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Al Kaye & Emily Mankin Phys 273; Spring 2009 BRIEF ARTICLE NONLINEAR OPTICS THE AUTHOR BRIEF ARTICLE BRIEF ARTICLE BRIEF ARTICLE THE AUTHOR Nonlinear optics refers to the alteration of THE theAUTHOR optical properties of a BRIEF material by light (Boyd THE AUTHOR ARTICLE 2003). As light propagates through a material, it generates oscillating dipoles, which in turn  give rise BRIEF AUTHOR described by the BRIEF ARTICLE to electromagnetic radiation, allowing the light to ARTICLE continue. This process isTHEtypically BRIEF ARTICLE P = χE following simple relation: THE AUTHOR THE AUTHOR , P χE = χE P = P = χE THE AUTHOR P = χE (2)polarization 2+ where P is the of the material (dipole moment per unit volume),   is a constant P = χ(1) ⊗ E + χ ⊗ E · · · called the susceptibility, and E is the(1) electrical field generated light. However, this (2) ⊗ 2E 2by+the· ·incident P+ =(2) (1)= PχP = χ⊗χE ⊗and E · approximation Pχ =χχE⊗ χE linear relationship between polarization electric field is a+ first· ·order to a more P = E + E · (1) (2) 2 P = χ complicated ⊗ E + functional χ ⊗relationship, E + · ·which · is better approximated (1) (2) in the P = χ by⊗theEhigher + χorder⊗terms E2 + ··· (2) (1) (2) 2 series χTaylor = Nexpansion: "β# s P = χ ⊗(2) E + χ ⊗ E + ··· χ =N s"β# (2) (1) (2) 2"β# (1) ⊗ (2) χ = N (2) P = χ ⊗ E + χ ⊗ E + · ·⊗ · E 2 +(2) P = χ E + χ ··· s χ = Ns"β# χ = Ns"β# (2)!= N "β#" (n) χ n−1 (n) where χ refers the n-th order susceptibility. Assuming that the magnitude of this χnonlinear Es BRIEF ARTICLE ∝ ! "n−1 (n) (2) = N (2) (n) (n) χ (1)s"β# χ E χ E χ = N "β# χ atom s we see second constant,  harmonic generation only (n) why ! χ , decreases "n−1 with n, it becomesχclear THE∝ AUTHOR (n) (1) to theEfollowing χ in very strong E a perturbation approach leads atom −1χW ∝ ! electric"fields. Applying 400mW 4 × 10 ! " 9 W/m2 (n)(n) χ(1)≈(1) n−1 (1)(n) n−1 I = = 4×10 Eatom χχ laser χ approximation for the ratio of   to    as a function of electric field strength: χ E −1W E 2 χ (1 × 10−5 10s of µm 400mW 4 × 10 m) ∝ ∝ Ilaser = ≈ = 4×109 W P = χE (1) (1) −5 2 −1 Eatom χ (n)! "! 9E 10s of µm (1 × 10 m) atom χ 400mW 4 ×(n) 10 W "" ! n−1 17 2 2 n−1 n−1 (n) χ E =E 10 W/m W/m ≈ χ −5χ 2Iatom = 4×10 E r= ∝ −1 10s400mW of µm (1 × −1W= 1017 W/m2 1010 (1) ∝ m) Eatom 4 (1) × W χ∝ 400mW 4 × 10 I (1) (1) 9 2 E atom P≈=W/m χ ⊗ atom E + χ(2) ⊗ E 2 9+W/m ··· 2 ≈ Iχlaser −5 = 4×10 =χ Eatom = 4×10 er = 10s of µm (1 × 10 m) 10s2 of µm (1 × 10−5m)2 −1W gives rise to the phenomenon of second As we will describe shortly, the second4 order 400mW × 10polarization 9 W/m2 (2) Iharmonic ≈ = 4×10 17 2 laser = χ = generation. Therefore we calculate this−5 fraction mode-locked Ti-sapphire 1+ cos(2ωt) for aN s"β# 2for  = 10 W/m 210 10s of µm Iatom (1 × m) cos (ωt) = 1 emitting + cos(2ωt) laser with an average power2of 400 mW 100 femtosecond pulses at a rate of 80 MHz. 2 cos (ωt) = The power in a single pulse is then equal to the2(2) average power over a second divided by the χ E 2(1 + cos(2ωt))(n) 1 1 1 1 1 1 BRIEF ARTICLE 3 BRIEF ARTICLE 1 1 fraction of time spent in pulses: 3 (2) χ 0 P = χ(2)E02(1 + cos(2ωt))2 (2) P = 2 1 × 10−13 seconds (1) 8 × 107 pulses = 8×10−6 Fraction of time pulsing = χ pulse 8 × 107 pulses 1 second × 10−13 seconds = second Power during pulse  = pulse = 8×10−6 χ(n) 0.4W 4W (1) ∝ = 5 × 10 χ 8 × 106 ! E Eatom 1 "n−1 E = E0 cos(ωt) BRIEF ARTICLE THE AUTHOR P = ser = cos2 (ωt) = 1 + cos(2ωt) 2 Al Kaye & Emily Mankin Phys 273; Spring 2009 (2) from In order to determine the intensity ofχlight incoming laser, we must also know the size of E02 (1 +the cos(2ωt)) (2) P = the beam at the focus. We will ignore the issues involved in the laser having a Gaussian intensity 2 profile and approximate it as a collimated beam with a diameter of 4 mm. Following Mansuripur 7 8 ×of10 pulses 1 ×focus 10−13 1998, we take the size the spot at the as seconds the wavelength of light −6(800nm) divided by the P = χE = The numerical aperture of an ordinary glass = 8 lens × 10 numerical aperture. is simply the index of second pulse 2 THE AUTHOR refraction  (which we take to be 1.6) multiplied by the sine of the angle between the edge of the 0.4W beam and the focal point. Using=a lens withr 2a=focal length 38 mm, we calculate that the size of =5 0.9871 × 104 WofARTICLE BRIEF 6 (1) (2) 2 χ the⊗spotEwill+beχ13 micrometers ⊗ E +in8·diameter. ·×·10 The intensity at the focal point is then (2) 4 (2)THE2 AUTHOR =W χ E power 5P× 10 Intensity = = = 9.4 × 1013 W/m2 −6 2 area π × (13 × 10 , ) χ(2) = Ns"β# E = E0 sin(ωt) P = χE which is approximately 1000 times smaller than the atomic electric field, so that each 1 + cos(2ωt) 2 cos (ωt) = (n) successive χ  will be 1000 times smaller before it. From P =than χ(1)the⊗one E2 + χ(2) ⊗ E 2 +this · · ·calculation, we can χ 2 χ(2) E0of (1incident + cos(2ωt)) see the importance of maximizing (2) the intensity light during the pulses if we are to P = (2) ! " generate an appreciable second n−1 order polarization, P2 = χ(2) E 2 (n) 2 ∝2 E THE AUTHOR −13 AUTHOR 8 × 107 pulses2 1THE × 10 seconds (2) (1) = = 8 × 10−6 r = 0.9871 Eatom χ = Ns "β# χ 2 THE AUTHOR 2 second pulse r = 0.9871 0.4W (2) 2 r2 = 0.9871 4 (n) = 6χ = E 5 × 10 W 400mW 4 × 10−1W= P8(2)× (2) χ 2 9E 2 10 = χ(2) P 4×10 ≈ = W/m 10s Sof µm (1 × 10−5m)2E = E sin(ωt)4 (1) P (2) = χ(2)E 2 ECOND HARMONIC GENERATION 05 × 10 W χ power 1 Intensity = = E = E0 sin(ωt) = 9.4 × 1013 W/m2 −6 , )2 area π × (13 × 10 ! E= "n−1 As a wave propagating in space, E has the form (Hecht 2001) E0 sin(ωt) , the polarization 1+ cos(2ωt) 2 χ1(n) E cos (ωt) = + cos(2ωt) 2 2 therefore has the form P = χ(1) E 2∝ χ(2) E02 sin (ωt) + · · · . However, since we cos (ωt) =+χ(1) 0 sin(ωt) E atom 1 + cos(2ωt) 2 2 cos (ωt) 1 − cos(2ωt) −1 = 2 400mW 4 × 10 W 2 polarization = 1 − cos(2ωt) know from trigonometry that sin (ωt) , the second order 2 Ilaser = ≈2 = 4 × 109 W/m2 sin10s (ωt) = −5 2 of µm (1 × 10 χ(2) m)E 2 (1 + cos(2ωt)) 2 (2) bcomes 0 (2) 2 P = χ E (1 + cos(2ωt)) 0 2 (2) 17 (2) P (2) = χ E0Iatom = 10 W/m2 2 2 −7cos(2ωt)) P = (1 −13 8 × 10c =pulses 1 × 10 seconds 2 a = .000282; b = -.005367; .007951; rsquare: 0.9932 = −13 seconds = 8 × 10−6 8 × 107 pulses 1 × 10 −6 7 −13 second pulse = 8 × 10 pulses 1 × 10 = 8 × 10 −6 seconds 2 pulse dipole = in second =induced 8× To rephrase this result English, we have electric being in the medium at y = an .000282x − .005367x + 10 .007951 0.4W second pulse 2 double the frequency (half the wavelength) laser light. This = 5is×similar 104 Wto the 0.4W of the incoming 4== r 0.9932 6 × 10 with a fluorophore in a = = 5two × 10 W4 8collide 0.4W technique of two-photon fluorescence photons 6 8 ×in 10which = = 5 × 10 W narrow window of time, causing it to become and to emit a photon4 of light at slightly less 8 × 106excited power 5 × 10 Wand second y = 1.968x 4 than double the frequencypower of the incoming photons. Two-photon fluorescence Intensity =W = − 8.081 = 9.4 × 1013 W/m2 5 × 10 13 −6 ,2)2 1 4 area π × (13 × 10 Intensity = = = 9.4 × 10 W/m harmonic generation are depicted Figure powerschematically ×in10 10 W, )21 −6 π × (135 × Intensity =area = = 9.4 × 1013 W/m2 −6 2 area π × (13 × 10 , ) (1) (2) 2 2 P =χ E (ωt) + · · · 0 sin(ωt) + χ E 0 sin (1) (2) 2 P = χ E0 sin(ωt) + χ E0 sin2 (ωt) + · · · Al Kaye & Emily Mankin Phys 273; Spring 2009 . Figure 1 - Two-photon fluorescence versus second harmonic generation, Jablonski diagram BRIEF ARTICLE Although these two techniques are superficially similar, they arise from very different physical THE AUTHOR processes - two-photon fluorescence requires a fluorophore, which relaxes from the excited state through vibrational states before emitting a photon in a random direction. In contrast, second harmonic generation does not include relaxation through vibrational states, so no energy is lost to heat, a significant problem in temperature-sensitive biological tissues. Furthermore, secondharmonic generation produces a coherent P = χE beam of light in the forward direction only. Since it does not rely on excitation of a larger fluorophore, it is also much faster, and ought not to be as susceptible to photobleaching. P = χ(1) ⊗ E + χ(2) ⊗ E 2 + · · · There are, however, certain strict conditions on the generation of second harmonics. The first is that only noncentrosymmetric objects can generate second harmonics, because the second-order nonlinear susceptibility  χ (2) = N "β#  is a vector s average over the hyperpolarizability of the medium, which cancels out when the hyperpolarizability is the same in all directions.  χ(n) The second condition is that of phase matching - in order to avoid destructive interference between second harmonics generated at different positions in the medium, which depletes an already weak signal, the original wavelength and the halved second harmonic wavelength can be (1) made to propagate at the sameχspeed. This is achieved despite the normal variation of the index of refraction with wavelength (e.g. prisms) by means of a birefringent crystal, which is a crystal with different indices of refraction ! at different polarization "n−1 angles. For example, for a crystal which (n) χ E propagates light more slowly at lower wavelengths and more slowly on the ordinary axis of the ∝ it is possible to achieve this at a tunable angle of the crystal crystal (negative uniaxial crystal), (1) χ light of E atom relative to the polarized the laser, ensuring constructive interference in the forward 1 direction. Al Kaye & Emily Mankin Phys 273; Spring 2009 EXPERIMENTAL SETUP Second harmonic generation begins with a mode-locked Ti:Sapphire laser, the set-up of which is beyond the scope of this paper. Using a glass slide, we divert approximately 4% of the laser beam to a spectrometer that is used to measure the spectrum of the emitted light. When the laser is mode-locked, the spectrum is broad, usually spanning the wavelengths 770-790 nm. Passing through the glass slide, the rest of the beam is directed to the optical axis by a set of mirrors. Using two flipping-mirrors (c and d in Figure 2), we are able to create two paths to the optic axis. When mirror c is down and d is up, there is a static path that can be used for aligning elements of the optic axis and checking on the generation of the second harmonics. Switching the status of the mirrors directs the light path to a set of scanning mirrors (e), which are controlled via a laptop running Matlab and connected to a data acquisition card. The beam next passes through a beam expander. We used a telescope system with a -25.4 mm lens and a 75.6 mm lens, expanding the beam by approximately 3 times. The goal of the beam expander is to get the beam to fill the back of the objective. The 3x expansion made our beam approximately 6 mm in diameter. After the expander, the beam passes through a wave plate and polarizing beam splitter. The wave plate adjusts the polarization of the light, and the beam splitter diverts light of a particular polarization away while letting the remaining light through. The wave plate and polarizing beam splitter, then, work together to serve as an attenuator for the beam. Rotating the wave plate allows different amounts of light to pass through the beam splitter. This is important, as we found that at full power the laser could burn biological tissue if left in one place. p a q o f h { e d b c g i j k l m n Figure 2: Experimental Setup. The setup is described in the text. a. Pumping laser. b. Ti:Sapphire Laser. c,d. Flipping mirrors. e. Scanning mirrors. f. Beam expander. g. Rotating wave plate. h. Polarizing beam splitter. i. Beam block for beam deflected from h. j. Objective lens. k. Sample. l. Collecting lens. m. Blue filter. n. Photodiode. o. Oscilloscope. p. Laptop. q. Data acquisition card. Al Kaye & Emily Mankin Phys 273; Spring 2009 An objective or lens is used to collect the remaining light and focus it onto the sample. We used a 38.1 mm lens. For our sample, we began with BBO crystal. We also tried to generate harmonics from a piece of bone and from a section of tendon from a rat tail. Unfortunately, only the BBO crystal was successful in producing second harmonics. Successful second harmonic generation produces a violet light. Because most photons do not get converted by second harmonic generation, a strong red beam will also come through. A blue filter blocks this beam. We used a 25.4 mm collecting lens to focus the beam onto a photo diode. By attaching a narrow band-pass filter to the photo diode and adjusting the wavelength of the mode-locked laser, we were able to determine that the light being generated was second-harmonic light, not autofluorescence of the sample. The photo-diode was connected to an oscilloscope, and the magnitude of the signal was recorded using the laptop and data acquisition card. We were successful in generating a second harmonic signal from BBO crystal, however we were not able to record a signal from biological tissue as we had originally hoped. Here are some ideas for making this successful in the future. First, make sure the mirrors used will reflect almost all the light with wavelength 750-800 nm. This is especially important to keep in mind when selecting a set of scanning mirrors: the first set we tried transmitted the majority of the light. We also had our pumping laser at a 90° angle to the Ti:Sapphire laser. A representative of the Tsunami Laser Company told us that this was bad practice and beam from the pumping laser should go directly into the second laser. It is possible that this would make the Ti:Sapphire laser easier to get into mode lock. We chose to use a 38.1 mm lens in place of a traditional objective in our setup. This worked well for the BBO crystal, but may have lost too much power for other samples. The goal of the beam expander is to cover the back aperture of the objective. Because we used a lens, we may have needed a larger expansion. On the other hand, the back of an objective is generally smaller than 6 mm, so a smaller expansion should be used. We used only one photo diode to collect the second harmonic light, and we found that our eyes were more sensitive than the photo diode. That is, we could see a clear signal even when the oscilloscope reading was indistinguishable from its reading when the beam was blocked entirely. That we never a saw second harmonic signal from the tendon may not mean that no signal was generated. We had hoped to be able to use the photodiode and oscilloscope to detect signals that were not visible to our eyes. With one photodiode this is not possible. Two potential solutions come to mind. First, it would be helpful to collect the portion of the signal that is back propagated from the sample. This may be achieved by inserting a dichroic mirror between the beam splitter and the objective lens. The dichroic mirror should transmit light in the range of the red beam and reflect light in the range of the violet beam. The reflected light from this mirror could then be collected by a second photodiode, and the signal added to that of the first. Another potentially useful solution would be to try using photo multiplier tubes instead of photodiodes; these should be more sensitive than the human eye. Al Kaye & Emily Mankin Phys 273; Spring 2009 RESULTS Although we were unable to generate second harmonics in tendon, as desired, we were able to generate them from BBO crystal, and could explore some properties of this signal. First, we showed that the second harmonic signal in BBO crystal is strongly dependent on the rotation of the crystal. The amount of second harmonic signal depends on the degree of phasematching between the red and violet light. As the crystal is rotated, the angle, theta, between the ordinary axis of the crystal and polarization of the light varies, which varies the index of refraction for the 400 nm light. Thus, the degree of phase-matching between the two wavelengths varies with the angle of the crystal, so the strength of the signal should vary as well. μW Results are shown in Figure 3. The blue line shows our first set of measurements. We expect the results to be periodic, with period some factor of 360°. We began measuring at 160°, which is where we had found maximal signal strength. We were surprised that when we had rotated the crystal 360°, our signal had diminished from 32 μW to 6 μW. To check our measurements, we repeated the experiment, rotating 45° at a time. Results, plotted as green dots, show the same pattern. Looking at the pattern of results, it appears that there are three levels at which the signal may plateau: no signal (0-1.5 μW)1 , weak signal (5-8 μW), and strong signal (28-32 μW). Between these plateaus are fairly sharp transitions. Figure 3. Strength of second harmonic signal is dependent on the rotation of the crystal, but may be sensitive to yet undetermined properties of the laser. 1 We hypothesize that the plateaus are relatively stable, but the location of the transition could be sensitive to parameters of the laser. In particular, notice that the first sharp descent is “interrupted” briefly, but if the first five measurements were shifted by 20°, the curve would be smooth and monotonic. Similarly, the descent at the end would line up nicely Note that the signal never disappeared completely to our eyes. The signal on the oscilloscope, however, was indistinguishable from a blocked beam when it reached 1.3 mW. All such values are plotted as 0 on the graph. Al Kaye & Emily Mankin Phys 273; Spring 2009 with the descent at the beginning if we imagined this as having a period of 330°. It is not clear what is causing the instability in where the transitions occur. We monitored the input power carefully during this experiment; all measurements were made with the mode-locked laser at a power of 420-430 mW. We often noticed a seeming “bistability” in the oscilloscope signal that would persist for a moment when we turned the crystal. That is, for a moment the oscilloscope would vacillate between two different amplitudes. This usually resolved itself quickly and we took the data point once it settled. We found that, in general, the mode-locking of the laser was unstable, and we suspect this instability could have contributed to the bistable oscilloscope signal and to the instability in where the transitions between strong and very weak signals occurred. For our next experiment, we set the crystal at a rotation of 200° (at the time, this was the strongest signal point) and then measured the power of the input laser and the output second harmonic signal as we rotated the wave plate. This data (see figure 4) shows a robust periodic effect with period of 90°. This shows, first, that the wave plate, indeed, attenuates the power of the mode-locked laser, as predicted. The periodicity of the second harmonic response also demonstrates that the dependence on the input intensity of the second harmonic signal is robust. Finally, we can see that the input intensity BRIEF ARTICLE varies between 15 and 350 mW, while the second harmonic signal varies between 0 and 30 μW. THE AUTHOR μW Theoretical calculations, P = χ(1) ⊗ E + χ(2) ⊗ E 2 + · · · , suggest that the strength of the second harmonic χ(2) = Nssignal "β# will vary with the intensity of the input ! To " signal,χ(n) squared. test n−1this, we E ∝ input intensity vs. plotted(1)the Eatom χ intensity output and fitted the data 400mW with a quadratic function 4 × 10−1 W Ilaser(see = Figure 5).≈ Looking−5 = 4×109 W/m2 at a 2log10s of µm (1 × 10 m) log plot of the same data reveals 2 the Iquadr relationship 1017 W/m atom =atic intensity and a =between .000282; b input = -.005367; c = .007951; second harmonic generation.  rsquare: 0.9932 The least-squares fit for the logtransformed data is y = 1.968x 8.081, with an r-squared value of .987.  The 95% confidence interval for the slope is (1.877, Figure 4. Rotating the wave plate causes a robust, periodic effect 2.059), so 2 is included. We on the amplitude of each signal. Note that the input signal is conclude that there is good several orders of magnitude greater than the output. evidence that the output intensity does vary quadratically with input intensity. 1 μW Al Kaye & Emily Mankin Phys 273; Spring 2009 2 2 2 THE AUTHOR 2 THE AUTHOR THE AUTHOR THE AUTHOR 2 − 2.005367x = .000282x − .005367x + .007951 y =y .000282x + .007951 2 2 = 0.9932 r =r 0.9932 2 −+ y = .000282xy2=−.000282x .005367x .007951 .005367x + .007951 2 2 r = 0.9932r = 0.9932 = 1.968x − 8.081 y =y1.968x − 8.081 r2 =r20.9871 = 0.9871 (2) (2) 2 P P=(2)χ= E χ(2)E 2 y = 1.968x − 8.081 2 r = 0.9871 Figure 5. Input intensity vs. output intensity (regular and log-log plots). Blue dots were taken from measurements E =EE= 0 cos(ωt) E0 cos(ωt) made every 10 degrees of wave plate rotation and were used for curve-fitting. Red curves are best fits, P (2) = χ(2)E 2 1 + cos(2ωt) calculated with the Matlab curve fitting toolbox. Green dotscos were taken 2 degrees of rotation between 2(ωt) 1 + cos(2ωt) = =every 2(ωt) 2 to lie along the best fit curve. The 300 and 310 degrees. They were not used in determining best cos fit, but appear 2 E = E0 cos(ωt) 2(1 + cos(2ωt)) χ(2)quadratically E(2) figures show that the intensity of the second harmonic signal(2)varies with the intensity of the 2 0 P = (2) = χ E02(1 + cos(2ωt)) incoming beam. P 1 + cos(2ωt) 2 cos2(ωt) = 2 CONCLUSION P (2) = χ(2)E02(1 + cos(2ωt)) 2 We have generated second harmonics by collimating and then focusing a mode-locked Ti-sapphire laser onto a birefringent crystal. We obtained experimental verification of several important features of second harmonic generation: namely, that the strength of the light varies quadratically as a function of the intensity of the incident beam, and that the polarization of the incident light relative to the ordinary axis of the birefringent crystal determines the strength of the second harmonic, in keeping with theoretical predictions based on phase matching. We furthermore showed that the light generated was at precisely half the wavelength by means of a narrow band filter,  and we were able to generate second harmonics in a scanning configuration. It is our hope that future students will be able to build upon these results by imaging the collagen fibrils in a rat tail, replicating published results from Freund et al (1986). Although we were successful in exploring some of the fundamental physical aspects of nonlinear optics with this project, we were not successful in our ultimate goal of applying this tool to neurobiology. With the scanning configuration in place, it ought to be possible to study the polarity of filaments in cells, including the microtubules in neuronal processes (Kwan et al 2008). The use of styryl dyes to optically record action potentials in neurons using second harmonic generation offers the possibility of a fast readout of neural activity that could penetrate deeply into tissue without the sort of heating encountered in traditional two-photon microscopy (Helmchen and Denk 2006). This might not be beyond future generations of students in this Biophysics course. Al Kaye & Emily Mankin Phys 273; Spring 2009 Bibliography Freund, I., Deutsch, M. & Sprecher, A. Connective tissue polarity. Optical second-harmonic 1. microscopy, crossed-beam summation, and small-angle scattering in rat-tail tendon. Biophysical Journal 50, 693-712(1986).   Helmchen, F. & Denk, W. Deep tissue two-photon microscopy. Nat. Methods 2, 2. 932-940(2005).   3. Boyd, R.W. Nonlinear optics. 578(Academic press: 2002).   Dombeck, D.A., Blanchard-Desce, M. & Webb, W.W. Optical Recording of Action 4. Potentials with Second-Harmonic Generation Microscopy. J. Neurosci. 24, 999-1003(2004).   5. Hecht, E. Optics. 680(Pearson Education, Limited: 2001).   Kwan, A.C., Dombeck, D.A. & Webb, W.W. Polarized microtubule arrays in apical 6. dendrites and axons. Proceedings of the National Academy of Sciences 105, 11370-11375(2008).   Campagnola, P.J. & Loew, L.M. Second-harmonic imaging microscopy for visualizing 7. biomolecular arrays in cells, tissues and organisms. Nat Biotech 21, 1356-1360(2003).   Mansuripur, M. The Physical Principles of Magneto-Optical Recording. 776(Cambridge 8. University Press: 1998).