Evolution of the frequency chirp of Gaussian pulses and beams when passing through a pulse compressor
|
|
- Susan Flynn
- 6 years ago
- Views:
Transcription
1 Evolution of the frequency chirp of Gaussian pulses and beams when passing through a pulse compressor Derong Li, 3, Xiaohua Lv *, Pamela Bowlan, Rui Du, Shaoqun Zeng, Qingming Luo Britton Chance Center for Biomedical Photonics, Wuhan National Laboratory for Optoelectronics, Huazhong University of Science & Technology, Wuhan, , China Georgia Institute of Technology, School of Physics 837 State Street NW, Atlanta, Georgia 3033 USA 3 Key Lab for Biomedical Informatics and Health Engineering, Institute of Biomedical and Health Engineering, Shenzhen Institute of Advanced Technology, Chinese Academy of Sciences, Shenzhen, 58055, China *xhlv@mail.hust.edu.cn Abstract: The evolution of the frequency chirp of a laser pulse inside a classical pulse compressor is very different for plane waves and Gaussian beams, although after propagating through the last (4th) dispersive element, the two models give the same results. In this paper, we have analyzed the evolution of the frequency chirp of Gaussian pulses and beams using a method which directly obtains the spectral phase acquired by the compressor. We found the spatiotemporal couplings in the phase to be the fundamental reason for the difference in the frequency chirp acquired by a Gaussian beam and a plane wave. When the Gaussian beam propagates, an additional frequency chirp will be introduced if any spatiotemporal couplings (i.e. angular dispersion, spatial chirp or pulse front tilt) are present. However, if there are no couplings present, the chirp of the Gaussian beam is the same as that of a plane wave. When the Gaussian beam is well collimated, the introduced frequency chirp predicted by the plane wave and Gaussian beam models are in closer agreement. This work improves our understanding of pulse compressors and should be helpful for optimizing dispersion compensation schemes in many applications of femtosecond laser pulses. 009 Optical Society of America OCIS codes: (30.030) Ultrafast optics; (30.550) Pulse compression; (60.030) Dispersion References and links. E. B. Treacy, Optical pulse compression with diffraction gratings, IEEE J. Quantum Electron. 5(9), (969).. R. L. Fork, O. E. Martinez, and J. P. Gordon, Negative dispersion using pairs of prisms, Opt. Lett. 9(5), 50 5 (984). 3. J. Squier, F. Salin, G. Mourou, and D. Harter, 00-fs pulse generation and amplification in Ti:AI O 3, Opt. Lett. 6(5), (99). 4. C. L. Blanc, G. Grillon, J. P. Chambaret, A. Migus, and A. Antonetti, Compact and efficient multipass Ti:sapphire system for femtosecond chirped-pulse amplification at the terawatt level, Opt. Lett. 8(), 40 4 (993). 5. O. E. Martinez, 3000 times grating compressor with positive group velocity dispersion: application to fiber compensation in.3-.6 um region, IEEE J. Quantum Electron. 3(), (987). 6. S. Zeng, D. Li, X. Lv, J. Liu, and Q. Luo, Pulse broadening of the femtosecond pulses in a Gaussian beam passing an angular disperser, Opt. Lett. 3(9), 80 8 (007). 7. D. Li, X. Li, S. Zeng, and Q. Luo, A generalized analysis of femtosecond laser pulse broadening after angular dispersion, Opt. Express 6(), (008). 8. D. Li, S. Zeng, Q. Luo, P. Bowlan, V. Chauahan, and R. Trebino, Propagation dependence of chirp in Gaussian pulses and beams due to angular dispersion, Opt. Lett. 34(7), (009). 9. S. Szatmari, G. Kuhnle, and P. Simon, Pulse compression and traveling wave excitation scheme using a single dispersive element, Appl. Opt. 9(36), (990). 0. S. Szatmári, P. Simon, and M. Feuerhake, Group velocity dispersion compensated propagation of short pulses in dispersive media, Opt. Lett. (5), (996). #0 - $5.00 USD Received Jun 009; revised 4 Sep 009; accepted 5 Sep 009; published 0 Sep 009 (C) 009 OSA 4 September 009 / Vol. 7, No. 9 / OPTICS EXPRESS 7070
2 . S. Akturk, X. Gu, M. Kimmel, and R. Trebino, Extremely simple single-prism ultrashort- pulse compressor, Opt. Express 4(), (006).. M. Nakazawa, T. Nakashima, and H. Kubota, Optical pulse compression using a TeO acousto-optical light deflector, Opt. Lett. 3(), 0 (988). 3. S. Zeng, X. Lv, C. Zhan, W. R. Chen, W. Xiong, S. L. Jacques, and Q. Luo, Simultaneous compensation for spatial and temporal dispersion of acousto-optical deflectors for two-dimensional scanning with a single prism, Opt. Lett. 3(8), (006). 4. Y. Kremer, J. F. Léger, R. Lapole, N. Honnorat, Y. Candela, S. Dieudonné, and L. Bourdieu, A spatiotemporally compensated acousto-optic scanner for two-photon microscopy providing large field of view, Opt. Express 6(4), (008). 5. O. E. Martinez, Grating and prism compressors in the case of finite beam size, J. Opt. Soc. Am. B 3(7), (986). 6. O. E. Martinez, Pulse distortions in tilted pulse schemes for ultrashort pulses, Opt. Commun. 59(3), 9 3 (986). 7. Z. L. Horváth, Z. Benkö, A. P. Kovács, H. A. Hazim, and Z. Bor, Propagation of femtosecond pulses through lenses, gratings, and slits, Opt. Eng. 3(0), (993). 8. K. Varjú, A. P. Kovács, K. Osvay, and G. Kurdi, Angular dispersion of femtosecond pulses in a Gaussian beam, Opt. Lett. 7(), (00). 9. J. C. Diels, and W. Rudolph, Ultrashort Laser Pulse Phenomena (Academic, San Diego, Calif., 996). 0. C. Fiorini, C. Sauteret, C. Rouyer, N. Blanchot, S. Seznec, and A. Migus, Temporal aberrations due to misalignments of a stretcher-compressor system and compensation, IEEE J. Quantum Electron. 30(7), (994).. K. Osvay, A. P. Kovács, Z. Heiner, G. Kurdi, J. Klebniczki, and M. Csatári, Angular dispersion and temporal change of femtosecond pulses from misaligned pulse compressors, IEEE J. Sel. Top. Quantum Electron. 0(), 3 0 (004).. K. Osvay, A. P. Kovacs, G. Kurdi, Z. Heiner, M. Divall, J. Klebniczki, and I. E. Ferincz, Measurement of noncompensated angular dispersion and the subsequent temporal lengthening of femtosecond pulses in a CPA laser, Opt. Commun. 48(-3), 0 09 (005). 3. A. E. Siegman, Lasers, (University Science, Mill Valley, CA, 986). 4. X. Gu, S. Akturk, and R. Trebino, Spatial chirp in ultrafast optics, Opt. Commun. 4(4-6), (004). 5. S. Akturk, M. Kimmel, P. O Shea, and R. Trebino, Measuring spatial chirp in ultrashort pulses using single-shot Frequency-Resolved Optical Gating, Opt. Express (), (003). 6. S. Akturk, X. Gu, E. Zeek, and R. Trebino, Pulse-front tilt caused by spatial and temporal chirp, Opt. Express (9), (004). 7. S. Akturk, X. Gu, P. Gabolde, and R. Trebino, The general theory of first-order spatio-temporal distortions of Gaussian pulses and beams, Opt. Express 3(), (005). 8. P. Gabolde, D. Lee, S. Akturk, and R. Trebino, Describing first-order spatio-temporal distortions in ultrashort pulses using normalized parameters, Opt. Express 5(), 4 5 (007).. Introduction A classical pulse compressor (commonly including four gratings or prisms, or a pair of gratings and prisms with a double-pass configuration) [,] can stretch or compress a femtosecond laser pulse by introducing variable amounts of positive or negative frequency chirp (also referred to as just the chirp ). Pulse compressors are ubiquitous in ultrafast optics because of their ability to tailor the pulse s temporal duration, which is essential for making and maintaining intense, short pulses. Important applications include chirped-pulse amplification (CPA) and material-dispersion compensation, the latter of which is necessary for generating ultrashort pulses [3 5]. Since the duration of the pulse after the compressor depends on its group-delay dispersion, it is vital to able to accurately calculate this quantity [6 8]. In some practical applications, the classical 4-dispersive element compressor is not always suitable, and instead, a single angular dispersion element [9 ] or a pair of angular dispersion elements (also just referred to as an element ) in a single-pass configuration [ 4] are used to for dispersion control. In these cases, spatiotemporal couplings (meaning that there are x-ω, or equivalently x-t cross terms in the field) such as angular dispersion, spatial chirp, and pulse front tilt are present in the output pulse, making it even more difficult to calculate the chirp that is added to the pulse by the compressor. To model compressors, either a plane wave [,,9 ], or a Gaussian beam [5 8,5 8]) model is usually used. Martinez first showed that these two models sometimes give very different results. Namely, the propagation dependence of the chirp of a pulse while propagating inside a classical pulse compressor is very different for the two types of beams. #0 - $5.00 USD Received Jun 009; revised 4 Sep 009; accepted 5 Sep 009; published 0 Sep 009 (C) 009 OSA 4 September 009 / Vol. 7, No. 9 / OPTICS EXPRESS 707
3 Interestingly, however, after propagating through the 4th dispersive element (i.e., once all of the angular dispersion is removed), both models predict the same results [5]. Similarly it has been shown several times, that, when a pulse passes through a single angular disperser, the chirp of a Gaussian beam increases nonlinearly with propagation distance away from the disperser, while that of a plane wave increases linearly [8]. Though compressors are very commonly used, it seems that there is still more to learn about how they affect ultrashort pulses. Usually, the diffraction integral is used to investigate the propagation dependence of an ultrashort pulse, but this approach is quite complex making it difficult to use to understand how compressors work and why they effect plane waves differently than they do Gaussian beams [5]. Actually, the evolution of the chirp is entirely caused by a spatio-spectral phase φ(x,z,ω) that is added to the pulse by the compressor (since the chirp is defined as the second order derivative of the pulse s spectral phase with respect to angular frequency) [9]. Therefore, it is sufficient to determine the chirp s evolution by calculating this phase change that is acquired by the pulse due to propagation through the compressor. In this paper, we have derived expressions for the propagation dependence of the frequency chirp of a femtosecond Gaussian laser pulse after passing through each element of a classical four-element compressor. These expressions were derived by calculating the phase change acquired by the Gaussian beam after propagating through each element of the compressor. These calculations reveal the physical mechanism by which chirp is acquired by an ultrashort pulse in a pulse compressor. As the effects of misalignment of the dispersive elements are very complex [0 ], we consider only perfectly aligned compressors in this paper.. Phase acquired by a Gaussian pulse in a compressor A steadily propagating electromagnetic field in free space is governed by the scalar approximation of the Helmholtz equation. The plane wave is the simplest solution to this equation, and the Gaussian beam is also a special solution which is obtained if the slowly varying amplitude (SVA) approximation is made. The Gaussian beam is a very good model for realistic laser beams, so it is frequently adopted for modeling optics experiments. The phase of a Gaussian beam propagating in free space is given by [3]: φ kr z = + () R( z) zr ( r, z) kz tan ( ). where k is the wave-number, z is the propagation distance away from the beam waist on the axis, r = (x + y ) / is the distance from the z axis, R(z) = z + z R /z is the radius of curvature of the wave front, z R = kw o / is the Rayleigh range of the Gaussian beam, and w 0 is the beam waist size. Equation () describes the phase shift of a Gaussian beam at the point (r, z) relative to the original point (0, 0). In this equation, the first term is the geometrical phase shift which is the same as the phase of a plane wave. The second term represents a phase shift relative to the radial position. This is due to the finite beam size of the Gaussian beam. No such term exists in the expression for plane waves because they are infinite in space. The third term is the Gouy phase shift which is relative to the geometrical phase shift, and is also unique to a Gaussian beam [3]. #0 - $5.00 USD Received Jun 009; revised 4 Sep 009; accepted 5 Sep 009; published 0 Sep 009 (C) 009 OSA 4 September 009 / Vol. 7, No. 9 / OPTICS EXPRESS 707
4 Fig.. (a) A classical pulse compressor, which consists of four identical prisms (or other angular dispersers, such as gratings). (b) Optical path diagram of a femtosecond laser pulse passing through a classical pulse compressor. For two adjacent elements, the spacings are L, L, L 3, respectively. Due to angular dispersion, in the system, different spectral components have different paths. Taking the entrance vertices (O ~O 4) of each element as the original point, four reference distances can be established: they are x -z through x 4-z 4. The distances for any spectral component are defined as x ω-z ω through x 4 ω-z 4 ω. In this figure, the dashed line between the second and third element is the wave front of the pulse. When the alignment is perfect, θ = θ 3 and L = L 3. BW is the beam waist, and d is the distance between the beam waist and the first dispersive element. As shown in Fig., in order to study the chirp evolution of a femtosecond Gaussian laser pulse propagating through a pulse compressor, we first need to define all of the system parameters. For a single angular dispersion element, the deflection angle of a femtosecond laser pulse after passing through the element is described by [5,6]: θ θ θ ( γ, ω) = α γ + β ω = γ + ω. γ ω where α is the angular magnification, β is the angular dispersion, γ is the angle of incidence, and ω is the frequency. For the four elements in a perfectly aligned pulse compressor, the parameters must obey the following equations [5]:, β, α α () β α = = (3) α = α, β = β, (4) 3 3 β α = = (5) 3 4, β4. α3 α3 where the subscripts on each parameter correspond to the number of the element. We will use α and β to denote these parameters for the first element. It is assumed that the angular dispersion only occurs in the x-z plane and the Gaussian beam still follows the propagation rules of free space in the y direction. In this case, we omit the information for the y axis in order to simplify our calculation. When an arbitrary spectral component ω of a femtosecond Gaussian laser pulse passes through the first element and arrives at the point (x ω, z ω ), the #0 - $5.00 USD Received Jun 009; revised 4 Sep 009; accepted 5 Sep 009; published 0 Sep 009 (C) 009 OSA 4 September 009 / Vol. 7, No. 9 / OPTICS EXPRESS 7073
5 waist position is at a distance d/α + z ω relative to the apex of the element for the opposite propagation direction for which the equivalent Rayleigh range is z R = z R /α, where d is the distance between the beam waist and the first dispersive element [8]. Thus the phase expression for the Gaussian beam can be written as: kx d / z φ (,, ω) ( / α ) tan ( ). ω α + ω x z = k d + z + ω ω ω R( d / α + z ) z ω R In the same way, when an arbitrary spectral component ω passes through the second, third and fourth element and arrives at the locations (x ω, z ω ), (x 3ω, z 3ω ) and (x 4ω, z 4ω ), the corresponding waist position is equivalent to d + α z ω + z ω, d/α + z ω + z ω /α + z 3ω, and d + α z ω + z ω + α z 3ω + z 4ω with respect to the apex of the first element for the opposite propagation direction, and the corresponding Rayleigh ranges are of z R = z R,z R3 = z R /α, and z R4 = z R. Then the corresponding phase expressions for the Gaussian beam are: (6) kx d + α z + z φ (,, ω) ( α ) tan ( ), ω ω ω x z = k d+ z + z + ω ω ω ω R( d + α z + z ) z ω ω R (7) 3ω 3( x3 ω, z3 ω, ) = k( d / + z ω + zω / + z3 ω ) + R d α + z ω + zω α + z3 ω φ ω α α d / α + z ω + zω / α + z3 ω tan ( ), z R3 kx ( / / ) (8) kx4ω φ4 ( x4 ω, z4ω, ω) = k( d+ α z ω + zω + α z3 ω + z4 ω ) + R( d+ α z ω + zω + α z3 ω + z4ω ) (9) d+ α z ω + zω + α z3 ω + z4 ω tan ( ). z R4 where the subscripts of the phase functions represent the number of elements. From the above analysis, we obtain the spatio-spectral phase of the Gaussian beam at any position when passing through a pulse compressor. The corresponding frequency chirp can be obtained by calculating the second order derivative of the phase with respect to the frequency ω. 3. Chirp evolution of a Gaussian pulse in a compressor As shown in Fig., after passing through the first angular dispersion element, each spectral component of the femtosecond laser pulse is separated in space, and propagates a different distance. The reference spectral component propagates a distance z, while an arbitrary spectral component ω propagates a distance z ω, which can be related to z using the angle θ. The relevant formulae are [7]: z ω = z cosθ+ x sin θ, (0) x ω = z sinθ+ x cos θ. () And the corresponding first and second order derivatives are: ' z ω = 0, () '' dθ ω z, z = dω (3) x dθ = z (4) d ω ' ω, #0 - $5.00 USD Received Jun 009; revised 4 Sep 009; accepted 5 Sep 009; published 0 Sep 009 (C) 009 OSA 4 September 009 / Vol. 7, No. 9 / OPTICS EXPRESS 7074
6 d θ = z (5) d ω ''. ω x where β = dθ /dω is the angular dispersion introduced by the element. The second order derivative of the expression forφ with respect to spectral frequency ω (which describes the introduced chirp of the Gaussian beam passing through the first element) can then be obtained as follows: kβ z α z ( d+ α z ) φ ( ) β β [ ]. '' G z = k z + = k z R( d / α + z ) ( d+ α z) + zr Comparing the above expression with the phase function φ (Eq. (6)), we can see that the first term is the chirp introduced by the geometrical phase shift, which is negative. The second term is the chirp introduced by the radially dependent phase term which has an x-ω coupling, and this is positive. The chirp introduced by the Guoy phase shift is usually far less than the first two terms, so for simplicity we have left it out of Eq. (6) and will neglect it for the rest of our discussion. After the pulse has passed through the second element (which is anti-parallel with the first element and separated by a distance L ), the angular dispersion is totally eliminated but different spectral component of the pulse are still separated transversely in space, or spatial chirp, and also the x- ω coupling term in the phase (wave front tilt dispersion) are present [4 8]. In this interval, the relevant distances are: (6) z = z + ω const, (7) x = x + ξω+ const (8) ω. where ξ = αβl is the spatial chirp present in the pulse before entering the second element. Then the corresponding first and second derivatives of z ω and x ω with respect to ω can be obtained as follows: x ' z ω = 0, (9) '' z ω = 0, (0) = αβ () ' L, ω '' x ω = () 0. And thus the chirp of the pulse after passing through the second element is: '' α L ( d + α L + z ) φg ( z ) = kβ L + kα β L = kβ L [ ]. (3) R( d + α L + z ) ( d + α L + z ) + z R Comparing Eq. (3) with the phase function φ (Eq. (7)), we can see that the first term is the chirp introduced by the geometrical phase shift, which is negative. As the second element removes angular dispersion, no extra chirp will be introduced by the geometrical phase shift after passing through the second element. The second term represents the frequency chirp introduced by the radially dependent phase when the pulse travels the distance z, which is positive. Here we can see that though the angular dispersion has been eliminated after passing through the second element, x-ω couplings, namely, spatial chirp and wave-front-curvature dispersion are still present, and these combined with a changing beam spot size introduce additional frequency chirp as the pulse propagates through this region of the compressor. After the pulse has passed through the third element, angular dispersion is present again. As shown in Fig., the distance for any spectral component between the third and the fourth elements is L 3 cosθ 3 [], and thus the relevant formulae are: #0 - $5.00 USD Received Jun 009; revised 4 Sep 009; accepted 5 Sep 009; published 0 Sep 009 (C) 009 OSA 4 September 009 / Vol. 7, No. 9 / OPTICS EXPRESS 7075
7 z3 ω = z3 cosθ3 x3 sin θ3, (4) x3 ω = ( L3 z3)sinθ3+ x3 cos θ3. (5) Then the first and second derivatives of z 3ω and x 3ω with respect to frequency ω can be obtained as follows: ' z3 ω = 0, (6) z '' dθ3 3ω z3, = dω (7) ' dθ3 x3 ω = ( L3 z3), (8) dω '' d θ3 x ω = ( L3 z3). (9) dω where the angular dispersion introduced by the third element is dθ 3 /dω = -β. Also, the chirp of the femtosecond laser pulse after passing through the third element is: '' ( ) = + ( ) φ z k G β L k β z k β L z R( d / a + L + L / α + z ) ( ) 3 α ( d + α L + L + α z 3) ( d + α L + L + α z 3) + z R = kβ L kβ z + kβ L z Comparing the above expression with the phase function φ 3 in Eq. (8), we see that the first term is the chirp from the geometrical phase term acquired from propagating between the first and the second elements, which is negative. The second term is the chirp from the geometrical phase shift due to propagation between the third and the fourth elements, which is also negative. Finally, the third term is the chirp introduced by the radially dependent phase shift due to propagation through the same distance, which is positive. If the propagation distance z 3 = L 3, i.e. when the pulse arrives at the entrance of the fourth element, the chirp from the last term becomes zero. Provided that z 3 = L 3, and the angular dispersion from the first and third elements are equal and opposite, after the pulse has passed through the fourth element, all of the spectral components overlap transversely in space (i.e. no spatiotemporal couplings are present), and both the angular dispersion and all other spatiotemporal couplings have been removed, giving:. (30) z = z, (3) 4ω 4 x = x. (3) 4ω 4 So, after passing through the fourth element, neither the geometrical phase shift nor the radially dependent phase shifts introduce any chirp and then we get [5]: φ '' ( z G 4 ) = kβ L kβ L 3 = kβ L. (33) In this formula we can see that the final frequency chirp of the Gaussian beam is the same as that of a plane wave after passing through the pulse compressor. This final expression depends only on the geometrical phase shift and has no dependence on the radially dependent phase shift. From the above analysis of the chirp of the Gaussian beam, we can easily obtain the corresponding chirp of the plane wave by considering only the geometrical phase term. As the term describing the geometrical phase shift of the Gaussian beam is equal to the phase term of the plane wave, the chirp introduced by this term is the corresponding chirp of the plane wave: #0 - $5.00 USD Received Jun 009; revised 4 Sep 009; accepted 5 Sep 009; published 0 Sep 009 (C) 009 OSA 4 September 009 / Vol. 7, No. 9 / OPTICS EXPRESS 7076
8 φ ( z P ) = kβ z, (34) '' φ ( z P ) = kβ L, (35) '' φ ( z P ) = kβ L kβ z, (36) '' 3 3 φ '' ( z P 4 ) = kβ L. (37) From the above analysis, we can see that, for the Gaussian beam, the chirp introduced by the geometrical phase term is negative while that introduced by the radially dependent phase shift is positive at any position after passing through the element; it becomes zero when no ω- x coupling terms are present in the phase. Also, the radially dependent phase shift in the phase function of the Gaussian beam is the fundamental reason for the difference between the frequency chirp of a Gaussian beam and plane wave when passing through the classical pulse compressor. The frequency chirp evolution of the Gaussian beam and plane wave over the whole propagation process are shown in Fig., using the parameters α =, β = 0. rad/µm, d = z R = L = L = L 3 = m. Fig.. Comparison of the chirp evolution of a Gaussian beam (green) and plane wave (black) when passing through a 4-dispersive element pulse compressor. The red arrows in the figure show the difference in the two models. As shown in Fig., though the frequency chirp evolution of Gaussian beams is different from that of plane waves, the final result is the same for both models. In the following section, we provide a detailed analysis of the physical mechanism of this phenomenon. 4. Comparison of the Gaussian beam and plane wave models for a pulse compressor Here we compare the propagation dependent frequency chirps predicted by the two models in order to better understand their differences. For the plane wave model, in each interval of propagation, the following expressions are obtained: '' φ ( ) = β, (38) P z k z φ '' z P ( ) = 0, (39) φ P ( z ) = kβ z, (40) '' 3 3 #0 - $5.00 USD Received Jun 009; revised 4 Sep 009; accepted 5 Sep 009; published 0 Sep 009 (C) 009 OSA 4 September 009 / Vol. 7, No. 9 / OPTICS EXPRESS 7077
9 ( ) = 0. (4) φ '' z P 4 where the z s are the corresponding propagation intervals as shown in Fig.. We can see that the frequency chirp of the plane waves only changes in the interval where the angular dispersion is nonzero (e.g. after the pulse has passed through the first and third elements, as shown in Fig. ). The corresponding chirp evolution of the Gaussian beam in each interval is: '' α z ( d + α z ) φ ( z ) = k z [ ], G β ( d + α z ) + z R (4) φ '' ( d + α L + z ) ( d + α L ) G α β ( z ) = k L [ ] ( d + α L + z ) + z ( d + α L ) + z R = [ ], ( ) ( ) kα β L R d + α L + z R d + α L R (43) '' α ( d+ α L + L + α z ) α L ( d+ α L + L ) φg ( z ) = k z + k ( L z ) k L ( ) ( ) 3 β β β d+ α L + L + α z + z d+ α L + L + z 3 R R L z = kβ z kα β L [ ( ) ], ( ) ( ) R d+ α L + L L R d+ α L + L + α z 3 3 (44) φ '' G ( z ) = 0. (45) In contrast to the plane wave model, for Gaussian beams, the chirp changes as the pulse propagates even when the angular dispersion is zero. When the angular dispersion is zero but the x-ω coupling in the phase is nonzero, the frequency chirp of the Gaussian beam still changes along the propagation distance (as shown in Fig., after the pulse has passed through the second element). After the pulse has passed through the first and third elements (where the angular dispersion is not zero), the changes in the frequency chirp consist of two parts; one is introduced by the geometrical phase shift (corresponding to the first term in Eqs. (4) and (44)) and the other is due to the radially dependent phase shift (corresponding to the second term in Eqs. (4) and (44)). Calculating the difference in the chirp predicted by the two models over corresponding intervals (Eq. (4) is substracted from Eq. (38), and, etc.), we get the following results: 4 φ φ = '' PG ( z ) k α β z R d + α z ( ), '' ( z ) k L φ = α β, PG R ( d L z α ) R ( d α L ) ( ), R( d + α L + L ) L R( d + α L + L + α z ) 3 3 '' L z 3 3 PG z = kα β L 3 (46) (47) (48) φ '' PG ( z ) = 0. (49) 4 From the above equations we can clearly see that over any propagation interval, the fundamental reason for the difference between frequency chirp of the Gaussian beam and plane wave is due to the radially dependent, or the x-ω coupling term in the phase of the Gaussian beam. As shown in Fig., (A) In the interval 0-L, the chirp of the Gaussian beam increases less rapidly than that of the plane wave because the radially dependent phase shift introduces a positive change in the chirp (see Eq. (46)). (B) In the interval L to L, there is no further #0 - $5.00 USD Received Jun 009; revised 4 Sep 009; accepted 5 Sep 009; published 0 Sep 009 (C) 009 OSA 4 September 009 / Vol. 7, No. 9 / OPTICS EXPRESS 7078
10 change of the chirp of the plane wave while the chirp of Gaussian beam continues to change, due to the spatiotemporal couplings in the radially dependent term. However, these changes can be either positive or negative, as determined by the parameters of the beam itself (such as the Rayleigh length z R ) and the propagation distances (such as d, L, L ) (see Eq. (47)). (C) In the interval L to L 3, the chirp of the Gaussian beam increases faster than that of the plane wave, because in this interval, different spectral components of the pulse show a tendency to converge, that is to say, the spatial chirp is becoming smaller, which is the opposite of what happens between 0 to L (where the beam is diverging and the spatial chirp is increasing), so the chirp introduced by radially dependent phase shift becomes negative (see Eq. (48)). When the propagation distance z 3 = L 3 (at the entrance of the fourth element), the changes in the difference of the chirp between the Gaussian beam and the plane wave in this interval offsets their difference in the intervals 0 to L and L to L exactly, leading to the final chirp of Gaussian beam and plane wave being the same. Actually, Eqs. (46) ~(48) also reveal this rule: '' '' '' φpg ( L ) + φ ( ) ( ) 0. PG L + φ PG L = (50) 3 Thus, for both the Gaussian beam and the plane wave, the final chirps are the same. The fundamental reason for this difference is that the Gaussian beam has an x-ω coupling (or a spatiotemporal coupling) in its phase which introduces some chirp, and this term is absent in the phase of plane wave. If the spot size of the Gaussian beam increases, or if the Rayleigh range is longer and the beam is better collimated, the propagation dependence of the chirp predicted by the Gaussian beam is closer to that of the plane wave, as shown in Fig. 3. Fig. 3. The chirp evolution of a femtosecond laser pulse when passing through the pulse compressor. The Gaussian beam (green line) compared with a plane wave (black line). Each of the subfigures has different Rayleigh range: from (a) to (d), they are m, 3m, 5m, and 0m, respectively. The other parameters are the same as those of Fig.. When the Rayleigh range increases, the chirp evolution of the Gaussian beam becomes closer to that of a plane wave. This phenomenon is also shown in Eqs. (46) ~(49) where you can see that if the Gaussian beam is well collimated, that is to say, d, L, L, L 3 << z R,(which is called the approximate '' condition for good collimation [5,6]), = 0. Also, we can see that the chirp changes of φ PG #0 - $5.00 USD Received Jun 009; revised 4 Sep 009; accepted 5 Sep 009; published 0 Sep 009 (C) 009 OSA 4 September 009 / Vol. 7, No. 9 / OPTICS EXPRESS 7079
11 the Gaussian beam can be either positive (see subfigures b and c) or negative (see figure a) in the interval L to L, and depends on the parameters of the Gaussian beam and the propagation distance. 5. Discussion Using the Kirchhoff-Fresnel diffraction integral, we can obtain the complete electric field of the pulse while propagating through compressor which will tell us not only the frequency chirp, but also the spatial chirp, pulse front tilt, spot size, and etc [6 8,5,6]. However, for studying the evolution of the chirp, an analysis of the whole electric field, is not necessary, as the chirp is directly determined by the phase shift of the laser pulse. In this paper, we used a much simpler and more straightforward method, by analyzing the phase acquired by a Gaussian pulse due to propagation through a compressor, and directly calculated the second order derivative of the phase with respect to the spectral frequency ω. This gives us the chirp evolution of the pulse when passing through a pulse compressor. We found the x-ω coupling term in the phase of the Gaussian beam to be the fundamental reason for its chirp evolution inside a pulse compressor being different from that of a plane wave. When the spot size of the Gaussian beam increases, or when it is well collimated and has a longer Rayleigh range, the chirp acquired by the Gaussian beam is much closer to that of a plane wave, as shown in Fig. 3. The reason for this is that the radially dependent phase term of the Gaussian beam becomes less significant (i.e. the curvature of the wave front increases). If the Rayleigh range is long enough, no radially dependent phase term exists, leading to equivalent chirps for the Gaussian beam and the plane wave models. Although the spot size of the Gaussian beam has no influence on the final chirp after propagation through the 4th dispersive element (assuming a well aligned compressor), it could significantly influence the value of the chirp when the pulse is passing through a single angular dispersion element, a pair of elements in a single-pass structure, or when the pulse compressor is not perfectly aligned. This of course affects the duration of the output laser pulse. Another phenomenon worth noting is that for Gaussian beams, as long as spatiotemporal couplings exist (i.e. x-ω cross terms), even though there is no angular dispersion, an additional frequency chirp will be introduced as the pulse propagates. Martinez implied such a phenomenon but provided no explanation [5]. In essence, propagation of an ultrashort pulse in the presence of spatiotemporal couplings is a three-dimensional (x, ω and z) effect, and it causes the spatial terms of the Gaussian to mix with the frequency terms, and phase terms are transferred to the intensity (and vice versa). Namely, the first dispersive element introduces angular dispersion which propagation changes into spatial chirp, wave-front-curvature dispersion (or pulse front tilt, if viewed in the time domain), frequency chirp, and others. And once the angular dispersion vanishes after the second prism, there are still coupling terms remaining the spatial chirp and the wave-front-tilt dispersion so propagation again transfers these terms into frequency chirp. Specificially it is the radially dependent phase term in the Gaussian model that allows for this mixing, and if this term vanishes, then the frequency chirp of Gaussian beam will be the same as that of plane wave (such as after the pulse has passed through the fourth element, which also makes this term vanish). The spatial chirp and frequency chirp are commonly regarded as independent parameters; the spatial chirp describes the transverse separation (perpendicular to the propagation direction) of different spectral components while the frequency chirp describes the longitudinal delay (along the propagation direction). Here we again illustrate that this is not the case [6 8], and we have shown that these two quantities can be coupled by propagation. 6. Conclusion In this paper, we studied the chirp evolution of a Gaussian beam when passing through a classical pulse compressor by directly calculating the acquired spatio-spectral phase. Compared with the chirp evolution predicted by the plane wave model, we found that a spatiotemporal coupling or an x-ω dependent term in the phase of the Gaussian beam is the fundamental reason for the difference in these two models predictions for pulse compressors. #0 - $5.00 USD Received Jun 009; revised 4 Sep 009; accepted 5 Sep 009; published 0 Sep 009 (C) 009 OSA 4 September 009 / Vol. 7, No. 9 / OPTICS EXPRESS 7080
12 For a Gaussian beam, the existence of spatiotemporal couplings also introduces an additional frequency chirp even when no angular dispersion exists. If the Gaussian beam is well collimated, the frequency chirp evolution is closer to that of a plane wave after passing through the angular dispersion elements. This work provides a deeper understanding of the physical mechanism of the frequency chirp evolution when a laser pulse passes through a classical pulse compressor. Our analysis will also be helpful for optimization of dispersion compensation schemes in many applications of femtosecond laser pulses. Acknowledgments This work was supported by the National Natural Science Foundation (NSFC) ( , ), and Program for Changjiang Scholars and Innovative Research Team in University. #0 - $5.00 USD Received Jun 009; revised 4 Sep 009; accepted 5 Sep 009; published 0 Sep 009 (C) 009 OSA 4 September 009 / Vol. 7, No. 9 / OPTICS EXPRESS 708
Describing first-order spatio-temporal distortions in ultrashort pulses using normalized parameters
Describing first-order spatio-temporal distortions in ultrashort pulses using normalized parameters Pablo Gabolde, Dongjoo Lee, Selcuk Akturk and Rick Trebino Georgia Institute of Technology, 837 State
More informationSwamp Optics Tutorial. Pulse Compression
Swamp Optics, LLC. 6300 Powers Ferry Rd. Suite 600-345 Atlanta, GA 30339 +1.404.547.9267 www.swamoptics.com Swamp Optics Tutorial Pulse Compression Recall that different colors propagate at different velocities
More informationDispersion and how to control it
Dispersion and how to control it Group velocity versus phase velocity Angular dispersion Prism sequences Grating pairs Chirped mirrors Intracavity and extra-cavity examples 1 Pulse propagation and broadening
More informationThe general theory of first-order spatio-temporal distortions of Gaussian pulses and beams
The general theory of first-order spatio-temporal distortions of Gaussian pulses and beams Selcuk Akturk, Xun Gu, Pablo Gabolde and ick Trebino School of Physics, Georgia Institute of Technology, Atlanta,
More informationNovel method for ultrashort laser pulse-width measurement based on the self-diffraction effect
Novel method for ultrashort laser pulse-width measurement based on the self-diffraction effect Peng Xi, Changhe Zhou, Enwen Dai, and Liren Liu Shanghai Institute of Optics and Fine Mechanics, Chinese Academy
More informationStrongly enhanced negative dispersion from thermal lensing or other focusing effects in femtosecond laser cavities
646 J. Opt. Soc. Am. B/ Vol. 17, No. 4/ April 2000 Paschotta et al. Strongly enhanced negative dispersion from thermal lensing or other focusing effects in femtosecond laser cavities R. Paschotta, J. Aus
More informationCourse Secretary: Christine Berber O3.095, phone x-6351,
IMPRS: Ultrafast Source Technologies Franz X. Kärtner (Umit Demirbas) & Thorsten Uphues, Bldg. 99, O3.097 & Room 6/3 Email & phone: franz.kaertner@cfel.de, 040 8998 6350 thorsten.uphues@cfel.de, 040 8998
More informationFocal shift in vector beams
Focal shift in vector beams Pamela L. Greene The Institute of Optics, University of Rochester, Rochester, New York 1467-186 pgreene@optics.rochester.edu Dennis G. Hall The Institute of Optics and The Rochester
More informationParametric Spatio-Temporal Control of Focusing Laser Pulses
Parametric Spatio-Temporal Control of Focusing Laser Pulses Matthew A. Coughlan, Mateusz Plewicki, and Robert J. Levis* Department of Chemistry, Center for Advanced Photonics Research, Temple University,
More informationLong- and short-term average intensity for multi-gaussian beam with a common axis in turbulence
Chin. Phys. B Vol. 0, No. 1 011) 01407 Long- and short-term average intensity for multi-gaussian beam with a common axis in turbulence Chu Xiu-Xiang ) College of Sciences, Zhejiang Agriculture and Forestry
More informationDirect spatiotemporal measurements of accelerating ultrashort Bessel-type light bullets
Direct spatiotemporal measurements of accelerating ultrashort Bessel-type light bullets Heli Valtna-Lukner, 1,* Pamela Bowlan, 2 Madis Lõhmus, 1 Peeter Piksarv, 1 Rick Trebino, 2 and Peeter Saari 1,* 1
More informationExtreme pulse-front tilt from an etalon
2322 J. Opt. Soc. Am. B/ Vol. 27, No. 11/ November 2010 P. Bowlan and R. Trebino Extreme pulse-front tilt from an etalon Pamela Bowlan 1,2, * and Rick Trebino 1,2 1 Swamp Optics, LLC, 6300 Powers Ferry
More informationNear-field diffraction of irregular phase gratings with multiple phase-shifts
References Near-field diffraction of irregular phase gratings with multiple phase-shifts Yunlong Sheng and Li Sun Center for optics, photonics and laser (COPL), University Laval, Quebec City, Canada, G1K
More informationMeasuring the spatiotemporal electric field of ultrashort pulses with high spatial and spectral resolution
Bowlan et al. Vol. 25, No. 6/ June 2008/J. Opt. Soc. Am. B A81 Measuring the spatiotemporal electric field of ultrashort pulses with high spatial and spectral resolution Pamela Bowlan, 1, * Pablo Gabolde,
More informationModeling microlenses by use of vectorial field rays and diffraction integrals
Modeling microlenses by use of vectorial field rays and diffraction integrals Miguel A. Alvarez-Cabanillas, Fang Xu, and Yeshaiahu Fainman A nonparaxial vector-field method is used to describe the behavior
More informationOptical Spectroscopy of Advanced Materials
Phys 590B Condensed Matter Physics: Experimental Methods Optical Spectroscopy of Advanced Materials Basic optics, nonlinear and ultrafast optics Jigang Wang Department of Physics, Iowa State University
More informationPROCEEDINGS OF SPIE. Advanced laboratory exercise: studying the dispersion properties of a prism pair
PROCEEDINGS OF SPIE SPIEDigitalLibrary.org/conference-proceedings-of-spie Advanced laboratory exercise: studying the dispersion properties of a prism pair T. Grósz, L. Gulyás, A. P. Kovács T. Grósz, L.
More informationDispersion Compensation with a Prism-pair
arxiv:1411.0232v1 [physics.optics] 2 Nov 2014 Dispersion Compensation with a Prism-pair Yaakov Shaked, Shai Yefet and Avi Pe er Department of physics and BINA Center of nano-technology, Bar-Ilan university,
More informationDirect measurement of spectral phase for ultrashort laser pulses
Direct measurement of spectral phase for ultrashort laser pulses Vadim V. Lozovoy, 1 Bingwei Xu, 1 Yves Coello, 1 and Marcos Dantus 1,2,* 1 Department of Chemistry, Michigan State University 2 Department
More informationThe structure of laser pulses
1 The structure of laser pulses 2 The structure of laser pulses Pulse characteristics Temporal and spectral representation Fourier transforms Temporal and spectral widths Instantaneous frequency Chirped
More informationThe spectrogram in acoustics
Measuring the power spectrum at various delays gives the spectrogram 2 S ω, τ = dd E t g t τ e iii The spectrogram in acoustics E ssssss t, τ = E t g t τ where g t is a variable gating function Frequency
More informationUncertainty Principle Applied to Focused Fields and the Angular Spectrum Representation
Uncertainty Principle Applied to Focused Fields and the Angular Spectrum Representation Manuel Guizar, Chris Todd Abstract There are several forms by which the transverse spot size and angular spread of
More informationExperimental studies of the coherence of microstructure-fiber supercontinuum
Experimental studies of the coherence of microstructure-fiber supercontinuum Xun Gu, Mark Kimmel, Aparna P. Shreenath and Rick Trebino School of Physics, Georgia Institute of Technology, Atlanta, GA 30332-0430,
More informationgives rise to multitude of four-wave-mixing phenomena which are of great
Module 4 : Third order nonlinear optical processes Lecture 26 : Third-order nonlinearity measurement techniques: Z-Scan Objectives In this lecture you will learn the following Theory of Z-scan technique
More informationLaser Optics-II. ME 677: Laser Material Processing Instructor: Ramesh Singh 1
Laser Optics-II 1 Outline Absorption Modes Irradiance Reflectivity/Absorption Absorption coefficient will vary with the same effects as the reflectivity For opaque materials: reflectivity = 1 - absorptivity
More informationNo. 9 Experimental study on the chirped structure of the construct the early time spectra. [14;15] The prevailing account of the chirped struct
Vol 12 No 9, September 2003 cfl 2003 Chin. Phys. Soc. 1009-1963/2003/12(09)/0986-06 Chinese Physics and IOP Publishing Ltd Experimental study on the chirped structure of the white-light continuum generation
More informationIMPRS: Ultrafast Source Technologies
IMPRS: Ultrafast Source Technologies Fran X. Kärtner & Thorsten Uphues, Bldg. 99, O3.097 & Room 6/3 Email & phone: fran.kaertner@cfel.de, 040 8998 6350 Thorsten.Uphues@cfel.de, 040 8998 706 Lectures: Tuesday
More information1 Mathematical description of ultrashort laser pulses
1 Mathematical description of ultrashort laser pulses 1.1 We first perform the Fourier transform directly on the Gaussian electric field: E(ω) = F[E(t)] = A 0 e 4 ln ( t T FWHM ) e i(ω 0t+ϕ CE ) e iωt
More informationVector diffraction theory of refraction of light by a spherical surface
S. Guha and G. D. Gillen Vol. 4, No. 1/January 007/J. Opt. Soc. Am. B 1 Vector diffraction theory of refraction of light by a spherical surface Shekhar Guha and Glen D. Gillen* Materials and Manufacturing
More informationEffects of self-steepening and self-frequency shifting on short-pulse splitting in dispersive nonlinear media
PHYSICAL REVIEW A VOLUME 57, NUMBER 6 JUNE 1998 Effects of self-steepening and self-frequency shifting on short-pulse splitting in dispersive nonlinear media Marek Trippenbach and Y. B. Band Departments
More informationIntrinsic beam emittance of laser-accelerated electrons measured by x-ray spectroscopic imaging
Intrinsic beam emittance of laser-accelerated electrons measured by x-ray spectroscopic imaging G. Golovin 1, S. Banerjee 1, C. Liu 1, S. Chen 1, J. Zhang 1, B. Zhao 1, P. Zhang 1, M. Veale 2, M. Wilson
More informationAn alternative method to specify the degree of resonator stability
PRAMANA c Indian Academy of Sciences Vol. 68, No. 4 journal of April 2007 physics pp. 571 580 An alternative method to specify the degree of resonator stability JOGY GEORGE, K RANGANATHAN and T P S NATHAN
More informationPropagation dynamics of abruptly autofocusing Airy beams with optical vortices
Propagation dynamics of abruptly autofocusing Airy beams with optical vortices Yunfeng Jiang, 1 Kaikai Huang, 1,2 and Xuanhui Lu 1, * 1 Institute of Optics, Department of Physics, Zhejiang University,
More informationAnalysis of second-harmonic generation microscopy under refractive index mismatch
Vol 16 No 11, November 27 c 27 Chin. Phys. Soc. 19-1963/27/16(11/3285-5 Chinese Physics and IOP Publishing Ltd Analysis of second-harmonic generation microscopy under refractive index mismatch Wang Xiang-Hui(
More informationFIBER OPTICS. Prof. R.K. Shevgaonkar. Department of Electrical Engineering. Indian Institute of Technology, Bombay. Lecture: 07
FIBER OPTICS Prof. R.K. Shevgaonkar Department of Electrical Engineering Indian Institute of Technology, Bombay Lecture: 07 Analysis of Wave-Model of Light Fiber Optics, Prof. R.K. Shevgaonkar, Dept. of
More informationSpatial evolution of laser beam profiles in an SBS amplifier
Spatial evolution of laser beam profiles in an SBS amplifier Edward J. Miller, Mark D. Skeldon, and Robert W. Boyd We have performed an experimental and theoretical analysis of the modification of the
More informationPolarization Mode Dispersion
Unit-7: Polarization Mode Dispersion https://sites.google.com/a/faculty.muet.edu.pk/abdullatif Department of Telecommunication, MUET UET Jamshoro 1 Goos Hänchen Shift The Goos-Hänchen effect is a phenomenon
More informationTransverse Coherence Properties of the LCLS X-ray Beam
LCLS-TN-06-13 Transverse Coherence Properties of the LCLS X-ray Beam S. Reiche, UCLA, Los Angeles, CA 90095, USA October 31, 2006 Abstract Self-amplifying spontaneous radiation free-electron lasers, such
More informationOptical Circular Deflector with Attosecond Resolution for Ultrashort Electron. Abstract
SLAC-PUB-16931 February 2017 Optical Circular Deflector with Attosecond Resolution for Ultrashort Electron Zhen Zhang, Yingchao Du, Chuanxiang Tang 1 Department of Engineering Physics, Tsinghua University,
More informationLecture 19 Optical MEMS (1)
EEL6935 Advanced MEMS (Spring 5) Instructor: Dr. Huikai Xie Lecture 19 Optical MEMS (1) Agenda: Optics Review EEL6935 Advanced MEMS 5 H. Xie 3/8/5 1 Optics Review Nature of Light Reflection and Refraction
More informationarxiv: v1 [physics.optics] 30 Mar 2010
Analytical vectorial structure of non-paraxial four-petal Gaussian beams in the far field Xuewen Long a,b, Keqing Lu a, Yuhong Zhang a,b, Jianbang Guo a,b, and Kehao Li a,b a State Key Laboratory of Transient
More information1 ESO's Compact Laser Guide Star Unit Ottobeuren, Germany Beam optics!
1 ESO's Compact Laser Guide Star Unit Ottobeuren, Germany www.eso.org Introduction Characteristics Beam optics! ABCD matrices 2 Background! A paraxial wave has wavefronts whose normals are paraxial rays.!!
More informationOffset Spheroidal Mirrors for Gaussian Beam Optics in ZEMAX
Offset Spheroidal Mirrors for Gaussian Beam Optics in ZEMAX Antony A. Stark and Urs Graf Smithsonian Astrophysical Observatory, University of Cologne aas@cfa.harvard.edu 1 October 2013 This memorandum
More information3.5 Cavities Cavity modes and ABCD-matrix analysis 206 CHAPTER 3. ULTRASHORT SOURCES I - FUNDAMENTALS
206 CHAPTER 3. ULTRASHORT SOURCES I - FUNDAMENTALS which is a special case of Eq. (3.30. Note that this treatment of dispersion is equivalent to solving the differential equation (1.94 for an incremental
More informationAssessment of Threshold for Nonlinear Effects in Ibsen Transmission Gratings
Assessment of Threshold for Nonlinear Effects in Ibsen Transmission Gratings Temple University 13th & Norris Street Philadelphia, PA 19122 T: 1-215-204-1052 contact: johanan@temple.edu http://www.temple.edu/capr/
More informationCompression and broadening of phase-conjugate pulses in photorefractive self-pumped phase conjugators
1390 J. Opt. Soc. Am. B/ Vol. 17, No. 8/ August 000 Yang et al. Compression and broadening of phase-conjugate pulses in photorefractive self-pumped phase conjugators Changxi Yang and Min Xiao Department
More information21. Propagation of Gaussian beams
1. Propagation of Gaussian beams How to propagate a Gaussian beam Rayleigh range and confocal parameter Transmission through a circular aperture Focusing a Gaussian beam Depth of field Gaussian beams and
More informationSupplemental material for Bound electron nonlinearity beyond the ionization threshold
Supplemental material for Bound electron nonlinearity beyond the ionization threshold 1. Experimental setup The laser used in the experiments is a λ=800 nm Ti:Sapphire amplifier producing 42 fs, 10 mj
More informationSupplementary Materials for
wwwsciencemagorg/cgi/content/full/scienceaaa3035/dc1 Supplementary Materials for Spatially structured photons that travel in free space slower than the speed of light Daniel Giovannini, Jacquiline Romero,
More informationLight as a Transverse Wave.
Waves and Superposition (Keating Chapter 21) The ray model for light (i.e. light travels in straight lines) can be used to explain a lot of phenomena (like basic object and image formation and even aberrations)
More informationPulse front tilt measurement of femtosecond laser pulses
Pulse front tilt measurement of femtosecond laser pulses Nikolay Dimitrov a, Lyubomir Stoyanov a, Ivan Stefanov a, Alexander Dreischuh a,, Peter Hansinger b,c, Gerhard G. Paulus b,c a Department of Quantum
More informationElectron acceleration by tightly focused radially polarized few-cycle laser pulses
Chin. Phys. B Vol. 1, No. (1) 411 Electron acceleration by tightly focused radially polarized few-cycle laser pulses Liu Jin-Lu( ), Sheng Zheng-Ming( ), and Zheng Jun( ) Key Laboratory for Laser Plasmas
More informationLinear pulse propagation
Ultrafast Laser Physics Ursula Keller / Lukas Gallmann ETH Zurich, Physics Department, Switzerland www.ulp.ethz.ch Linear pulse propagation Ultrafast Laser Physics ETH Zurich Superposition of many monochromatic
More informationAlignment of chirped-pulse compressor
Quantum Electronics 42 (11) 996 1001 (2012) 2012 Kvantovaya Elektronika and Turpion Ltd Alignment of chirped-pulse compressor PACS numbers: 42.65.Re; 42.60.By; 42.79.Dj DOI: 10.1070/QE2012v042n11ABEH014966
More informationDouble-distance propagation of Gaussian beams passing through a tilted cat-eye optical lens in a turbulent atmosphere
Double-distance propagation of Gaussian beams passing through a tilted cat-eye optical lens in a turbulent atmosphere Zhao Yan-Zhong( ), Sun Hua-Yan( ), and Song Feng-Hua( ) Department of Photoelectric
More informationElectromagnetic fields and waves
Electromagnetic fields and waves Maxwell s rainbow Outline Maxwell s equations Plane waves Pulses and group velocity Polarization of light Transmission and reflection at an interface Macroscopic Maxwell
More informationRevival Structures of Linear Molecules in a Field-Free Alignment Condition as Probed by High-Order Harmonic Generation
Journal of the Korean Physical Society, Vol. 49, No. 1, July 2006, pp. 337 341 Revival Structures of Linear Molecules in a Field-Free Alignment Condition as Probed by High-Order Harmonic Generation G.
More informationSpatiotemporal coupling in dispersive nonlinear planar waveguides
2382 J. Opt. Soc. Am. B/Vol. 12, No. 12/December 1995 A. T. Ryan and G. P. Agrawal Spatiotemporal coupling in dispersive nonlinear planar waveguides Andrew T. Ryan and Govind P. Agrawal The Institute of
More information37. 3rd order nonlinearities
37. 3rd order nonlinearities Characterizing 3rd order effects The nonlinear refractive index Self-lensing Self-phase modulation Solitons When the whole idea of χ (n) fails Attosecond pulses! χ () : New
More informationSpatio-temporal Coupling of Random Electromagnetic Pulses Interacting With Reflecting Gratings
University of Miami Scholarly Repository Physics Articles and Papers Physics -- Spatio-temporal Coupling of Random Electromagnetic Pulses Interacting With Reflecting Gratings Min Yao Yangjian Cai Olga
More informationWaveplate analyzer using binary magneto-optic rotators
Waveplate analyzer using binary magneto-optic rotators Xiaojun Chen 1, Lianshan Yan 1, and X. Steve Yao 1, 1. General Photonics Corp. Chino, CA, 91710, USA Tel: 909-590-5473 Fax: 909-90-5535. Polarization
More informationSome Topics in Optics
Some Topics in Optics The HeNe LASER The index of refraction and dispersion Interference The Michelson Interferometer Diffraction Wavemeter Fabry-Pérot Etalon and Interferometer The Helium Neon LASER A
More information37. 3rd order nonlinearities
37. 3rd order nonlinearities Characterizing 3rd order effects The nonlinear refractive index Self-lensing Self-phase modulation Solitons When the whole idea of χ (n) fails Attosecond pulses! χ () : New
More informationPhysics 3312 Lecture 7 February 6, 2019
Physics 3312 Lecture 7 February 6, 2019 LAST TIME: Reviewed thick lenses and lens systems, examples, chromatic aberration and its reduction, aberration function, spherical aberration How do we reduce spherical
More informationA Single-Beam, Ponderomotive-Optical Trap for Energetic Free Electrons
A Single-Beam, Ponderomotive-Optical Trap for Energetic Free Electrons Traditionally, there have been many advantages to using laser beams with Gaussian spatial profiles in the study of high-field atomic
More informationEffects of resonator input power on Kerr lens mode-locked lasers
PRAMANA c Indian Academy of Sciences Vol. 85, No. 1 journal of July 2015 physics pp. 115 124 Effects of resonator input power on Kerr lens mode-locked lasers S KAZEMPOUR, A KESHAVARZ and G HONARASA Department
More informationUsing phase diversity for the measurement of the complete spatiotemporal electric field of ultrashort laser pulses
244 J. Opt. Soc. Am. B / Vol. 29, No. 2 / February 2012 P. Bowlan and R. Trebino Using phase diversity for the measurement of the complete spatiotemporal electric field of ultrashort laser pulses Pamela
More informationDiffraction gratings. B.Tech-I
Diffraction gratings B.Tech-I Introduction Diffraction grating can be understood as an optical unit that separates polychromatic light into constant monochromatic composition. Uses are tabulated below
More informationThomson Scattering from Nonlinear Electron Plasma Waves
Thomson Scattering from Nonlinear Electron Plasma Waves A. DAVIES, 1 J. KATZ, 1 S. BUCHT, 1 D. HABERBERGER, 1 J. BROMAGE, 1 J. D. ZUEGEL, 1 J. D. SADLER, 2 P. A. NORREYS, 3 R. BINGHAM, 4 R. TRINES, 5 L.O.
More informationDigital Holographic Measurement of Nanometric Optical Excitation on Soft Matter by Optical Pressure and Photothermal Interactions
Ph.D. Dissertation Defense September 5, 2012 Digital Holographic Measurement of Nanometric Optical Excitation on Soft Matter by Optical Pressure and Photothermal Interactions David C. Clark Digital Holography
More informationMODELLING PLASMA FLUORESCENCE INDUCED BY FEMTOSECOND PULSE PROPAGATION IN IONIZING GASES
MODELLING PLASMA FLUORESCENCE INDUCED BY FEMTOSECOND PULSE PROPAGATION IN IONIZING GASES V. TOSA 1,, A. BENDE 1, T. D. SILIPAS 1, H. T. KIM, C. H. NAM 1 National Institute for R&D of Isotopic and Molecular
More informationSupplementary Figure 1. Illustration of the angular momentum selection rules for stimulated
0 = 0 1 = 0 0 = 0 1 = 1 0 = -1 1 = 1 0 = 1 1 = 1 k φ k φ k φ k φ a p = 0 b p = -1 c p = - d p = 0 Supplementary Figure 1. Illustration of the angular momentum selection rules for stimulated Raman backscattering
More informationVectorial structure and beam quality of vector-vortex Bessel Gauss beams in the far field
COL (Suppl., S6( CHINESE OPTICS LETTERS June 3, Vectorial structure and beam quality of vector-vortex Bessel Gauss beams in the far field Lina Guo (, and Zhilie Tang ( School of Physics and Telecommunication
More informationAccumulated Gouy phase shift in Gaussian beam propagation through first-order optical systems
90 J. Opt. Soc. Am. A/Vol. 4, No. 9/September 997 M. F. Erden and H. M. Ozaktas Accumulated Gouy phase shift Gaussian beam propagation through first-order optical systems M. Fatih Erden and Haldun M. Ozaktas
More informationThe Gouy phase shift in nonlinear interactions of waves
The Gouy phase shift in nonlinear interactions of waves Nico Lastzka 1 and Roman Schnabel 1 1 Institut für Gravitationsphysik, Leibniz Universität Hannover and Max-Planck-Institut für Gravitationsphysik
More informationMotion control of the wedge prisms in Risley-prism-based beam steering system for precise target tracking
Motion control of the wedge prisms in Risley-prism-based beam steering system for precise target tracking Yuan Zhou,,2, * Yafei Lu, 2 Mo Hei, 2 Guangcan Liu, and Dapeng Fan 2 Department of Electronic and
More informationarxiv: v1 [math-ph] 3 Nov 2011
Formalism of operators for Laguerre-Gauss modes A. L. F. da Silva (α), A. T. B. Celeste (β), M. Pazetti (γ), C. E. F. Lopes (δ) (α,β) Instituto Federal do Sertão Pernambucano, Petrolina - PE, Brazil (γ)
More informationFIBER OPTICS. Prof. R.K. Shevgaonkar. Department of Electrical Engineering. Indian Institute of Technology, Bombay. Lecture: 15. Optical Sources-LASER
FIBER OPTICS Prof. R.K. Shevgaonkar Department of Electrical Engineering Indian Institute of Technology, Bombay Lecture: 15 Optical Sources-LASER Fiber Optics, Prof. R.K. Shevgaonkar, Dept. of Electrical
More informationSpatiotemporal Amplitude and Phase Retrieval of Bessel-X pulses using a Hartmann-Shack Sensor.
Spatiotemporal Amplitude and Phase Retrieval of Bessel-X pulses using a Hartmann-Shack Sensor. F. Bonaretti, D. Faccio,, M. Clerici, J. Biegert,3, P. Di Trapani,4 CNISM and Department of Physics and Mathematics,
More informationSpectral Fraunhofer regime: time-to-frequency conversion by the action of a single time lens on an optical pulse
Spectral Fraunhofer regime: time-to-frequency conversion by the action of a single time lens on an optical pulse José Azaña, Naum K. Berger, Boris Levit, and Baruch Fischer We analyze a new regime in the
More informationSelf-Phase-Modulation of Optical Pulses From Filaments to Solitons to Frequency Combs
Self-Phase-Modulation of Optical Pulses From Filaments to Solitons to Frequency Combs P. L. Kelley Optical Society of America Washington, DC and T. K. Gustafson EECS University of California Berkeley,
More informationOptical time-domain differentiation based on intensive differential group delay
Optical time-domain differentiation based on intensive differential group delay Li Zheng-Yong( ), Yu Xiang-Zhi( ), and Wu Chong-Qing( ) Key Laboratory of Luminescence and Optical Information of the Ministry
More informationTwo-Dimensional simulation of thermal blooming effects in ring pattern laser beam propagating into absorbing CO2 gas
Two-Dimensional simulation of thermal blooming effects in ring pattern laser beam propagating into absorbing CO gas M. H. Mahdieh 1, and B. Lotfi Department of Physics, Iran University of Science and Technology,
More information: Imaging Systems Laboratory II. Laboratory 6: The Polarization of Light April 16 & 18, 2002
151-232: Imaging Systems Laboratory II Laboratory 6: The Polarization of Light April 16 & 18, 22 Abstract. In this lab, we will investigate linear and circular polarization of light. Linearly polarized
More informationOptics. n n. sin c. sin
Optics Geometrical optics (model) Light-ray: extremely thin parallel light beam Using this model, the explanation of several optical phenomena can be given as the solution of simple geometric problems.
More informationUsing GRENOUILLE to characterize asymmetric femtosecond pulses undergoing self- and cross-phase modulation in a polarization-maintaining optical fiber
Using GRENOUILLE to characterize asymmetric femtosecond pulses undergoing self- and cross-phase modulation in a polarization-maintaining optical fiber Bhaskar Khubchandani and A. Christian Silva Department
More informationLecture notes 5: Diffraction
Lecture notes 5: Diffraction Let us now consider how light reacts to being confined to a given aperture. The resolution of an aperture is restricted due to the wave nature of light: as light passes through
More informationEngineering Physics 1 Prof. G.D. Vermaa Department of Physics Indian Institute of Technology-Roorkee
Engineering Physics 1 Prof. G.D. Vermaa Department of Physics Indian Institute of Technology-Roorkee Module-04 Lecture-02 Diffraction Part - 02 In the previous lecture I discussed single slit and double
More informationHIGH-POWER THIRD-HARMONIC FLAT LASER PULSE GENERATION. Abstract
SPARC-LS-07/001 23 May 2007 HIGH-POWER THIRD-HARMONIC FLAT LASER PULSE GENERATION C. Vicario (INFN/LNF), M. Petrarca. (INFN/Roma1), S. Cialdi (INFN/Milano) P. Musumeci (UCLA). Abstract The generation of
More informationAn electric field wave packet propagating in a laser beam along the z axis can be described as
Electromagnetic pulses: propagation & properties Propagation equation, group velocity, group velocity dispersion An electric field wave packet propagating in a laser beam along the z axis can be described
More informationOptics.
Optics www.optics.rochester.edu/classes/opt100/opt100page.html Course outline Light is a Ray (Geometrical Optics) 1. Nature of light 2. Production and measurement of light 3. Geometrical optics 4. Matrix
More informationLecture 2: Geometrical Optics 1. Spherical Waves. From Waves to Rays. Lenses. Chromatic Aberrations. Mirrors. Outline
Lecture 2: Geometrical Optics 1 Outline 1 Spherical Waves 2 From Waves to Rays 3 Lenses 4 Chromatic Aberrations 5 Mirrors Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Astronomical Telescopes
More informationWaves & Oscillations
Physics 42200 Waves & Oscillations Lecture 32 Electromagnetic Waves Spring 2016 Semester Matthew Jones Electromagnetism Geometric optics overlooks the wave nature of light. Light inconsistent with longitudinal
More informationSTUDY OF FEMTOSECOND LASER BEAM FOCUSING IN DIRECT-WRITE SYSTEM
MSc in Photonics Universitat Politècnica de Catalunya (UPC) Universitat Autònoma de Barcelona (UAB) Universitat de Barcelona (UB) Institut de Ciències Fotòniques (ICFO) PHOTONICSBCN http://www.photonicsbcn.eu
More informationCanalization of Sub-wavelength Images by Electromagnetic Crystals
Progress In Electromagnetics Research Symposium 2005, Hangzhou, China, August 22-26 37 Canalization of Sub-wavelength Images by Electromagnetic Crystals P. A. Belov 1 and C. R. Simovski 2 1 Queen Mary
More informationAs a partial differential equation, the Helmholtz equation does not lend itself easily to analytical
Aaron Rury Research Prospectus 21.6.2009 Introduction: The Helmhlotz equation, ( 2 +k 2 )u(r)=0 1, serves as the basis for much of optical physics. As a partial differential equation, the Helmholtz equation
More informationOPTI 511L Fall A. Demonstrate frequency doubling of a YAG laser (1064 nm -> 532 nm).
R.J. Jones Optical Sciences OPTI 511L Fall 2017 Experiment 3: Second Harmonic Generation (SHG) (1 week lab) In this experiment we produce 0.53 µm (green) light by frequency doubling of a 1.06 µm (infrared)
More informationJitter measurement by electro-optical sampling
Jitter measurement by electro-optical sampling VUV-FEL at DESY - Armin Azima S. Duesterer, J. Feldhaus, H. Schlarb, H. Redlin, B. Steffen, DESY Hamburg K. Sengstock, Uni Hamburg Adrian Cavalieri, David
More informationEffect of cross-phase modulation on supercontinuum generated in microstructured fibers with sub-30 fs pulses
Effect of cross-phase modulation on supercontinuum generated in microstructured fibers with sub-30 fs pulses G. Genty, M. Lehtonen, and H. Ludvigsen Fiber-Optics Group, Department of Electrical and Communications
More informationControl of the filamentation distance and pattern in long-range atmospheric propagation
Control of the filamentation distance and pattern in long-range atmospheric propagation Shmuel Eisenmann, Einat Louzon, Yiftach Katzir, Tala Palchan, and Arie Zigler Racah Institute of Physics, Hebrew
More information