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Speckle beam-oriented schlieren technique

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An advanced background-oriented schlieren (BOS) method, named as the speckle beam-oriented schlieren technique, was newly developed to measure the distribution of refraction angles in transparent media. A speckle pattern is generated by passing a coherent laser beam through a holographic diffuser, a pinhole, and a lens, generating a collimated background image that is projected directly onto the image sensors. Since the intensity of the background image is maintained at a high level, this method is, in principle, useful for diagnosing fast and/or low signal-to-noise phenomena, such as high-temperature gasses with radiation emission. Moreover, by splitting the background beam into two imaging paths with different focal lengths, the refraction angles can be measured for a schlieren object with uncertain location, and the depth position of the refraction angles can be resolved. This technique was demonstrated by measuring the refraction angle and the depth position distribution in a sonic jet with different injected locations.

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Availability of data and material

The datasets analyzed during this study available from the corresponding author on reasonable request.


f :


\(f_{\text{even}}\), \(f_{\text{odd}}\) :

Even/odd number component of f

\(f_{\text{L}}\) :

Focal length of lens L

G :

Gladstone–Dale constant

\(I_{i,j}\) :


\(\hat{I}_{i,j}\) :

Zero-mean normalized image

\(I_{{{\text{L}}1}}\), \(I_{{{\text{L}}2}}\) :

Images by L1 and L2

\(I_{\text{raw}}\) :

Raw image

\(i\), \(j\) :

Pixel position on image plane

\(i_{{{\text{L}}1, 0}}\), \(j_{{{\text{L}}1,0}}\) :

Reference position for image by L1

\(i_{{{\text{L}}2, 0}}\), \(j_{{{\text{L}}2,0}}\) :

Reference position for image by L2

\(i_{\text{M}}\), \(j_{\text{M}}\) :

Matched position

\(i_{\text{sub}}\), \(j_{\text{sub}}\) :

Sub-pixel estimated position

\(i_{\text{T}}\), \(j_{\text{T}}\) :

Template originated position

L1, L2 :


\(l_{\text{pix}}\) :

Length corresponding to one pixel

R i,j :


\(T_{m,n}\) :

Template image

\(n_{0}\) :

Reference refractive index

m, n :

Pixel position on template image

\(S_{\text{T}}\) :

Template size

\(\varvec{u}_{{{\text{L}}1}}\), \(\varvec{v}_{{{\text{L}}1}}\) :

Vectors for correcting image by L1

\(\varvec{u}_{{{\text{L}}2}}\), \(\varvec{v}_{{{\text{L}}2}}\) :

Vectors for correcting image by L2

\(x\), \(y\), \(z\) :


\(z_{\text{c}}\) :

Correction parameter

\(z_{\text{eff}}\) :

Effective refracted position

\(z_{\text{jet}}\) :

Jet injected position

\(z_{\text{r}}\) :

Position of refraction

\(z_{\text{s}}\) :

Center position of schlieren object

\(\Delta z\) :

Distance between two focal planes

\(\varvec{\delta}\) :


\(\varvec{\delta}_{1}\), \(\varvec{\delta}_{2}\) :

Displacement at focal plane of L1, L2

\(\delta_{1}\), \(\delta_{2}\) :

Displacement defined as Eq. (7)

\(\varvec{\delta}_{{1,{\text{r}}}}\), \(\varvec{\delta}_{{2,{\text{r}}}}\) :

Displacement by refraction

\(\varvec{\delta}_{\text{err}}\) :

Error at displacement

\(\varvec{\varepsilon}\) :

Refractive angle

\(\varvec{\varepsilon}_{\text{ray}}\) :

Angle of light ray

\(\lambda\) :


\(\rho\) :



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This work was supported by JSPS Grant-in-Aid for Scientific Research: Grant no. 18H03813.

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Correspondence to Yusuke Nakamura.

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The codes used during this study available from the corresponding author on reasonable request.

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Nakamura, Y., Suzuki, T., Kinefuchi, K. et al. Speckle beam-oriented schlieren technique. Exp Fluids 62, 13 (2021).

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