Abstract

Two-dimensional chromatic aberrations are characterized by a single-shot scheme based on a simultaneous measurement of chromatically diversified focal spots. The chromatic diversity is introduced by a 2-D grating with holographic defocus terms. The chromatic aberrations in the beam are either subtracted or added by the additional known chromatic aberrations in the grating, depending on the diffraction order. By analyzing the asymmetry in the size of diffracted focal spots, input beam chromatic aberrations can be deduced. Theoretical discussions and experimental results are presented.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2016 (3)

2015 (1)

2014 (1)

2008 (2)

2006 (2)

2005 (1)

2003 (1)

2002 (2)

2000 (2)

1993 (1)

Aasen, M. D.

Akturk, S.

Alessi, D. A.

Arnold, C. L.

Bahk, S.-W.

Begishev, I. A.

Bergkoetter, M.

M. Bergkoetter and J. R. Fienup, “Phase retrieval with linear chromatic dispersion,” in Imaging and Applied Optics 2016, OSA Technical Digest (online) (Optical Society of America, 2016), paper CT4C.5.
[Crossref]

Blanchard, P. M.

Borot, A.

G. Pariente, V. Gallet, A. Borot, O. Gobert, and F. Quéré, “Space-time characterization of ultra-intense femtosecond laser beams,” Nat. Photonics 10(8), 547–553 (2016).
[Crossref]

Britten, J. A.

Bromage, J.

Dorrer, C.

Fienup, J. R.

J. R. Fienup, “Phase-retrieval algorithms for a complicated optical system,” Appl. Opt. 32(10), 1737–1746 (1993).
[Crossref] [PubMed]

M. Bergkoetter and J. R. Fienup, “Phase retrieval with linear chromatic dispersion,” in Imaging and Applied Optics 2016, OSA Technical Digest (online) (Optical Society of America, 2016), paper CT4C.5.
[Crossref]

Fisher, D. J.

Gabolde, P.

Gallet, V.

G. Pariente, V. Gallet, A. Borot, O. Gobert, and F. Quéré, “Space-time characterization of ultra-intense femtosecond laser beams,” Nat. Photonics 10(8), 547–553 (2016).
[Crossref]

Gobert, O.

G. Pariente, V. Gallet, A. Borot, O. Gobert, and F. Quéré, “Space-time characterization of ultra-intense femtosecond laser beams,” Nat. Photonics 10(8), 547–553 (2016).
[Crossref]

Greenaway, A. H.

Gu, X.

Guo, C.

Haefner, C.

Harth, A.

Heuck, H.-M.

H.-M. Heuck, P. Neumayer, T. Kühl, and U. Wittrock, “Chromatic aberration in petawatt-class lasers,” Appl. Phys. B 84, 421–428 (2006).
[Crossref]

Hill, E. M.

Jain, P.

P. Jain and J. Schwiegerling, “RGB Shack-Hartmann wavefront sensor,” J. Mod. Opt. 55(4–5), 737–748 (2008).
[Crossref]

Kalb, A.

Kasper, A.

G. Pretzler, A. Kasper, and K. Witte, “Angular chirp and tilted light pulses in CPA lasers,” Appl. Phys. B 70, 1–9 (2000).
[Crossref]

Kosik, E. M.

Kotur, M.

Kühl, T.

H.-M. Heuck, P. Neumayer, T. Kühl, and U. Wittrock, “Chromatic aberration in petawatt-class lasers,” Appl. Phys. B 84, 421–428 (2006).
[Crossref]

L’Huillier, A.

Mileham, C.

Miranda, M.

Neumayer, P.

H.-M. Heuck, P. Neumayer, T. Kühl, and U. Wittrock, “Chromatic aberration in petawatt-class lasers,” Appl. Phys. B 84, 421–428 (2006).
[Crossref]

Nguyen, H. T.

Pariente, G.

G. Pariente, V. Gallet, A. Borot, O. Gobert, and F. Quéré, “Space-time characterization of ultra-intense femtosecond laser beams,” Nat. Photonics 10(8), 547–553 (2016).
[Crossref]

Pretzler, G.

G. Pretzler, A. Kasper, and K. Witte, “Angular chirp and tilted light pulses in CPA lasers,” Appl. Phys. B 70, 1–9 (2000).
[Crossref]

Quéré, F.

G. Pariente, V. Gallet, A. Borot, O. Gobert, and F. Quéré, “Space-time characterization of ultra-intense femtosecond laser beams,” Nat. Photonics 10(8), 547–553 (2016).
[Crossref]

Roides, R. G.

Rosso, P. A.

Rudawski, P.

Schwiegerling, J.

P. Jain and J. Schwiegerling, “RGB Shack-Hartmann wavefront sensor,” J. Mod. Opt. 55(4–5), 737–748 (2008).
[Crossref]

Stoeckl, C.

Storm, M.

Trebino, R.

Walmsley, I. A.

Waxer, L. J.

Witte, K.

G. Pretzler, A. Kasper, and K. Witte, “Angular chirp and tilted light pulses in CPA lasers,” Appl. Phys. B 70, 1–9 (2000).
[Crossref]

Wittrock, U.

H.-M. Heuck, P. Neumayer, T. Kühl, and U. Wittrock, “Chromatic aberration in petawatt-class lasers,” Appl. Phys. B 84, 421–428 (2006).
[Crossref]

Woods, S. C.

Zuegel, J. D.

Appl. Opt. (4)

Appl. Phys. B (2)

G. Pretzler, A. Kasper, and K. Witte, “Angular chirp and tilted light pulses in CPA lasers,” Appl. Phys. B 70, 1–9 (2000).
[Crossref]

H.-M. Heuck, P. Neumayer, T. Kühl, and U. Wittrock, “Chromatic aberration in petawatt-class lasers,” Appl. Phys. B 84, 421–428 (2006).
[Crossref]

J. Mod. Opt. (1)

P. Jain and J. Schwiegerling, “RGB Shack-Hartmann wavefront sensor,” J. Mod. Opt. 55(4–5), 737–748 (2008).
[Crossref]

J. Opt. Soc. Am. A (1)

Nat. Photonics (1)

G. Pariente, V. Gallet, A. Borot, O. Gobert, and F. Quéré, “Space-time characterization of ultra-intense femtosecond laser beams,” Nat. Photonics 10(8), 547–553 (2016).
[Crossref]

Opt. Express (4)

Opt. Lett. (3)

Other (1)

M. Bergkoetter and J. R. Fienup, “Phase retrieval with linear chromatic dispersion,” in Imaging and Applied Optics 2016, OSA Technical Digest (online) (Optical Society of America, 2016), paper CT4C.5.
[Crossref]

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Figures (7)

Fig. 1
Fig. 1 (a) Example of quadratically distorted 2-D grating (actual grid density is higher); (b) setup introducing chromatic diversity
Fig. 2
Fig. 2 Ray-traced spot diagrams at the detector plane. Blue and red correspond to shorter and longer wavelengths, respectively.
Fig. 3
Fig. 3 The presence of group delay breaks the symmetry of focal spot dispersion in the opposite orders. (a) The originally symmetric dispersed focal spots in (−1,1) and (1,−1) orders (represented by blue diagonal ellipses) are shortened in (−1,1) order and lengthened in (1,−1) order (red ellipses). The relative differences in this dimensional change are used to derive linear group delay coefficients. (b) The defocused and horizontally dispersed focal spots of a square beam in (−1,0) and (1,0) orders are shown as blue boxes at the extremes of the spectrum and the connecting lines representing the tip of the boxes in the intermediate spectral range. The inner boxes correspond to shorter wavelength. The quadratic group delay increases the size of the outer box of the (−1,0) order and the inner box of the (1,0) order whereas it decreases the size of the inner box of the (−1,0) order and the outer box of the (1,0) order. The opposite trend is possible in both (a) and (b).
Fig. 4
Fig. 4 Calculation of dynamic range based on ray tracing of (a) β and (b) γ. The red and blue lines correspond to ideal and actual responses. The vertical dashed lines correspond to ±|β0| and ±2|γ0|λ0λ, respectively.
Fig. 5
Fig. 5 Effect of different beam shapes and spectra on the RGD coefficient estimation. S4, S20 correspond to the 4th- and 20th-order super-Gaussian spectra in the case of (a) no aberrations and (b) 0.3-waves peak-to-valley aberrations.
Fig. 6
Fig. 6 (a) Experimental setup (A:square apodizer, F:band-pass filter, W:wedge, L1/L2: lenses, G: 2D grating, D: achromatic lens, SLED:superluminescent light-emitting diode). (b) camera image in logarithmic scale.
Fig. 7
Fig. 7 (a) Two-dimensional representation of the measured pulse front tilt delay coefficients (red dots) against the expected values (black dots). The dashed black circle represents the trajectory of possible values that can be introduce by the in-plane rotation of the wedge. The black circle and dots are shifted by a constant offset defined to be same as the average of the measurements at all angles. (b) retrieved narrowband wavefront.

Tables (1)

Tables Icon

Table 1 Sensitivity of PFTD error on alignment error. Δβ: error in the PFTD coefficient; Δtg, Δxg, and Δθg: tilt, position, and in-plane rotation error of the grating, respectively; Δtl, Δxl: tilt and position error of the focusing lens, respectively.

Equations (26)

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a ( x , y , Ω ) = a 0 [ ( 1 + m x Ω ) x + s x Ω , ( 1 + m y Ω ) y + s y Ω ] × [ 1 + ( p x x + p y y + q r 2 ) Ω ] ,
ϕ ( x , y , Ω ) = ϕ 0 [ ( 1 + m x Ω ) x + s x Ω , ( 1 + m y Ω ) y + s y Ω ] × [ β x x + β y y + γ r 2 ) Ω ] ,
Δ ϕ ( x , y , Ω ) = ( β x x + β y y + γ r 2 ) Ω ,
Ψ n x , n y = 2 π n x ( x Λ x + h x r 2 ) + 2 π n y ( y Λ y + h y r 2 ) ,
Δ ϕ = Δ ϕ + Ψ n x , n y = { ( β x + n x β 0 ) x + ( β y n y β 0 ) y + [ γ + ( n x + n y ) γ 0 ] r 2 } Ω + 2 π λ ( λ 0 2 π Ψ n x , n y ) ,
β x = β x + n x β 0 ,
β y = β y + n y β 0 ,
γ = γ + ( n x + n y ) γ 0 .
Δ x 1 , 1 = c F 0 Δ λ λ 0 ( β 0 + β x ) ,
Δ x 1 , 1 = c F 0 Δ λ λ 0 ( β 0 β x ) ,
β x = 1 2 λ 0 c F 0 Δ λ ( Δ x 1 , 1 Δ x 1 , 1 ) .
β y = 1 2 λ 0 c F 0 Δ λ ( Δ y 1 , 1 Δ y 1 , 1 ) .
D 1 , 0 = 2 c F 0 D 0 [ λ λ 0 ( γ 0 + γ ) γ 2 c L γ 0 γ λ λ 0 ( λ λ 0 1 ) ] ,
D 1 , 0 = 2 c F 0 D 0 [ λ λ 0 ( γ 0 γ ) + γ 2 c L γ 0 γ λ λ 0 ( λ λ 0 1 ) ] ,
Δ D 1 , 0 D 1 , 0 ( λ max ) D 1 , 0 ( λ min ) = 2 c F 0 D 0 Δ λ λ 0 [ ( γ 0 + γ ) 2 c L γ 0 γ ] ,
Δ D 1 , 0 D 1 , 0 ( λ max ) D 1 , 0 ( λ min ) = 2 c F 0 D 0 Δ λ λ 0 [ ( γ 0 γ ) 2 c L γ 0 γ ] ,
γ = 1 4 λ 0 c F 0 D 0 Δ λ ( Δ D 1 , 0 Δ D 1 , 0 ) .
I 1 , 1 ( x , y ) d x d y max I 1 , 1 ( x , y ) d y = η c F 0 λ 0 S ( λ ) d λ max S ( λ ) ( β 0 + β x ) ,
I 1 , 1 ( x , y ) d x d y max I 1 , 1 ( x , y ) d y = η c F 0 λ 0 S ( λ ) d λ max S ( λ ) ( β 0 β x ) ,
β x = β 0 ( Δ x 1 , 1 ¯ Δ x 1 , 1 ¯ ) / ( Δ x 1 , 1 ¯ + Δ x 1 , 1 ¯ ) ,
β y = β 0 ( Δ y 1 , 1 ¯ Δ y 1 , 1 ¯ ) / ( Δ y 1 , 1 ¯ + Δ y 1 , 1 ¯ ) ,
D ˜ ( x ) = I ( x , y ) d y max x I ( x , y ) ,
( x x ¯ 1 , 0 ) D ˜ 1 , 0 ( x ) d x D ˜ 1 , 0 ( x ) d x = κ 2 D 0 c F 0 1 λ 0 S ( λ ) d λ max S ( λ ) ( γ 0 + γ 2 c L γ 0 γ ) ,
( x x ¯ 1 , 0 ) D ˜ 1 , 0 ( x ) d x D ˜ 1 , 0 ( x ) d x = κ 2 D 0 c F 0 1 λ 0 S ( λ ) d λ max S ( λ ) ( γ 0 γ 2 c L γ 0 γ ) ,
γ = γ 0 ( Δ D ˜ 1 , 0 Δ D ˜ 1 , 0 ) ( Δ D ˜ 1 , 0 + Δ D ˜ 1 , 0 ) 1 [ 1 + 2 c L γ 0 ( Δ D ˜ 1 , 0 Δ D ˜ 1 , 0 ) ( Δ D ˜ 1 , 0 + Δ D ˜ 1 , 0 ) ] ,
γ measured = γ incident + γ lens .

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