.14 shows the optical spectrum corresponding to Fig. 3.12. We can see that at least four modes were populated. This figure has the same vertical scale as in Figs. 3.3 and 3.6, so we can see that the most intense mode has about half the intensity of the single mode that arises in the unmodulated case, but considering all four modes, there is a bit more total power in the spectrum now than there was in 3.3. Modelocking Behavior 45 Intensity HmVL 1.5 1.0 0.5 0.0 -20 -10 0 Time HnsL 10 20 Figure 3.12: Timeseries waveform: Modelocked. Modelocked output at 91 cm. FWHM of pulses is about 2 ns. -60 RF spectrum HdBL -70 -80 -90 -100 -110 -120 163.5 163.6 163.7 Frequency HMHzL 163.8 Figure 3.13: ESA spectrum: Modelocked. Beat signal corresponding to Fig. 3.12. This data was used to find 2Ω ≈ 163.698 MHz. Dashed line shows ∆ν ≈ 163.684 MHz. 46 Chapter 3. Results and Discussion -500 Relative frequency HHzL 0 2W 4W 500 1000 -1 0 1500 Optical spectrum HmVL 40 30 20 10 0 1 2 Scan time HmsL 3 Figure 3.14: FPI spectrum: Modelocked. Here 2Ω ≈ 163.698 MHz. Note that vertical scale is same as in Figs. 3.3 and 3.6. the unmodulated case. The selection of the zero-point on the horizontal axis of Fig. 3.14 is not arbitrary. In fact, there is a very weak mode centered at approximately this point, and this mode lines up with the single primary mode that is visible when the laser is unmodulated. This is shown in Fig. 3.15, where a modelocked spectrum is shown as a solid line, and the corresponding unmodulated spectrum is shown as a dashed line, which was acquired by turning off the AOM. In order to show the detailed structure of the modelocked spectrum, the FPI’s amplification of the spectrum was increased tenfold above the other FPI plots in this thesis. In order to show the (far more intense) unmodulated signal on the same plot, both sets of data were individually rescaled, so the scales of the two lines should not be considered accurate. What is important, though, is that the unmodulated mode overlaps with the the first, very weak mode in the modelocked spectrum. This may seem counterintuitive: modelocking has caused higher-frequency modes to become more intense than the central primary mode. We expect that this behavior can be attributed to two facts. The first is the fact that, because our experimental set up only utilized light that was up-shifted in frequency, the flow of optical power due to modulation is unidirectional. This creates a Gaussianshaped envelope of mode intensities which becomes wider and is pushed rightwards in frequency-space as time progresses. The second fact we must consider is the effect of gain, gain saturation, and cavity losses. While power is being transferred to higher frequencies, lower frequencies (closer to the gain peak) experiences more gain than higher frequencies, and this increases the intensity of lower-frequency modes over time. Uniform cavity losses prevent modes far from the gain peak from becoming very intense. By considering all of these effects together, it is plausible that the modelocked laser can achieve the steady states indicated in Figs. 3.14 and 3.15. 3.3. Modelocking Behavior Optical spectrum Ha.u.L -250 1.0 0 47 Relative frequency HMHzL 2W 4W 500 750 1000 0.8 0.6 0.4 0.2 0.0 -0.5 0.0 0.5 1.0 Scan time HmsL 1.5 2.0 Figure 3.15: FPI spectrum: Modelocked vs. unmodulated. Amplified modelocked spectrum (solid line) and unmodulated spectrum (dashed line), showing relative frequencies of modulated and unmodulated modes. Shows distribution of modelocked mode power relative to primary mode. Note that the two plots are scaled for simpler comparison. It is important to reiterate that the existence of a ‘modelocking range,’ a range of detunings over which the laser stays modelocked, is a direct result of injection locking. If the laser system were completely linear, no injection locking would occur, and modelocking would require a perfectly tuned modulation frequency as well as a stabilization equipment to reduce phase noise. It is difficult to show the frequency pulling associated with injection locking directly, because the FPI signal is noisy and the horizontal scale is too coarse to notice small frequency shifts. Frequency pulling can, however, be seen as a shift of the beat frequency between primary modes on the ESA, as in Fig. 3.16. The thin line in this figure shows the primary-primary beat frequency when the AOM is turned off, while the thicker line shows the modelocked beat signal when the AOM is turned on and correctly tuned. Readers will have to trust the author that the unmodulated primary-primary beat frequency was never seen as long as the laser was modelocked, because the injection of frequency-shifted light pulled the frequencies of the primary cavity modes to coincide with the injected modes. The best evidence that the thicker line represents modelocking is the fact that the modulated beat frequency is about 9 decibels greater in magnitude than the unmodulated beat, indicating that there are more modes, and that the phase differences between mode pairs is more uniform. This coupling of mode phases was another result of injection locking. Assuming that the modelocking range for our laser was about 20 kHz above and below ∆ν, we can use the injection locking range equation 1.64, r Iinj ≈ 20 kHz, (3.1) =γ I0 48 Chapter 3. Results and Discussion RF spectrum HdBL -60 -70 -80 -90 -100 -110 -120 163.60 163.65 163.70 Frequency HMHzL 163.75 163.80 Figure 3.16: ESA spectrum: Modelocked vs. unmodulated. Unmodulated beat signal (thin line) compared to modelocked beat signal (thick line). The displacement of the peaks indicates that the primary modes were frequency-shifted by about 17.4 kHz in the presence of the injected modes. This is evidence of the frequency-pulling associated with injection locking. to learn more about how our laser operated. Here where Iinj is the intensity of the injected signal, I0 is the intensity of the undisturbed mode, both measured inside δ the cavity. The cavity decay rate γ ≈ TRT , where δ is the percentage of light that is emitted from the output coupler mirror, and TRT ≈ 6.25 ns is the round-trip time through the cavity. Estimating δ ≈ 1%, the ratio of injected mode intensity to undisturbed mode intensity was about 0.02%. This means that very little light, after being split by the AOM, reflecting off the third mirror, being split by the AOM a second time, and transmitting through the ouput coupler, made it back into the primary cavity. However, this was enough light to broaden the injection locking range to a point where modelocking could be achieved without too much difficulty. Conclusion In this thesis, we have introduced the theoretical components necessary for discussing modelocking, utilized the theory to motivate the design of a particular modelocked laser, and presented experimental evidence that such a laser operates as intended. Although the depth of theoretical material presented may be exhausting, it is hardly exhaustive. A more general and rigorous form of pulse-propagation modeling can be accomplished by using Haus’ ‘Master equation of modelocking’ [Haus et al., 1991], which is a partial differential equation that can accommodate active and passive modelocking, and take into account complicated effects like intracavity dispersion and nonlinearities of the gain medium that arise for high peak intensities. This equation also allows for modeling of pulse shapes other than Gaussian pulses. Another type of modeling in the frequency domain, using ‘coupled-mode’ equations, presents analytical solutions that track how power is transferred between modes over time in an actively-modelocked laser. This frequency-domain modeling can easily account for detuning of the modulation frequency. If we wanted a more accurate model of how we should expect our laser to operate, we could have applied both the Master equation and the coupled-mode equations to our particular setup. Our laser succeeded in producing pulses that were about 2 nanoseconds long, although the low intensity of the laser suggests that we could not expect to use its output for industrial or research applications. There are many other designs of modelocked lasers that could have been studied and assembled, the limiting factors being time available to the student and funds available from the institution. 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