Abstract

Schlieren systems with a coherent light source were investigated by the Fourier optics technique. The imaging properties of the systems with various cutoff filters were studied. Systems with a graded piecewise linear filter and a Gaussian step function convolution (graded) filter are considered, demonstrating that the image can be approximated by the geometrical-optics theory of conventional schlieren systems. A nonlinear phase contribution was estimated, allowing for the measurement of strong phase objects. Within the framework of linear approximation the results are described by the phase derivative point-spread function, introduced in this paper as the schlieren point-spread function. In addition, modification of the Lopez cutoff filter is proposed, demonstrating its superiority over the piecewise linear and the Gaussian step convolution filters. Simulations of coherent schlieren imaging as well as phase derivative measurements were performed. Finally, the imaging properties of the schlieren systems with the different filters are compared.

© 2004 Optical Society of America

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References

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  1. A. K. Oppenheim, P. A. Urtiew, F. J. Weinberg, “On the use of laser light sources in schlieren-interferometer systems,” Proc. R. Soc. London Ser. A 291, 279–290 (1966).
    [CrossRef]
  2. D. W. Holder, R. J. North, “Schlieren methods,” No. 31 of Notes on Applied Science (National Physical Laboratory, Middlesex, UK, 1963).
  3. G. S. Settles, Schlieren and Shadowgraph Techniques (Springer-Verlag, New York, 2001).
    [CrossRef]
  4. C. A. Lopez, “Numerical simulation of a schlieren system from the Fourier optics perspective,” paper AIAA-94-2618, presented at the Eighteenth Aerospace Ground Testing Conference, Colorado Springs, Colo., 20–23 June 1994 (American Institute of Aeronautics and Astronautics, Reston, Va., 1994).
  5. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).
  6. B. G. Boon, Signal Processing Using Optics (Oxford University, London, 1998).
  7. B. Zakharin, J. Stricker, “Fourier optics analysis of schlieren images,” in Proceedings of the Ninth Millenium International Symposium on Flow Visualization, I. Grant, G. M. Carlomagno, eds. (n.p., 2000), available at http;//www.ode-web.demon.co.uk/9misfv .
  8. L. M. Weinstein, “An improved large-field focusing schlieren system,” paper AIAA-91-0567, presented at the 29th Aerospace Sciences Meeting, Reno, Nev., 7–10 January 1991 (American Institute of Aeronautics and Astronautics, Reston, Va., 1991).
  9. R. N. Bracewell, Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, New York, 1978).

1966 (1)

A. K. Oppenheim, P. A. Urtiew, F. J. Weinberg, “On the use of laser light sources in schlieren-interferometer systems,” Proc. R. Soc. London Ser. A 291, 279–290 (1966).
[CrossRef]

Boon, B. G.

B. G. Boon, Signal Processing Using Optics (Oxford University, London, 1998).

Bracewell, R. N.

R. N. Bracewell, Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, New York, 1978).

Carlomagno, G. M.

B. Zakharin, J. Stricker, “Fourier optics analysis of schlieren images,” in Proceedings of the Ninth Millenium International Symposium on Flow Visualization, I. Grant, G. M. Carlomagno, eds. (n.p., 2000), available at http;//www.ode-web.demon.co.uk/9misfv .

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

Grant, I.

B. Zakharin, J. Stricker, “Fourier optics analysis of schlieren images,” in Proceedings of the Ninth Millenium International Symposium on Flow Visualization, I. Grant, G. M. Carlomagno, eds. (n.p., 2000), available at http;//www.ode-web.demon.co.uk/9misfv .

Holder, D. W.

D. W. Holder, R. J. North, “Schlieren methods,” No. 31 of Notes on Applied Science (National Physical Laboratory, Middlesex, UK, 1963).

Lopez, C. A.

C. A. Lopez, “Numerical simulation of a schlieren system from the Fourier optics perspective,” paper AIAA-94-2618, presented at the Eighteenth Aerospace Ground Testing Conference, Colorado Springs, Colo., 20–23 June 1994 (American Institute of Aeronautics and Astronautics, Reston, Va., 1994).

North, R. J.

D. W. Holder, R. J. North, “Schlieren methods,” No. 31 of Notes on Applied Science (National Physical Laboratory, Middlesex, UK, 1963).

Oppenheim, A. K.

A. K. Oppenheim, P. A. Urtiew, F. J. Weinberg, “On the use of laser light sources in schlieren-interferometer systems,” Proc. R. Soc. London Ser. A 291, 279–290 (1966).
[CrossRef]

Settles, G. S.

G. S. Settles, Schlieren and Shadowgraph Techniques (Springer-Verlag, New York, 2001).
[CrossRef]

Stricker, J.

B. Zakharin, J. Stricker, “Fourier optics analysis of schlieren images,” in Proceedings of the Ninth Millenium International Symposium on Flow Visualization, I. Grant, G. M. Carlomagno, eds. (n.p., 2000), available at http;//www.ode-web.demon.co.uk/9misfv .

Urtiew, P. A.

A. K. Oppenheim, P. A. Urtiew, F. J. Weinberg, “On the use of laser light sources in schlieren-interferometer systems,” Proc. R. Soc. London Ser. A 291, 279–290 (1966).
[CrossRef]

Weinberg, F. J.

A. K. Oppenheim, P. A. Urtiew, F. J. Weinberg, “On the use of laser light sources in schlieren-interferometer systems,” Proc. R. Soc. London Ser. A 291, 279–290 (1966).
[CrossRef]

Weinstein, L. M.

L. M. Weinstein, “An improved large-field focusing schlieren system,” paper AIAA-91-0567, presented at the 29th Aerospace Sciences Meeting, Reno, Nev., 7–10 January 1991 (American Institute of Aeronautics and Astronautics, Reston, Va., 1991).

Zakharin, B.

B. Zakharin, J. Stricker, “Fourier optics analysis of schlieren images,” in Proceedings of the Ninth Millenium International Symposium on Flow Visualization, I. Grant, G. M. Carlomagno, eds. (n.p., 2000), available at http;//www.ode-web.demon.co.uk/9misfv .

Proc. R. Soc. London Ser. A (1)

A. K. Oppenheim, P. A. Urtiew, F. J. Weinberg, “On the use of laser light sources in schlieren-interferometer systems,” Proc. R. Soc. London Ser. A 291, 279–290 (1966).
[CrossRef]

Other (8)

D. W. Holder, R. J. North, “Schlieren methods,” No. 31 of Notes on Applied Science (National Physical Laboratory, Middlesex, UK, 1963).

G. S. Settles, Schlieren and Shadowgraph Techniques (Springer-Verlag, New York, 2001).
[CrossRef]

C. A. Lopez, “Numerical simulation of a schlieren system from the Fourier optics perspective,” paper AIAA-94-2618, presented at the Eighteenth Aerospace Ground Testing Conference, Colorado Springs, Colo., 20–23 June 1994 (American Institute of Aeronautics and Astronautics, Reston, Va., 1994).

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

B. G. Boon, Signal Processing Using Optics (Oxford University, London, 1998).

B. Zakharin, J. Stricker, “Fourier optics analysis of schlieren images,” in Proceedings of the Ninth Millenium International Symposium on Flow Visualization, I. Grant, G. M. Carlomagno, eds. (n.p., 2000), available at http;//www.ode-web.demon.co.uk/9misfv .

L. M. Weinstein, “An improved large-field focusing schlieren system,” paper AIAA-91-0567, presented at the 29th Aerospace Sciences Meeting, Reno, Nev., 7–10 January 1991 (American Institute of Aeronautics and Astronautics, Reston, Va., 1991).

R. N. Bracewell, Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, New York, 1978).

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Figures (6)

Fig. 1
Fig. 1

Schematic of a general schlieren system: S, light source; L1, L2, L3, lenses; SF, schlieren filter.

Fig. 2
Fig. 2

Schlieren filters transparency functions: a 0 is the filter parameter.

Fig. 3
Fig. 3

Two sinc functions comprising integral I 3(y) [Eq. (34)]. The solid curve represents sinc[(a 0f)(ξ - y)], and the dashed curve represents sinc {1/[∂φ(y)/π ∂y](ξ - y)}: (A) |∂φ/∂y| < πa 0f, (B) |∂φ/∂y| > πa 0f, (C) |∂φ/∂y| = πa 0f.

Fig. 4
Fig. 4

Simulation of the PWL filter for a phase object, φ(ξ) = 150 sinc(ξ2/π). (A) Exact intensity calculation [Eq. (10)]. (B) The solid curve represents the exact phase derivative and the dashed curve is its linear approximation [Eq. (26)]. (C) The solid curve represents the exact phase derivative, and the dashed curve is its nonlinear approximation [Eq. (31)].

Fig. 5
Fig. 5

Demonstration of the saturation effect for the φ = A sin ξ phase object: (A), (B), (C), and (D) show A = 15, 600, 750, and 1500, respectively. (A1), (B1), (C1), (D1): —, phase object; ----, phase object derivative; (A2), (A3), (B2), (B3), (C3), (C2), (C3), (D2), (D3): —, precise numerical image intensity; (A2), (B2), (C2), (D2): ----, nonlinear image intensity approximation; (A3), (B3), (C3), (D3): ----, linear image intensity approximation.

Fig. 6
Fig. 6

Illustration of the imaging properties of the different filters by simulating the φ(ξ) = 12(1 + cos ξ) phase object: (A) the solid curve represents the phase of the object and the dashed curve is the phase object derivative, (B) image intensity of incoherent schlieren with the knife-edge filter, (C) image intensity of coherent schlieren with the knife-edge filter, (D) the dashed curve represents the image intensity with the Lopez filter and the solid curve is the image intensity with the ML filter, (E), (F) coherent schlieren image intensity with the PWL and GSC filters, respectively.

Tables (1)

Tables Icon

Table 1 Imaging Properties of the Coherent Light Source Schlieren with Different Filters

Equations (63)

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U0ξ- Usuexp-j 2πλfξuOξdu,
UfνFνU0ξΦν,
UiyFy-1FνU0ξΦν.
UiyU0ξFξ-1Φν,
Iiy-- OξO*ξSξS*ξ ×Kξ-yK*ξ-ydξdξ,
Sξ=- Usuexp-j kuξfdu,
Kξ=- Φνexpj kuξfdν.
UsuUs*u=δuδu.
SξS*ξ=-- UsuUs*uexp-j kuξf×expj kuξfdudu=1.
Iiy--expjφξ-φ(ξ×Kξ-yK*ξ-ydξdξ.
Φν=0ν<-a0/2ν+a0/2a0-a0/2<ν<a0/21ν>a0/2
Φν=θνrectνa0,
Kξ=- Φνexpj kvξfdν=- θνexpj kvξfdν-rectνa0×expj kvξfdν,
Kξ=λfa02δξ+jπ1ξsinca0ξλf.
I1y=--expj[φξ-φξ}δξ-y×δξ-ydξdξ,I2y=-jπ--expjφξ-φξδξ-y1ξ-y×sinca0ξ-yλfdξdξ,I3y=jπ--expjφξ-φξδξ-y1ξ-y×sinca0ξ-yλfdξdξ,I4y=1π2--expjφξ-φξ1ξ-y×sinca0ξ-yλf1ξ-ysinca0ξ-yλf×dξdξ.
I1y=1.
expjφξ-φξ1+jφξ-φξ,
φξφξ¯+φξ¯Δξ2, φξφξ¯-φξ¯Δξ2,
ξ¯=ξ+ξ2, Δξ=ξ-ξ,
φξ-φξφξ¯ Δξ.
I2y+I3y=-4π-φξ¯ξ¯sinc2a0ξ¯-yλfdξ¯.
I4y=λfπa0ϕyy2.
Iily=1-4π-φξ¯ξ¯ Pξ¯-ydξ¯,
Pξ¯=sinc2a0ξ¯λf.
Δy=λfa0.
Iily1-2λfπa0φyy,
Iiy1-2λfπa0φyy+λfπa0φyy2 =1-λfπa0φyy2.
φyyλfπa0<1, φyyλfπa0>1,
φyy=1Iiyπa0λf.
φyy<πa0λf,
φyy=1-Iiyπa0λf.
I3yj -expj[φξ-φy} 1ξ-y×sinca0ξ-yλfdξ.
I3yφyy-sinc1πφyyξ-y ×sinca0λfξ-ydξ.
φν=θνexp-πνa02=0.51+erfπνa0,
Pξ=exp-4π a0λf2ξ2,
Iiy=1-λfπa0φyy2,
-Oξξexp-j 2πνξλfdξ=j 2πνλf-Oξ ×exp-j 2πνξλfdξ.
λf Oyy=-j2πν- Oξexp-j 2πνξλfdξ×expj 2πνyλfdν.
Uiy=λfLOyy.
Uiy= jλfLexpjφyφyy,
Iiy=λfLφyy2,
Φν=2π ν-aL,
Uiy=expjφyφyyλfL-2πaLexpjφy.
Iiy=1-12πφyyλfa2,
φyyλfa<2π, φyyλfa>2π.
φyy=1Iiy2πaλf,
C=2πλfa0φyy=4fa0 Δα,
Δα=λ2πφyy.
Δαmin=a04f Cmin.
ΔyΔαλCmin4.
NL=12πϕyyλfa0,
NL=C4.
ϕξ=150 sincξ2π, -2π<ξ<2π.
ϕξ=A sin ξ, -3π<ξ<3π,
φξ0-2π<ξ<-π121+cos ξ-π<ξ<π0π<ξ<2π
Kξ=- Φνexpj kνξfdν=- θνexpj kνξfdν×-rectνa0expj kνξfdν.
Fθν=πδ2πξλf+j λf2π1ξ=λf2 δξ+j λf2π1ξ,
Kξ=λfa02δξ+jπ1ξsinca0ξλf.
I4y=1π-expjφξ1ξ-ysinca0ξ-yλfdξ2,
I4y=1π-expjφξ-φy1ξ-y×sinca0ξ-yλfdξ2.
expjφξ-φy1+jφξ-φy,
φξ-φyφyyξ-y.
I4yλfπa0φyy2.

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