Abstract

We present an in situ beam characterization technique to analyze femtosecond optical beams in a folded version of a 2f-2f setup. This technique makes use of a two-dimensional spatial light modulator (SLM) to holographically redirect radiation between different diffraction orders. This manipulation of light between diffraction orders is carried out locally within the beam. Because SLMs can withstand intensities of up to I1011W/cm2, this makes them suitable for amplified femtosecond radiation. The flexibility of the SLM was demonstrated by producing a diverse assortment of “soft apertures” that are mechanically difficult or impossible to reproduce. We test our method by holographically knife-edging and tomographically reconstructing both continuous wave and broadband radiation in transverse optical modes.

© 2011 OSA

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. Babiker, W. L. Power, and L. Allen, “Light-induced torque on moving atoms,” Phys. Rev. Lett. 73(9), 1239–1242 (1994).
    [CrossRef] [PubMed]
  2. A. Picón, J. Mompart, J. R. Vázquez de Aldana, L. Plaja, G. F. Calvo, and L. Roso, “Photoionization with orbital angular momentum beams,” Opt. Express 18(4), 3660–3671 (2010).
    [CrossRef] [PubMed]
  3. A. Alexandrescu, D. Cojoc, and E. Di Fabrizio, “Mechanism of angular momentum exchange between molecules and Laguerre-Gaussian beams,” Phys. Rev. Lett. 96(24), 243001 (2006).
    [CrossRef] [PubMed]
  4. E. Serabyn, D. Mawet, and R. Burruss, “An image of an exoplanet separated by two diffraction beamwidths from a star,” Nature 464(7291), 1018–1020 (2010).
    [CrossRef] [PubMed]
  5. J. H. Lee, G. Foo, E. G. Johnson, and G. A. Swartzlander., “Experimental verification of an optical vortex coronagraph,” Phys. Rev. Lett. 97(5), 053901 (2006).
    [CrossRef] [PubMed]
  6. J. Y. Vinet, “Thermal noise in advanced gravitational wave interferometer antennas: A comparison between arbitrary order Hermite and Laguerre Gasussian modes,” Phys. Rev. D Part. Fields Gravit. Cosmol. 82(4), 042003 (2010).
    [CrossRef]
  7. L. T. Vuong, T. D. Grow, A. Ishaaya, A. L. Gaeta, G. W. ’t Hooft, E. R. Eliel, and G. Fibich, “Collapse of optical vortices,” Phys. Rev. Lett. 96(13), 133901 (2006).
    [CrossRef] [PubMed]
  8. J. Ng, Z. Lin, and C. T. Chan, “Theory of optical trapping by an optical vortex beam,” Phys. Rev. Lett. 104(10), 103601 (2010).
    [CrossRef] [PubMed]
  9. M. A. Bandres and J. C. Gutiérrez-Vega, “Ince-Gaussian beams,” Opt. Lett. 29(2), 144–146 (2004).
    [CrossRef] [PubMed]
  10. J. Strohaber, C. Petersen, and C. J. G. J. Uiterwaal, “Efficient angular dispersion compensation in holographic generation of intense ultrashort paraxial beam modes,” Opt. Lett. 32(16), 2387–2389 (2007).
    [CrossRef] [PubMed]
  11. J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, “Laser beams: knotted threads of darkness,” Nature 432(7014), 165 (2004).
    [CrossRef] [PubMed]
  12. K. Bezuhanov, A. Dreischuh, G. G. Paulus, M. G. Schätzel, and H. Walther, “Vortices in femtosecond laser fields,” Opt. Lett. 29(16), 1942–1944 (2004).
    [CrossRef] [PubMed]
  13. I. G. Mariyenko, J. Strohaber, and C. J. G. J. Uiterwaal, “Creation of optical vortices in femtosecond pulses,” Opt. Express 13(19), 7599–7608 (2005).
    [CrossRef] [PubMed]
  14. I. Zeylikovich, H. I. Sztul, V. Kartazaev, T. Le, and R. R. Alfano, “Ultrashort Laguerre-Gaussian pulses with angular and group velocity dispersion compensation,” Opt. Lett. 32(14), 2025–2027 (2007).
    [CrossRef] [PubMed]
  15. J. Strohaber, T. D. Scarborough, and C. J. G. J. Uiterwaal, “Ultrashort intense-field optical vortices produced with laser-etched mirrors,” Appl. Opt. 46(36), 8583–8590 (2007).
    [CrossRef] [PubMed]
  16. J. A. Davis, D. M. Cottrell, J. Campos, M. J. Yzuel, and I. Moreno, “Encoding amplitude information onto phase-only filters,” Appl. Opt. 38(23), 5004–5013 (1999).
    [CrossRef] [PubMed]
  17. J. B. Bentley, J. A. Davis, M. A. Bandres, and J. C. Gutiérrez-Vega, “Generation of helical Ince-Gaussian beams with a liquid-crystal display,” Opt. Lett. 31(5), 649–651 (2006).
    [CrossRef] [PubMed]
  18. J.W. Goodman, Introduction to Fourier Optics, 2nd Ed. (McGraw-Hill,New York, l996).
  19. J. F. James, A Student’s Guide tothe Fourier Transform (Cambridge U. Press, 1995).
  20. J. M. Khosrofian and B. A. Garetz, “Measurement of a Gaussian laser beam diameter through the direct inversion of knife-edge data,” Appl. Opt. 22(21), 3406–3410 (1983).
    [CrossRef] [PubMed]
  21. O. Mendoza-Yero and M. Arronte, “Determination of Hermite Gaussian modes using moving knife-edge,” J. Phys: Conference Series 59, 497–500 (2007).
    [CrossRef]
  22. J. Soto, M. Rendón, and M. Martín, “Experimental demonstration of tomographic slit technique for measurement of arbitrary intensity profiles of light beams,” Appl. Opt. 36(29), 7450–7454 (1997).
    [CrossRef] [PubMed]
  23. S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light- theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).
    [CrossRef]

2010 (4)

A. Picón, J. Mompart, J. R. Vázquez de Aldana, L. Plaja, G. F. Calvo, and L. Roso, “Photoionization with orbital angular momentum beams,” Opt. Express 18(4), 3660–3671 (2010).
[CrossRef] [PubMed]

E. Serabyn, D. Mawet, and R. Burruss, “An image of an exoplanet separated by two diffraction beamwidths from a star,” Nature 464(7291), 1018–1020 (2010).
[CrossRef] [PubMed]

J. Y. Vinet, “Thermal noise in advanced gravitational wave interferometer antennas: A comparison between arbitrary order Hermite and Laguerre Gasussian modes,” Phys. Rev. D Part. Fields Gravit. Cosmol. 82(4), 042003 (2010).
[CrossRef]

J. Ng, Z. Lin, and C. T. Chan, “Theory of optical trapping by an optical vortex beam,” Phys. Rev. Lett. 104(10), 103601 (2010).
[CrossRef] [PubMed]

2007 (4)

2006 (4)

J. B. Bentley, J. A. Davis, M. A. Bandres, and J. C. Gutiérrez-Vega, “Generation of helical Ince-Gaussian beams with a liquid-crystal display,” Opt. Lett. 31(5), 649–651 (2006).
[CrossRef] [PubMed]

L. T. Vuong, T. D. Grow, A. Ishaaya, A. L. Gaeta, G. W. ’t Hooft, E. R. Eliel, and G. Fibich, “Collapse of optical vortices,” Phys. Rev. Lett. 96(13), 133901 (2006).
[CrossRef] [PubMed]

J. H. Lee, G. Foo, E. G. Johnson, and G. A. Swartzlander., “Experimental verification of an optical vortex coronagraph,” Phys. Rev. Lett. 97(5), 053901 (2006).
[CrossRef] [PubMed]

A. Alexandrescu, D. Cojoc, and E. Di Fabrizio, “Mechanism of angular momentum exchange between molecules and Laguerre-Gaussian beams,” Phys. Rev. Lett. 96(24), 243001 (2006).
[CrossRef] [PubMed]

2005 (1)

2004 (3)

2001 (1)

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light- theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).
[CrossRef]

1999 (1)

1997 (1)

1994 (1)

M. Babiker, W. L. Power, and L. Allen, “Light-induced torque on moving atoms,” Phys. Rev. Lett. 73(9), 1239–1242 (1994).
[CrossRef] [PubMed]

1983 (1)

’t Hooft, G. W.

L. T. Vuong, T. D. Grow, A. Ishaaya, A. L. Gaeta, G. W. ’t Hooft, E. R. Eliel, and G. Fibich, “Collapse of optical vortices,” Phys. Rev. Lett. 96(13), 133901 (2006).
[CrossRef] [PubMed]

Alexandrescu, A.

A. Alexandrescu, D. Cojoc, and E. Di Fabrizio, “Mechanism of angular momentum exchange between molecules and Laguerre-Gaussian beams,” Phys. Rev. Lett. 96(24), 243001 (2006).
[CrossRef] [PubMed]

Alfano, R. R.

Allen, L.

M. Babiker, W. L. Power, and L. Allen, “Light-induced torque on moving atoms,” Phys. Rev. Lett. 73(9), 1239–1242 (1994).
[CrossRef] [PubMed]

Arronte, M.

O. Mendoza-Yero and M. Arronte, “Determination of Hermite Gaussian modes using moving knife-edge,” J. Phys: Conference Series 59, 497–500 (2007).
[CrossRef]

Babiker, M.

M. Babiker, W. L. Power, and L. Allen, “Light-induced torque on moving atoms,” Phys. Rev. Lett. 73(9), 1239–1242 (1994).
[CrossRef] [PubMed]

Bandres, M. A.

Bentley, J. B.

Bezuhanov, K.

Burruss, R.

E. Serabyn, D. Mawet, and R. Burruss, “An image of an exoplanet separated by two diffraction beamwidths from a star,” Nature 464(7291), 1018–1020 (2010).
[CrossRef] [PubMed]

Calvo, G. F.

Campos, J.

Chan, C. T.

J. Ng, Z. Lin, and C. T. Chan, “Theory of optical trapping by an optical vortex beam,” Phys. Rev. Lett. 104(10), 103601 (2010).
[CrossRef] [PubMed]

Cojoc, D.

A. Alexandrescu, D. Cojoc, and E. Di Fabrizio, “Mechanism of angular momentum exchange between molecules and Laguerre-Gaussian beams,” Phys. Rev. Lett. 96(24), 243001 (2006).
[CrossRef] [PubMed]

Cottrell, D. M.

Courtial, J.

J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, “Laser beams: knotted threads of darkness,” Nature 432(7014), 165 (2004).
[CrossRef] [PubMed]

Davis, J. A.

Dennis, M. R.

J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, “Laser beams: knotted threads of darkness,” Nature 432(7014), 165 (2004).
[CrossRef] [PubMed]

Di Fabrizio, E.

A. Alexandrescu, D. Cojoc, and E. Di Fabrizio, “Mechanism of angular momentum exchange between molecules and Laguerre-Gaussian beams,” Phys. Rev. Lett. 96(24), 243001 (2006).
[CrossRef] [PubMed]

Dorn, R.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light- theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).
[CrossRef]

Dreischuh, A.

Eberler, M.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light- theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).
[CrossRef]

Eliel, E. R.

L. T. Vuong, T. D. Grow, A. Ishaaya, A. L. Gaeta, G. W. ’t Hooft, E. R. Eliel, and G. Fibich, “Collapse of optical vortices,” Phys. Rev. Lett. 96(13), 133901 (2006).
[CrossRef] [PubMed]

Fibich, G.

L. T. Vuong, T. D. Grow, A. Ishaaya, A. L. Gaeta, G. W. ’t Hooft, E. R. Eliel, and G. Fibich, “Collapse of optical vortices,” Phys. Rev. Lett. 96(13), 133901 (2006).
[CrossRef] [PubMed]

Foo, G.

J. H. Lee, G. Foo, E. G. Johnson, and G. A. Swartzlander., “Experimental verification of an optical vortex coronagraph,” Phys. Rev. Lett. 97(5), 053901 (2006).
[CrossRef] [PubMed]

Gaeta, A. L.

L. T. Vuong, T. D. Grow, A. Ishaaya, A. L. Gaeta, G. W. ’t Hooft, E. R. Eliel, and G. Fibich, “Collapse of optical vortices,” Phys. Rev. Lett. 96(13), 133901 (2006).
[CrossRef] [PubMed]

Garetz, B. A.

Glöckl, O.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light- theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).
[CrossRef]

Grow, T. D.

L. T. Vuong, T. D. Grow, A. Ishaaya, A. L. Gaeta, G. W. ’t Hooft, E. R. Eliel, and G. Fibich, “Collapse of optical vortices,” Phys. Rev. Lett. 96(13), 133901 (2006).
[CrossRef] [PubMed]

Gutiérrez-Vega, J. C.

Ishaaya, A.

L. T. Vuong, T. D. Grow, A. Ishaaya, A. L. Gaeta, G. W. ’t Hooft, E. R. Eliel, and G. Fibich, “Collapse of optical vortices,” Phys. Rev. Lett. 96(13), 133901 (2006).
[CrossRef] [PubMed]

Johnson, E. G.

J. H. Lee, G. Foo, E. G. Johnson, and G. A. Swartzlander., “Experimental verification of an optical vortex coronagraph,” Phys. Rev. Lett. 97(5), 053901 (2006).
[CrossRef] [PubMed]

Kartazaev, V.

Khosrofian, J. M.

Le, T.

Leach, J.

J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, “Laser beams: knotted threads of darkness,” Nature 432(7014), 165 (2004).
[CrossRef] [PubMed]

Lee, J. H.

J. H. Lee, G. Foo, E. G. Johnson, and G. A. Swartzlander., “Experimental verification of an optical vortex coronagraph,” Phys. Rev. Lett. 97(5), 053901 (2006).
[CrossRef] [PubMed]

Leuchs, G.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light- theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).
[CrossRef]

Lin, Z.

J. Ng, Z. Lin, and C. T. Chan, “Theory of optical trapping by an optical vortex beam,” Phys. Rev. Lett. 104(10), 103601 (2010).
[CrossRef] [PubMed]

Mariyenko, I. G.

Martín, M.

Mawet, D.

E. Serabyn, D. Mawet, and R. Burruss, “An image of an exoplanet separated by two diffraction beamwidths from a star,” Nature 464(7291), 1018–1020 (2010).
[CrossRef] [PubMed]

Mendoza-Yero, O.

O. Mendoza-Yero and M. Arronte, “Determination of Hermite Gaussian modes using moving knife-edge,” J. Phys: Conference Series 59, 497–500 (2007).
[CrossRef]

Mompart, J.

Moreno, I.

Ng, J.

J. Ng, Z. Lin, and C. T. Chan, “Theory of optical trapping by an optical vortex beam,” Phys. Rev. Lett. 104(10), 103601 (2010).
[CrossRef] [PubMed]

Padgett, M. J.

J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, “Laser beams: knotted threads of darkness,” Nature 432(7014), 165 (2004).
[CrossRef] [PubMed]

Paulus, G. G.

Petersen, C.

Picón, A.

Plaja, L.

Power, W. L.

M. Babiker, W. L. Power, and L. Allen, “Light-induced torque on moving atoms,” Phys. Rev. Lett. 73(9), 1239–1242 (1994).
[CrossRef] [PubMed]

Quabis, S.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light- theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).
[CrossRef]

Rendón, M.

Roso, L.

Scarborough, T. D.

Schätzel, M. G.

Serabyn, E.

E. Serabyn, D. Mawet, and R. Burruss, “An image of an exoplanet separated by two diffraction beamwidths from a star,” Nature 464(7291), 1018–1020 (2010).
[CrossRef] [PubMed]

Soto, J.

Strohaber, J.

Swartzlander, G. A.

J. H. Lee, G. Foo, E. G. Johnson, and G. A. Swartzlander., “Experimental verification of an optical vortex coronagraph,” Phys. Rev. Lett. 97(5), 053901 (2006).
[CrossRef] [PubMed]

Sztul, H. I.

Uiterwaal, C. J. G. J.

Vázquez de Aldana, J. R.

Vinet, J. Y.

J. Y. Vinet, “Thermal noise in advanced gravitational wave interferometer antennas: A comparison between arbitrary order Hermite and Laguerre Gasussian modes,” Phys. Rev. D Part. Fields Gravit. Cosmol. 82(4), 042003 (2010).
[CrossRef]

Vuong, L. T.

L. T. Vuong, T. D. Grow, A. Ishaaya, A. L. Gaeta, G. W. ’t Hooft, E. R. Eliel, and G. Fibich, “Collapse of optical vortices,” Phys. Rev. Lett. 96(13), 133901 (2006).
[CrossRef] [PubMed]

Walther, H.

Yzuel, M. J.

Zeylikovich, I.

Appl. Opt. (4)

Appl. Phys. B (1)

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light- theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).
[CrossRef]

J. Phys: Conference Series (1)

O. Mendoza-Yero and M. Arronte, “Determination of Hermite Gaussian modes using moving knife-edge,” J. Phys: Conference Series 59, 497–500 (2007).
[CrossRef]

Nature (2)

J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, “Laser beams: knotted threads of darkness,” Nature 432(7014), 165 (2004).
[CrossRef] [PubMed]

E. Serabyn, D. Mawet, and R. Burruss, “An image of an exoplanet separated by two diffraction beamwidths from a star,” Nature 464(7291), 1018–1020 (2010).
[CrossRef] [PubMed]

Opt. Express (2)

Opt. Lett. (5)

Phys. Rev. D Part. Fields Gravit. Cosmol. (1)

J. Y. Vinet, “Thermal noise in advanced gravitational wave interferometer antennas: A comparison between arbitrary order Hermite and Laguerre Gasussian modes,” Phys. Rev. D Part. Fields Gravit. Cosmol. 82(4), 042003 (2010).
[CrossRef]

Phys. Rev. Lett. (5)

L. T. Vuong, T. D. Grow, A. Ishaaya, A. L. Gaeta, G. W. ’t Hooft, E. R. Eliel, and G. Fibich, “Collapse of optical vortices,” Phys. Rev. Lett. 96(13), 133901 (2006).
[CrossRef] [PubMed]

J. Ng, Z. Lin, and C. T. Chan, “Theory of optical trapping by an optical vortex beam,” Phys. Rev. Lett. 104(10), 103601 (2010).
[CrossRef] [PubMed]

A. Alexandrescu, D. Cojoc, and E. Di Fabrizio, “Mechanism of angular momentum exchange between molecules and Laguerre-Gaussian beams,” Phys. Rev. Lett. 96(24), 243001 (2006).
[CrossRef] [PubMed]

J. H. Lee, G. Foo, E. G. Johnson, and G. A. Swartzlander., “Experimental verification of an optical vortex coronagraph,” Phys. Rev. Lett. 97(5), 053901 (2006).
[CrossRef] [PubMed]

M. Babiker, W. L. Power, and L. Allen, “Light-induced torque on moving atoms,” Phys. Rev. Lett. 73(9), 1239–1242 (1994).
[CrossRef] [PubMed]

Other (2)

J.W. Goodman, Introduction to Fourier Optics, 2nd Ed. (McGraw-Hill,New York, l996).

J. F. James, A Student’s Guide tothe Fourier Transform (Cambridge U. Press, 1995).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (9)

Fig. 1
Fig. 1

(a) Measured output power of a Michelson interferometer versus modulation depth (blue squares). The SLM was positioned in one arm of the interferometer and introduced a phase-modulation encoded as a grayscale value. The solid curve is the theoretically expected result. (b) Phase modulation retrieved from the data and theoretical curve shown in panel (a).

Fig. 2
Fig. 2

(a) Illustration of a blazed phase grating having modulation depth m and spatial period displayed on the SLM. (b) Blazing efficiency as a function of modulation depth measured as the ratio of the power in the first diffraction order to that in the zero order. Nine sets of data were obtained each having a different grating period and denoted by the number of pixels (np) used for the grating period. Each pixel is assumed to be equal to the pitch, which is 20 µm. All data sets show a peak at a grayscale value of ~100, which corresponds to a phase shift of 2π radian. The inset in panel (b) shows the efficiencies for the peak values (grayscale value of 100) demonstrating the best achieved diffraction efficiency for a grating period of 8 pixels.

Fig. 3
Fig. 3

(a) Illustration of a hologram used to create a holographic knife-edge. The solid black color on the left side of hologram denotes a constant phase modulation and the right side of the hologram is that of a blazed grating. (b) Measured power as a function of knife-edge position. The black crosses represent the measured power from a mechanical knife-edge position at the location of the SLM, and the red circles are that obtained from the holographic knife-edge. Both mechanical and holographic knife-edge measurements are in good agreement. The insets are fits of the data to theoretical curves.

Fig. 4
Fig. 4

Experimental setup. Laser radiation from either a He-Ne or Ti:sapphire laser enters the setup from the right. DL = 50 cm diverging lens, CL = 200 cm converging lens, SLM = spatial light modulator, FM = folding mirror placed a distance of f = 100 c m away from the SLM, PD = photodiode power meter head, PM = power meter. The upper left inset is an example-hologram to create a L G 2 , 2 o beam followed by an angular knife-edge.

Fig. 5
Fig. 5

Intensity profiles of a L G 2 , 2 o beam being azimuthally knife-edged. (a) This sequence of frames shows the holograms used to perform an angular knife-edge measurement with an angular step size of 90 degrees. (b) CCD images of radiation from a He-Ne source after passing through the corresponding grating in sequence (a). Each frame shows both the zero and first diffraction orders. As the area of constant phase (denoted by the blackened areas in (a)) increases, the corresponding local radiation in the first diffraction order is directed into the zero order until the radiation is gone.

Fig. 6
Fig. 6

Cartesian knife-edge measurements of the Hermite-Gaussian modes. The modality of each mode is given as the label of the panel. The black opened squares are data taken from knife-edge measurements in the x direction and the black opened circles are that in the y direction. The number of plateaus is equal to the mode number. The red curves were obtained by fitting the data with theoretical curves presented in the text. From this fit the beam size was determined.

Fig. 7
Fig. 7

Radial knife-edge measurements for an assortment of helical Laguerre-Gaussian beams. The number of plateaus is equal to the radial mode number p. The waist of the beams can be determined by fitting the data to the theoretical equations given in the text. The fits are shown by the solid red curves. Unlike the HG beams, the radial knife-edge measurements depend on both radial and azimuthal mode numbers p and l. The dependence of the curve on the azimuthal mode number can be seen by the size of the initial plateau increasing from the leftmost column to the rightmost column

Fig. 8
Fig. 8

Angular knife-edge measurements for an assortment of even LG beams having modalities as indicated. The number of plateaus is equal to twice the angular mode number 2l. The waist of the beam cannot be determined from an angular knife-edge measurement, but this measurement can give an indication of the quality of the modal lobes.

Fig. 9
Fig. 9

Tomographic reconstruction of a femtosecond LG p = 1 = 1 beam in a folded-2f setup. All images are 200-by-200 pixels and have the same vertical and horizontal scaling. The dimension of the images is given by the scale in panel (a). (a, c) Raw double-knife-edge data recorded by stepping a knife-edge in one direction (i.e., x) by a single step, completing a full knife-edge in the other direction (i.e., y), and repeating this process until finished. The raw data shown in panel (a) is that obtained by not correcting for angular dispersion in the folded-2f setup, while that in panel (c) has been corrected. Panels (b) and (d) were obtained by taking the partial derivatives (see text) of the measured double-knife-edge power.

Equations (16)

Equations on this page are rendered with MathJax. Learn more.

E holo = ( H ( ξ x ) + H ( x ξ ) q = J q ( m 2 ) e i q K x ) ψ ( x ) .
E HG n , m = N n . m E 0 w 0 w H n ( 2 x w ( z ) ) H m ( 2 y w ( z ) ) e ( r 2 / w 2 ) e i [ k r 2 2 R ( n + m + 1 ) φ G + k z ] .
P HG n ( x ) = x I HG n , m ( x , y , z ) d x d y .
P HG n ( x ) = N n . m 2 I 0 w 0 2 1 2 H m 2 ( η ) e η 2 d η x H n 2 ( ξ ) e ξ 2 d ξ = N n . m 2 I 0 w 0 2 1 2 I η I ξ ( x ) .
I ξ = ( 1 ) n x H n ( ξ ) d n d ξ n ( e ξ 2 ) d ξ .
I ξ = ( 1 ) n [ H n ( ξ ) d n 1 d ξ n 1 ( e ξ 2 ) 2 n x H n 1 ( ξ ) d n 1 d ξ n 1 ( e ξ 2 ) d ξ ] .
I ξ = ( 1 ) n [ ( 1 ) n e ξ 2 H n 1 ( ξ ) H n ( ξ ) 2 n x H n 1 ( ξ ) d n 1 d ξ n 1 ( e ξ 2 ) d ξ ] .
P HG n ( x ) = 1 2 P 0 { e ξ 2 1 π k = 1 n 2 k n ( n + 1 k ) ! H n k + 1 ( ξ ) H n k ( ξ ) + erfc ( ξ ) } .
E LG l , p = N l . p E 0 w 0 w ( 2 r w ( z ) ) l L p l ( 2 r 2 w 2 ( z ) ) e ( r 2 / w 2 ) e i l θ e i [ k r 2 2 R ( 2 p + l + 1 ) φ G + k z ] .
P LG ( r ) = 0 r 0 2 π I LG ( r ) r d r d θ .
P LG l , p ( r ) = π 2 I 0 w 0 2 N l . p 2 0 r ξ l L p l ( ξ ) L p l ( ξ ) e ξ d ξ = π 2 I 0 w 0 2 N l . p 2 I ξ ( r ) .
I ξ = 1 p ! 0 r d p d ξ p ( ξ p + l e ξ ) L p l ( ξ ) d ξ .
I ξ = 1 p ! [ L p l ( ξ ) d p 1 d ξ p 1 ( ξ p + l e ξ ) 0 r L p l ( ξ ) d p d ξ p 1 ( ξ p + l e ξ ) d ξ ] .
I ξ = 1 p ! [ ( p 1 ) ! ξ l + 1 e ξ L p l ( ξ ) L p 1 l + 1 ( ξ ) ( 1 ) 0 r L p 1 l + 1 ( ξ ) d p 1 d ξ p 1 ( ξ p + l e ξ ) d ξ ] .
P LG p l ( r ) = P 0 { e ξ k = 1 p ( p k ) ! ξ l + k L p k l + k ( ξ ) L p k + 1 l + k 1 ( ξ ) + γ ( p + l + 1 , ξ ) } .
P LG ( θ ) = P 0 { 1 θ 2 π [ 1 ± sin ( 2 l θ ) 2 l θ ] } ,

Metrics