## Abstract

We consider spatial shaping of broadband (either stationary or pulsed) spatially coherent light, comparing refractive, standard diffractive, and harmonic diffractive (modulo 2*πM*) elements. Considering frequency-integrated target profiles we show that, contrary to common belief, standard diffractive (*M* = 1) elements work reasonably well for, e.g., Gaussian femtosecond pulses and spatially coherent amplified-spontaneous-emission sources such as superluminescent diodes. It is also shown that harmonic elements with *M* ≥ 5 behave in essentially the same way as refractive elements and clearly outperform standard diffractive elements for highly broadband light.

© 2014 Optical Society of America

## 1. Introduction

Transformation of laser beams from the Gaussian shape to some more desirable shapes is an ever-more important problem in optics. Both freeform refractive optics and diffractive optics can be employed for this purpose [1]. It is generally believed, however, that diffractive elements work only for quasimonochromatic light whereas refractive elements can be used also for broadband illumination. Forbes et al. recently proposed an interesting idea that diffractive elements can be used for wavelength-tunable beams, at least over a limited spectral range, if the target plane is adjusted longitudinally according to the wavelength used [2]. In this paper we consider the situation where all wavelengths in a broadband spectrum are simultaneously present and show that standard diffractive elements work well over wavelength ranges extending over tens of nanometers, especially if the spectrum is symmetric. We then concentrate on so-called harmonic diffractive elements, which are generalizations of harmonic diffractive lenses [3,4]. The phase function of such an elements is a modulo 2*πM* version of the phase function of a refractive element, where the integer *M* represents the harmonic parameter (*M* = 1 for a standard diffractive element). We begin with the theory of beam shaping with general harmonic elements in a 2*F* Fourier-transform setup and then proceed to numerical analysis of elements with different harmonic orders. The analysis reveals that harmonic diffractive elements with *M* ≥ 5 produce high-quality shaped beam profiles for spectra that extend essentially over the entire visible region.

## 2. Theory of harmonic beam shaping elements

We consider the geometry of Fig. 1, assuming that the beam shaping element at *z* = 0 has a *y*-invariant transmission function *t*(*x*, *ω*); the extension to separable or rotationally symmetric transmission functions is obvious. The Fourier transform geometry is the most troublesome one if diffractive elements are used since the zeroth generalized order (which appears at frequencies other than the design frequency) is focused on-axis in the target plane and interferes strongly with the desired shaped beam as will become evident at the end of this section. The formulation given below can be straightforwardly modified to cover Fresnel-domain beam shaping problems with a finite free-space distance between the element and target planes.

Let us denote a single space–frequency-domain realization of the incident field in the plane *z* = 0 in Fig. 1 by *E*(*x*, *ω*) = *a*(*ω*)*V*(*x*, *ω*), where *a*(*ω*) is a complex random function and *V*(*x*, *ω*) is the deterministic spatial beam profile, which in general depends on frequency. After passage through a complex-amplitude transparency *t*(*x*, *ω*) and (paraxial) propagation into the Fourier plane *z* = 2*F*, the field takes on the form *E*(*u*, *ω*) = *a*(*ω*)*V*(*u*, *ω*), where

*z*= 2

*F*is given by

*S*(

*u*,

*ω*) = 〈

*E*

^{*}(

*u*,

*ω*)

*E*(

*u*,

*ω*)〉, where the sharp brackets denote ensemble averaging. Hence where

*S*

_{0}(

*ω*) = 〈

*a*

^{*}(

*ω*)

*a*(

*ω*)〉 defines the spectral density of the incident field.

We stress at this point that the results to be presented directly apply to stationary fields, for which the space–frequency domain realizations are defined in [5,6]. In addition the results apply also to non-stationary fields provided that each realization of such a field has a finite energy. These realizations can be, e.g., single pulses or finite sets of pulses in a pulse train originating from a primary source or obtained by modulating a stationary source [7, 8]. We concentrate on frequency-integrated target-plane profiles

If the field is stationary*S*(

*u*) simply equals the average intensity [6]. In the case of non-stationary fields defined above,

*S*(

*u*) is related to the time-integrated intensity

*I*(

*u*) by where and

*I*(

*u*,

*t*) is the spatiotemporal profile. Equation (4) follows from the generalized Wiener–Khintchine theorem for non-stationary fields (see Appendix A and In. [9], K. Saastamoinen et al., “Spectrally partially coherent propagation-invariant fields,” Phys. Rev. A

**80,**053804 (2009)). Hence, in this paper, we do not consider the spatiotemporal behavior of non-stationary fields at the target plane. Importantly, the spatial distribution

*S*(

*u*) is the same for stationary or non-stationary fields of any state of coherence with given

*S*

_{0}(

*ω*) and

*V*(

*x*,

*ω*), including quasi-stationary fields, fully coherent pulses, and fields with intermediate degrees of spectral coherence.

Although any spectral profile *S*_{0}(*ω*) could be used, we assume specifically that the incident field has a Gaussian spectrum

*ω*

_{0}and

*ω*

_{S}are the center frequency and the spectral width, respectively. We also assume an isodiffracting (the Rayleigh range is independent on frequency) spatial profile of the form

*w*

_{0}is the beam width at frequency

*ω*

_{0}. This form is appropriate for Gaussian beams and pulses generated in spherical-mirror resonators [10] typically used in mode-locked lasers, where the spectral dependence of the spot size is significant for ultrashort (wideband) pulses. It is also a decent approximation for superluminescent diodes, which typically generate nearly Gaussian (though anisotropic) single-spatial-mode beams, which are spatially nearly coherent but have low spectral coherence even in pulsed-mode operation.

The transmission function is of the phase-only form *t*(*x*, *ω*) = exp[i*ϕ*(*x*, *ω*)]. Considering first refractive beam shaping elements, the phase function at any frequency in the thin-element approximation is

*n*(

*ω*) is the refractive index of the material and

*ϕ*(

*x*,

*ω*

_{0}) is the phase function at

*ω*=

*ω*

_{0}, i.e., the ‘design’ phase function.

In the absence of the beam shaping element, *V*(*u*, *ω*_{0}) becomes a Gaussian function with 1/e half-width *w*_{F} = 2*cF/ω*_{0}*w*_{0} (the diffraction limit at the center frequency). Hence it is useful to define dimensionless position and frequency coordinates *X* = *x/w*_{0}, *U* = *u/w*_{F}, and Ω = *ω/ω*_{0}. Writing also the dispersion term in the form *D*(Ω) = [*n*(Ω*ω*_{0}) − 1]/[*n*(*ω*_{0}) − 1], the transmission function may be written as

_{S}=

*ω*

_{S}/

*ω*

_{0}.

Let us divert for a moment to consider the extension of the theory of generalized orders of diffractive elements [11] to the harmonic case. Ignoring the frequency dependence, we write the complex-amplitude transmission function in the form

where*α*is a constant and

*ϕ*(

*x*) is the phase function when

*α*= 1. Let us restrict the phase

*αϕ*(

*x*) into an interval [0, 2

*πM*], where

*M*is the harmonic parameter. Then

*t*(

*x*) may be expressed in the form of a generalized Fourier series where the Fourier coefficients

*α*= Ω

*D*(Ω) and hence the Fourier coefficients take on the form

*m*. At other frequencies the contributions from all orders with significant

*G*(Ω) interfere coherently.

_{m}## 3. Numerical results and interpretations

In numerical calculations we may either use Eq. (10) directly, with *ϕ*(*X*, *ω*_{0}) reduced to the interval [0, 2*πM*], or employ Eq. (17); the former approach is more efficient since the latter requires the inclusion of many generalized orders. However, as we will soon see, the generalized-order theory provides much more insight into the character of harmonic beam shaping elements.

In the examples we choose the phase function designed by the geometrical map-transform technique [12–14] for Gaussian to flat-top conversion [1, 15]:

*Q*is the ratio of the half-width of the intended flat-top profile and the diffraction-limited beam width

*w*

_{F}. High-quality flat-top profiles can be obtained only when

*Q*is several times greater than unity [1]; otherwise diffraction effects dominate. When

*Q*≫ 1, the width of the flat-top profile increases linearly with

*Q*, which is therefore treated as a convenient design parameter that should be chosen according to the needs of the application at hand. In all of our examples we choose

*Q*= 20. Finally, the dispersion data for polycarbonate (PC) taken from [16] is used in all numerical examples.

Figure 2 illustrates the chromatic behavior of standard and higher-harmonic diffractive elements. The profiles obtained at the *ω*_{0} = 2.51 × 10^{15} Hz (corresponding to *λ* = 750 nm) and certain other frequencies are shown, as well as the frequency-integrated profiles obtained by choosing *ω*_{S} = 10^{14} Hz (Ω_{S} = 0.0398). For the modulo 2*π* element, a central dip is seen for frequencies higher than the design frequency *ω*_{0}, whereas a central peak appears at frequencies below *ω*_{0}. However, when the frequency-integrated profile with the symmetric Gaussian spectrum is considered, the peaks and dips virtually cancel each other and we obtain a good approximation of a flat-top profile (apart from the ripple, which is also present for refractive elements). The chromatic effects in the beam profile are greatly reduced when the harmonic factor *M* is increases, as seen from Fig. 2(b). At *M* = 5 the dips and peaks have already disappeared and the element acts virtually as a refractive one although it is still diffractive (see Fig. 3).

The generalized-order expansion for *V*(*U*, Ω) in Eq. (17), with *G _{m}*(Ω) given by Eq. (15), provides more insight into the results shown in Fig. 2. The amplitudes of some generalized orders (with largest values of |

*G*|) are plotted for a standard diffractive element in Fig. 4(a) assuming that Ω ≠ 1. Most significantly, the zero order

_{m}*m*= 0 (which has the shape of the diffraction-limited spot) appears in the center of the target plane, causing the dips and peaks. Although each non-zero order alone produces a hight-quality flat-top, their widths vary widely. The reason for the sharp variations of the combined profiles outside the center is that the phases of generalized orders are radically different for

*M*= 1 elements. For arbitrary

*M*, all light goes to order

*m*for frequencies that satisfy Eq. (18); these values of Ω may be called resonant frequencies [4]. For large

*M*, the orders which satisfy Eq. (18) most closely have largest values of |

*G*|, and the target-plane profiles of these orders are nearly identical as shown in Fig. 4(b). Moreover, their phases become better matched with increasing

_{m}*M*(not shown), which reduces the interference effects in the generalized-order series and therefore leads to better shaped-beam quality. Finally, the efficiency of

*m*:th order has its first zeroes at spectral positions |

*M*Ω

*D*(Ω) −

*m*| =

*π*/2, i.e., the spectral width of the efficiency curve of order

*m*narrows down when

*M*increases. The zeroth order becomes insignificant for large

*M*, causing the disappearance of the dips and peaks. The combination of these reasons means that the harmonic element begins to behave essentially like a refractive one. Figure 4(b) shows that

*M*= 5 is already a ‘large’ value.

As the final example, we consider an RGB light source consisting of three discrete wavelengths *λ*_{R} = 633 nm (red), *λ*_{G} = 532 nm (green), and *λ*_{B} = 473 nm (blue). The corresponding frequencies are *ω*_{R} = 2.98 × 10^{15} Hz, *ω*_{G} = 3.54 × 10^{15} Hz, and *ω*_{B} = 3.99 × 10^{15} Hz. The relative intensities of these sources are chosen as *S*_{R} = 1, *S*_{G} = 0.6290, and *S*_{B} = 0.8177 in order to produce white in sRGB space. Figure 5 illustrates the individual profiles generated by these sources when the design wavelength is *λ* = 546 nm, as well as the frequency-integrated profile *S*(*U*) = *S*_{R}(*U*) + *S*_{G}(*U*) + *S*_{B}(*U*). Even though the spectrum in now non-symmetric and extends over most of the visible region, the standard diffractive element still works reasonably well if we consider only the frequency-integrated results shown by the black line in Fig. 5(a). As expected, the individual profiles at red and blue wavelengths are highly distorted. On the other hand, as illustrated in Fig. 5(b), the modulo *M* = 5 element produces rather high-quality flat-top profiles at each wavelength and an excellent integrated profile.

Because of highly different profile shapes at different wavelengths, the combined pattern generated by the standard diffractive element exhibits widely varying colors all over the flat-top region. The result, as would be seen by by the human eye, as illustrated in the lower part of Fig. 5(a). In the case of the modulo *M* = 5 element, the individual monochromatic profile are more identical, which reduces the color variations in the combined profile. The edges of the pattern produced by the standard diffractive element are distinctly red since the dispersion of the standard diffractive element spreads the pattern at large wavelengths more than at small wavelengths. For a purely refractive element the edges would be bluish because of the opposite sign of dispersion. In the case of the harmonic *M* = 5 element these color variations are in general reduced greatly (see the lower part of Fig. 5(b)). In fact, the effective widths of the red, green, and blue contributions are roughly the same; this behavior is reminiscent of the operation of hybrid diffractive-refractive lenses [17].

## 4. Conclusions

In conclusion, we have shown that standard diffractive beam shaping elements work well for broadband light if the spectrum is reasonably narrow and symmetric about the central frequency *ω*_{0}. Hence they can be used, e.g., for shaping pulsed femtosecond laser beams and light emitted by superluminescent diodes. In the case of highly broadband and non-symmetric spectra of, e.g., RGB or supercontinuum sources [18], low-harmonic diffractive elements with *M* > 4 are preferable, especially in color-sensitive applications.

## A Appendix

In this Appendix we show that a relation

connects the frequency-integrated and time-integrated quantities defined in Eqs. (3) and (5). The space-frequency and space-time realizations of an arbitrary non-stationary (spectrally and temporally partially coherent) field, denoted by*E*(

*u*,

*ω*) and

*E*(

*u*,

*t*), are connected by

*W*(

*u*

_{1},

*u*

_{2},

*ω*

_{1},

*ω*

_{2}) = 〈

*E*

^{*}(

*u*

_{1},

*ω*

_{1})

*E*(

*u*

_{2},

*ω*

_{2})〉 and the two-time mutual coherence function as Γ(

*u*

_{1},

*u*

_{2},

*t*

_{1},

*t*

_{2}) = 〈

*E*

^{*}(

*u*

_{1},

*t*

_{1})

*E*(

*u*

_{2},

*t*

_{2})〉, and inserting from Eq. (21) we arrive at the generalized Wiener–Khintchine theorem for non-stationary fields:

*S*(

*u*,

*ω*) =

*W*(

*u*,

*u*,

*ω*,

*ω*), is given by

*u*,

*u*,

*t*,

*t*) =

*I*(

*u*,

*t*).

## Acknowledgments

This work was supported by the Academy of Finland (project 252910)

## References and links

**1. **L. A. Romero and F. M. Dickey, “Lossless laser beam shaping,” J. Opt. Soc. Am. A **13**, 751–760 (1996). [CrossRef]

**2. **A. Forbes, F. Dickey, A. DeGama, and A. du Plessis, “Wavelength tunable laser beam shaping,” Opt. Lett. **37**, 49–51 (2012). [CrossRef] [PubMed]

**3. **D. Faklis and G. M. Morris, “Spectral properties of multiorder diffractive lenses,” Appl. Opt. **34**, 2462–2468 (1995). [CrossRef] [PubMed]

**4. **D. W. Sweeney and G. E. Sommargren, “Harmonic diffractive lenses,” Appl. Opt. **34**, 2469–2475 (1995). [CrossRef] [PubMed]

**5. **E. Wolf, “New theory of partial coherence in the space-frequency domain. Part I: Spectra and cross-spectra of steady-state sources,” J. Opt. Soc. Am. **72**, 343–351 (1982). [CrossRef]

**6. **L. Mandel and E. Wolf, *Optical Coherence and Quantum Optics* (Cambridge University Press, 1995). [CrossRef]

**7. **H. Lajunen, J. Tervo, J. Turunen, P. Vahimaa, and F. Wyrowski, Spectral coherence properties of temporally modulated stationary light sources,” Opt. Express **11**, 1894–1899 (2003). [CrossRef] [PubMed]

**8. **V. Torres-Company, H. Lajunen, and A.T. Friberg, ”Coherence theory of noise in ultrashort-pulse trains,” J. Opt. Soc. Am. B **24**, 1441–1450 (2007). [CrossRef]

**9. **K. Saastamoinen, J. Turunen, P. Vahimaa, and A. T. Friberg, “Spectrally partially coherent propagation-invariant fields,” Phys. Rev. A **80**, 053804 (2009). [CrossRef]

**10. **P. W. Milonni and J. H. Eberly, *Lasers* (Wiley, 1988), Sect. 14.7.

**11. **W.-H. Lee, “Computer-generated holograms: techniques and applications,” in *Progress in Optics*, Vol. XVI, E. Wolf, ed. (North-Holland, 1978), pp. 119–223. [CrossRef]

**12. **C. N. Kurtz, H. O. Hoadley, and J. J. dePalma, “Design and synthesis of random phase diffusers,” J. Opt. Soc. Am. **63**, 1080–1092 (1973). [CrossRef]

**13. **O. Bryngdahl, “Geometrical transformations in optics,” J. Opt. Soc. Am. **64**, 1092–1099 (1974). [CrossRef]

**14. **N. C. Roberts, “Multilevel computer-generated holograms with separable phase functions for beam shaping,” Appl. Opt. **31**, 3198 (1992).

**15. **F. M. Dickey and S. C. Holswade, “Gaussian laser beam profile shaping,” Opt. Eng. **35**, 3285–3295 (1996). [CrossRef]

**16. **S. N. Kasarova, Nina Georgieva Sultanova, Christo Dimitrov Ivanov, and Ivan Dechev Nikolov, “Analysis of the dispersion of optical plastic materials,” Opt. Mater. **29**, 1481–1490 (2007). [CrossRef]

**17. **T. Stone and N. Goerge, “Hybrid diffractive-refractive lenses and achromats,” Appl. Opt. **27**, 2960–2971 (1988) [CrossRef] [PubMed]

**18. **J. M. Dudley and J. R. Taylor, eds., *Supercontinuum Generation in Optical Fibers* (Cambridge University, 2010). [CrossRef]