## Theory for upconversion of incoherent images |

Optics Express, Vol. 20, Issue 2, pp. 1475-1482 (2012)

http://dx.doi.org/10.1364/OE.20.001475

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### Abstract

Upconversion of images is a generic method for shifting the spectral content of entire images. A comprehensive theory for upconversion of incoherent light images is presented and compared against experiments. In particular we consider the important case for upconversion of infinity corrected light. We show that the spatial resolution for upconversion of incoherent light images is better than for the corresponding coherent image upconversion case. The fundamental differences between upconversion of coherent and incoherent images are investigated theoretically and experimentally. The theory includes the general case of upconversion using *TEM _{nm}* modes.

© 2012 OSA

## 1. Introduction

12. J. S. Dam, C. Pedersen, and P. Tidemand-Lichtenberg, “High-resolution two-dimensional image upconversion of incoherent light,” Opt. Lett. **35**(22), 3796–3798 (2010). [CrossRef] [PubMed]

4. R. W. Boyd and C. H. Townes, “An infrared upconverter for astronomical imaging,” Appl. Phys. Lett. **31**(7), 440–442 (1977). [CrossRef]

13. S. Baldelli, “Sensing: Infrared image upconversion,” Nat. Photonics **5**(2), 75–76 (2011). [CrossRef]

14. C. Pedersen, E. Karamehmedović, J. S. Dam, and P. Tidemand-Lichtenberg, “Enhanced 2D-image upconversion using solid-state lasers,” Opt. Express **17**(23), 20885–20890 (2009). [CrossRef] [PubMed]

14. C. Pedersen, E. Karamehmedović, J. S. Dam, and P. Tidemand-Lichtenberg, “Enhanced 2D-image upconversion using solid-state lasers,” Opt. Express **17**(23), 20885–20890 (2009). [CrossRef] [PubMed]

14. C. Pedersen, E. Karamehmedović, J. S. Dam, and P. Tidemand-Lichtenberg, “Enhanced 2D-image upconversion using solid-state lasers,” Opt. Express **17**(23), 20885–20890 (2009). [CrossRef] [PubMed]

*TEM*modes. Examples of upconversion with higher order

_{nm}*TEM*modes are presented showing the predictive strength of the derived theoretical model. In particular we discuss how such modes may be used to optimize resolution.

_{nm}## 2. Theory using a *TEM*_{00} mode

3. J. F. Weller and R. A. Andrews, “Resolution measurements in parametric upconversion of images,” Opt. Quantum Electron. **2**(3), 171–176 (1970). [CrossRef]

9. W. Chiou, “Geometric optics theory of parametric image upconversion,” J. Appl. Phys. **42**(5), 1985–1993 (1971). [CrossRef]

*f*, from the object plane. Each plane wave is mixed inside the non-linear crystal with a Gaussian laser beam, producing an upconverted field at

*λ*

_{3}cropped by the Gaussian beam. This upconverted light is then Fourier transformed by a second lens,

*f*

_{1}, to the image plane producing an upconverted Gaussian spot in the image plane. As will be described in the following, this Gaussian spot is the point spread function of the image upconversion system.

**17**(23), 20885–20890 (2009). [CrossRef] [PubMed]

*f*

_{1}. Note that the spectral radiance,

*L*, is conserved in (lossless) geometrical optics. Thus, the spectral radiance in the image,

*L*, is conserved from the spectral radiance at the output of the non-linear crystal,

_{Image}*L*. In the setup, a position in the image plane (

_{SFG}*x, y*) corresponds to an angle (

*x / f*

_{1},

*y / f*

_{1}) at the output of the nonlinear crystal. Likewise, angles (

*θ*,

*ϕ*) in the image plane relate to positions (–

*θ f*

_{1}, –

*ϕ f*

_{1}) in the SFG plane, Eq. (2).

*L*is the spectral radiance at the object plane and is assumed to be composed of incoherent point sources. Point sources are by definition spherical emitters, and thus

_{Object}*L*will only be a function of position and wavelength. Consequently, the radiance at the crystal input plane

_{Object}*L*

_{2}

*is constant in transverse position, i.e. a plane wave.*

_{-f}*L*(

_{Image}*x, y, θ, ϕ, λ*) in the image plane and the object plane is found as detailed in Eq. (3).

_{3}*ε*

_{0}is the vacuum permeability,

*c*is the speed of light in vacuum,

*d*is the effective nonlinear coefficient of the crystal,

_{eff}*l*is the length of the crystal,

*n*is the refractive index corresponding to wavelength

_{i}*λ*.

_{i}*k*is the

_{i}*k*-vector corresponding to the three waves.

*θ*,

*ϕ*) do not enter the expression for

*L*, since we assume that light is emitted uniformly in all directions. Consequently,

_{object}*L*is independent of transverse position (plane wave).

_{2-f}*L*, into an intensity distribution,

_{Image}*I*and to include diffraction caused by the Gaussian cropping inside the nonlinear crystal.

_{Image}*θ*,

*ϕ*), is performed. Since the upconverted light from a single point in the object plane is Gaussian distributed when exiting the upconversion crystal, the diffraction can be described by a Fourier transform of the Gaussian intensity distribution. This gives the point spread function. Note, that this step is analog to the imaging process through a Gaussian aperture. Thus, the blurring of the image can be accurately described by a Gaussian convolution, Eq. (4).

*P*is the power of the laser beam.

_{Gauss}**17**(23), 20885–20890 (2009). [CrossRef] [PubMed]

*w*. The mixing laser beam size exists only as a part of the normalized convolution function which relates to resolution of the image. Since the convolution function approaches a delta function as

_{0}*w*increases, the conclusion is clear: The upconverted intensity is independent of beam waist size

_{0}*w*of the mixing laser field. The convolution function can be regarded as the imaging point spread function (PSF). The optical transfer function (OTF) is the Fourier transform of the PSF. This means that the laser beam shape inside the crystal defines the OTF.

_{0}*f*, except as a magnification factor of the image. This is also in contrast to the coherent case [14

**17**(23), 20885–20890 (2009). [CrossRef] [PubMed]

*λ*, being smaller than

_{3}*w*. Beyond this limit, the upconversion still works, but the modeling of the imaging properties require a more comprehensive theory to be developed. Furthermore, the phase matching condition, Δ

_{0}/ l*k*, is a function of propagation angles (

*x / f*

_{1},

*y / f*

_{1}) and refractive indices. The angular dependence of Δ

*k*is of second order, which shows that close to the optical axis there will only be small variations from Δ

*k*= 0. At larger angles (the edges of the image) the optimally upconverted wavelength will change. For a 10 mm long PP-LN crystal the typical angular acceptance (intra-crystal) is about 1°, outside this solid angle the phase-matched wavelength will change significantly.

**17**(23), 20885–20890 (2009). [CrossRef] [PubMed]

*e*

^{−2}diameter is

*QE*) is found for phase matched light travelling along the center of the nonlinear crystal, Eq. (5).

_{max}12. J. S. Dam, C. Pedersen, and P. Tidemand-Lichtenberg, “High-resolution two-dimensional image upconversion of incoherent light,” Opt. Lett. **35**(22), 3796–3798 (2010). [CrossRef] [PubMed]

*d*= 11 pm/V (PP:KTP). This difference can be explained by imperfect periodic poling, since the poling period is quite short (7.2 µm) and the crystal quite thick (1 mm).

_{eff}## 3. Comparison of upconversion of coherent and incoherent images

**17**(23), 20885–20890 (2009). [CrossRef] [PubMed]

**17**(23), 20885–20890 (2009). [CrossRef] [PubMed]

12. J. S. Dam, C. Pedersen, and P. Tidemand-Lichtenberg, “High-resolution two-dimensional image upconversion of incoherent light,” Opt. Lett. **35**(22), 3796–3798 (2010). [CrossRef] [PubMed]

**35**(22), 3796–3798 (2010). [CrossRef] [PubMed]

**17**(23), 20885–20890 (2009). [CrossRef] [PubMed]

## 4. Upconversion of incoherent light using higher order *TEM*_{nm} modes

_{nm}

*TEM*mode in the upconversion process yields the following result for the upconverted image,

_{nm}*H*is

_{j}*j*’th order Hermite polynomial. Analogous to Eq. (4) the right hand side of the convolution is the point spread function when using a

*TEM*mode.

_{nm}**17**(23), 20885–20890 (2009). [CrossRef] [PubMed]

*TEM*

_{01}in Fig. 4.

*TEM*

_{03}gives a significantly lower limit for the width of the line source. A slightly thicker line source would make the image copies overlap. This experiment can also be interpreted as a direct image of the point spread function.

*TEM*

_{01}mode. The resemblance between Fig. 5(a) the experimental image and Fig. 5(b) the theoretical image using incoherent theory is very good, showing many detailed features, underlining the predictive strength of the proposed theory. Figure 5(c) on the other hand, which has been calculated using the coherent theory, shows how the asymmetric electric field of the mixing laser, results in edge detection in the image.

## 5. Discussion and outlook

15. M. J. Missey, V. Dominic, L. E. Myers, and R. C. Eckardt, “Diffusion-bonded stacks of periodically poled lithium niobate,” Opt. Lett. **23**(9), 664–666 (1998). [CrossRef] [PubMed]

16. J. Hellström, V. Pasiskevicius, H. Karlsson, and F. Laurell, “High-power optical parametric oscillation in large-aperture periodically poled KTiOPO(4),” Opt. Lett. **25**(3), 174–176 (2000). [CrossRef] [PubMed]

## 6. Conclusion

## References and links

1. | J. E. Midwinter, “Image conversion from 1.6 µ to the visible in lithium niobate,” Appl. Phys. Lett. |

2. | J. Warner, “Spatial resolution measurements in up-conversion from 10.6 m to the visible,” Appl. Phys. Lett. |

3. | J. F. Weller and R. A. Andrews, “Resolution measurements in parametric upconversion of images,” Opt. Quantum Electron. |

4. | R. W. Boyd and C. H. Townes, “An infrared upconverter for astronomical imaging,” Appl. Phys. Lett. |

5. | R. A. Andrews, “IR image parametric up-conversion,” IEEE J. Quantum Electron. |

6. | A. H. Firester, “Image Upconversion: Part III,” J. Appl. Phys. |

7. | J. Falk and W. B. Tiffany, “Theory of parametric upconversion of thermal images,” J. Appl. Phys. |

8. | K. F. Hulme and J. Warner, “Theory of thermal imaging using infrared to visible image up-conversion,” Appl. Opt. |

9. | W. Chiou, “Geometric optics theory of parametric image upconversion,” J. Appl. Phys. |

10. | F. Devaux, A. Mosset, E. Lantz, S. Monneret, and H. Le Gall, “Image Upconversion from the Visible to the UV Domain: Application to Dynamic UV Microstereolithography,” Appl. Opt. |

11. | P. M. Vaughan and R. Trebino, “Optical-parametric-amplification imaging of complex objects,” Opt. Express |

12. | J. S. Dam, C. Pedersen, and P. Tidemand-Lichtenberg, “High-resolution two-dimensional image upconversion of incoherent light,” Opt. Lett. |

13. | S. Baldelli, “Sensing: Infrared image upconversion,” Nat. Photonics |

14. | C. Pedersen, E. Karamehmedović, J. S. Dam, and P. Tidemand-Lichtenberg, “Enhanced 2D-image upconversion using solid-state lasers,” Opt. Express |

15. | M. J. Missey, V. Dominic, L. E. Myers, and R. C. Eckardt, “Diffusion-bonded stacks of periodically poled lithium niobate,” Opt. Lett. |

16. | J. Hellström, V. Pasiskevicius, H. Karlsson, and F. Laurell, “High-power optical parametric oscillation in large-aperture periodically poled KTiOPO(4),” Opt. Lett. |

**OCIS Codes**

(110.3080) Imaging systems : Infrared imaging

(110.6820) Imaging systems : Thermal imaging

(190.7220) Nonlinear optics : Upconversion

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: December 15, 2011

Manuscript Accepted: December 19, 2011

Published: January 9, 2012

**Citation**

Jeppe Seidelin Dam, Christian Pedersen, and Peter Tidemand-Lichtenberg, "Theory for upconversion of incoherent images," Opt. Express **20**, 1475-1482 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-2-1475

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### References

- J. E. Midwinter, “Image conversion from 1.6 µ to the visible in lithium niobate,” Appl. Phys. Lett.12(3), 68–70 (1968). [CrossRef]
- J. Warner, “Spatial resolution measurements in up-conversion from 10.6 m to the visible,” Appl. Phys. Lett.13(10), 360–362 (1968). [CrossRef]
- J. F. Weller and R. A. Andrews, “Resolution measurements in parametric upconversion of images,” Opt. Quantum Electron.2(3), 171–176 (1970). [CrossRef]
- R. W. Boyd and C. H. Townes, “An infrared upconverter for astronomical imaging,” Appl. Phys. Lett.31(7), 440–442 (1977). [CrossRef]
- R. A. Andrews, “IR image parametric up-conversion,” IEEE J. Quantum Electron.6(1), 68–80 (1970). [CrossRef]
- A. H. Firester, “Image Upconversion: Part III,” J. Appl. Phys.41(2), 703–709 (1970). [CrossRef]
- J. Falk and W. B. Tiffany, “Theory of parametric upconversion of thermal images,” J. Appl. Phys.43(9), 3762–3769 (1972). [CrossRef]
- K. F. Hulme and J. Warner, “Theory of thermal imaging using infrared to visible image up-conversion,” Appl. Opt.11(12), 2956–2964 (1972). [CrossRef] [PubMed]
- W. Chiou, “Geometric optics theory of parametric image upconversion,” J. Appl. Phys.42(5), 1985–1993 (1971). [CrossRef]
- F. Devaux, A. Mosset, E. Lantz, S. Monneret, and H. Le Gall, “Image Upconversion from the Visible to the UV Domain: Application to Dynamic UV Microstereolithography,” Appl. Opt.40(28), 4953–4957 (2001). [CrossRef] [PubMed]
- P. M. Vaughan and R. Trebino, “Optical-parametric-amplification imaging of complex objects,” Opt. Express19(9), 8920–8929 (2011). [CrossRef] [PubMed]
- J. S. Dam, C. Pedersen, and P. Tidemand-Lichtenberg, “High-resolution two-dimensional image upconversion of incoherent light,” Opt. Lett.35(22), 3796–3798 (2010). [CrossRef] [PubMed]
- S. Baldelli, “Sensing: Infrared image upconversion,” Nat. Photonics5(2), 75–76 (2011). [CrossRef]
- C. Pedersen, E. Karamehmedović, J. S. Dam, and P. Tidemand-Lichtenberg, “Enhanced 2D-image upconversion using solid-state lasers,” Opt. Express17(23), 20885–20890 (2009). [CrossRef] [PubMed]
- M. J. Missey, V. Dominic, L. E. Myers, and R. C. Eckardt, “Diffusion-bonded stacks of periodically poled lithium niobate,” Opt. Lett.23(9), 664–666 (1998). [CrossRef] [PubMed]
- J. Hellström, V. Pasiskevicius, H. Karlsson, and F. Laurell, “High-power optical parametric oscillation in large-aperture periodically poled KTiOPO(4),” Opt. Lett.25(3), 174–176 (2000). [CrossRef] [PubMed]

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