Generating laser pulses and spectroscopy using the temporal talbot effect

Abstract

A method of generating laser pulses (1) includes: creating a circulating light field in resonator device (11) having resonator length L and an intra-cavity dispersion and configured for supporting light field resonator modes, and generating a pulse train of laser pulses (1) by a mode-locking mechanism. Laser pulses (1) are generated with a repetition frequency and provide a frequency comb with carrier frequency .sub.o and comb modes in frequency space. The intra-cavity dispersion is selected such that round trip phases have a dependency on frequency according to $\left(\right)=m\left(\sqrt{1+4\frac{-{}_0}{m{}_r}}-1\right)+\frac{L}{c}{}_0$
wherein m is an integer providing effective repetition rate (m.sub.r) in combination with mode spacing .sub.r at optical carrier frequency (.sub.o), and the mode-locking mechanism provides a coupling of the resonator modes whereby frequency difference (n=.sub.n+1.sub.n) between neighboring mode frequencies (.sub.n, .sub.n+1) is a linear function of mode frequency number n. Furthermore, a spectroscopy method for investigating a sample, a laser pulse source apparatus and a spectroscopy apparatus are described.

Claims

1. A method of generating laser pulses, comprising the steps of creating a circulating light field in a resonator device having a resonator length L and an intra-cavity dispersion and being configured for supporting a plurality of resonator modes of the light field, and generating a pulse train of the laser pulses by a mode-locking mechanism, wherein the laser pulses provide a frequency comb with a carrier frequency .sub.o and plurality of comb modes in frequency space, wherein the intra-cavity dispersion is selected such that round trip phases have a dependency on frequency according to $\left(\right)=m\left(\sqrt{1+4\frac{-{}_0}{m{}_r}}-1\right)+\frac{L}{c}{}_0$ wherein m is an integer that gives an effective repetition rate m.sub.r of the laser pulses in combination with a mode spacing .sub.r at the optical carrier frequency .sub.o, and the mode-locking mechanism provides a coupling of the resonator modes such that a frequency difference (.sub.n=.sub.n+1.sub.n) between neighboring mode frequencies (.sub.n, .sub.n+1) is a linear function of an integer mode frequency number n.

2. The method according to claim 1, wherein the mode frequency .sub.n with the mode frequency number n is given by ${}_n={}_0+\left(n+\frac{n^2}{m}\right){}_r.$

3. The method according to claim 2, wherein the intra-cavity dispersion is selected such that the k-th derivative of the comb mode phase at the carrier frequency .sub.o is given by ${}_{{}_0}^{\left(k\right)}={\left(-1\right)}^{k+1}\frac{2^k\left(2k-3\right)!!}{m^{k-1}{}_r^k}$ wherein k is the order of intra-cavity dispersion.

4. The method according to claim 1, wherein the frequency difference between neighboring comb frequencies (.sub.n=.sub.n+1.sub.n) is in a radio frequency range.

5. The method according to claim 1, wherein the intra-cavity dispersion is set with at least one fiber Bragg grating, at least one intracavity prism and/or at least one intracavity grating included in the resonator device.

6. The method according to claim 1, wherein the resonator device is a fiber ring laser.

7. A spectroscopy method for obtaining a spectral response of a sample, comprising the steps of generating a pulse train of laser pulses with a method according to claim 1, applying the laser pulses on the sample under investigation, detecting the laser pulses with a detector device, analyzing a detector signal of the detector device for obtaining beat signals created by the comb modes of the pulse train of laser pulses, and determining the spectral response of the sample from the beat signals.

8. The spectroscopy method according to claim 7, further comprising the steps of detecting a reference portion of the pulse train of laser pulses without an application on the sample with the detector device, and analyzing a reference detector signal of the detector device for obtaining reference beat signals created by the comb modes of the reference portion of the pulse train of laser pulses, wherein the spectral response of the sample is determined from the beat signals and the reference beat signals.

9. The spectroscopy method according to claim 7, wherein the detector device comprises at least one photodiode.

10. A laser pulse source apparatus, being configured for generating laser pulses, comprising a resonator device having a resonator length L and an intra-cavity dispersion and being configured for supporting a plurality of resonator modes of a circulating light field, and a mode-locking mechanism being arranged for generating the laser pulses providing a frequency comb with carrier frequency .sub.o and a plurality of comb modes in frequency space, wherein the intra-cavity group velocity dispersion is selected such that round trip phases have a dependency on frequency according to $\left(\right)=m\left(\sqrt{1+4\frac{-{}_0}{m{}_r}}-1\right)+\frac{L}{c}{}_0$ wherein m is an integer that gives an effective repetition rate m.sub.r of the laser pulses in combination with a mode spacing .sub.r at the optical carrier frequency .sub.o, and the mode-locking mechanism is arranged for providing a coupling of the resonator modes such that the frequency difference (.sub.n=.sub.n+1.sub.n) between neighboring comb frequencies .sub.n, .sub.n+1 is a linear function of an integer mode frequency number n.

11. The laser pulse source apparatus according to claim 10, having at least one of the features the mode frequency (.sub.n) with the mode frequency number n is given by ${}_n={}_0+\left(n+\frac{n^2}{m}\right){}_r,$ the intra-cavity dispersion is selected such that the k-th derivative of the comb mode phase at the carrier frequency (.sub.o) is given by ${}_{{}_0}^{\left(k\right)}={\left(-1\right)}^{k+1}\frac{2^k\left(2k-3\right)!!}{m^{k-1}{}_r^k}$ wherein k is the order of intra-cavity dispersion, and the frequency difference between neighboring comb frequencies (.sub.n=.sub.n+1.sub.n) is included in a radio frequency range.

12. The laser pulse source apparatus according to claim 10, wherein the resonator device includes at least one fiber Bragg grating, at least one intracavity prism and/or at least one intracavity grating being arranged for setting the intra-cavity dispersion.

13. The laser pulse source apparatus according to claim 10, wherein the resonator device is a fiber ring laser.

14. A spectroscopy apparatus being configured for obtaining a spectral response of a sample under investigation, comprising a laser pulse source apparatus according to claim 10, a sample holder being configured for accommodating the sample and applying the laser pulses on the sample, a detector device being configured for detecting the laser pulses, a spectrum analyzer device being configured for analyzing a detector signal of the detector device for obtaining beat signals created by the comb modes of the pulse train of laser pulses, and a calculation device being configured for determining the spectral response of the sample from the beat signals.

15. The spectroscopy apparatus according to claim 14, further comprising a beam splitting device being arranged for directing a reference portion of the laser pulses without an application on the sample to the detector device, wherein the spectrum analyzer device is configured for analyzing a reference detector signal of the detector device and for obtaining reference beat signals created by the comb modes of the reference portion of the pulse train of laser pulses, and the calculation device is configured for determining the spectral response of the sample from the beat signals and the reference beat signals.

16. The spectroscopy apparatus according to claim 14, wherein the detector device comprises at least one photodiode.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) Further details and advantages of the invention are described in the following with reference to the attached drawings, which show in:

(2) FIG. 1: a schematic illustration of a laser pulse source device according to a preferred embodiment of the invention;

(3) FIG. 2: a schematic illustration of a spectroscopy apparatus according to a preferred embodiment of the invention

(4) FIG. 3: an exemplary illustration of the temporal evolution of the light field amplitude of a periodic light field created in a resonator device according to the invention;

(5) FIG. 4: a graphical illustration of frequencies of comb modes of a Talbot frequency comb created with the method of generating laser pulses according to the invention;

(6) FIG. 5: a graphical illustration of the change of the spectral phase generating a periodic light field according to the invention.

DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS

(7) Features of the invention are described in the following with particular reference to the creation of the Talbot frequency comb using a fiber ring laser. The invention is not restricted to this embodiment but rather can be implemented with other types of laser resonators. Details of designing and operating the laser resonator are not described as far as they are known as such from prior art.

(8) According to the embodiment of FIG. 1, the laser pulse source apparatus 10 for generating and outputting laser pulses 1 comprises a fiber ring laser. The resonator device (resonator cavity) of the laser pulse source apparatus 10 is provided by a ring of an optical fiber (gain fiber), including a pump light combiner 12, in particular a wavelength division multiplexer (WDM), a circulator 13, a fiber Bragg grating section (FBG section) 14, an optical isolator 15 and an output coupler 16. The pump light combiner 12 is arranged for coupling pump light from a pump source 30 into the fiber and creating a circulating light field 2. With the circulator 13, the FBG section 14 is coupled with the fiber for adjusting the intra-cavity dispersion of the laser pulse source apparatus 10. For smaller values m and hence larger dispersion, several, e.g. two three or more FBG sections can be included via one or more circulators in the resonator device. The fiber comprises a gain material, like e.g. Ytterbium or Erbium doped fiber material. A mode-locking mechanism, e.g. based on nonlinear effects such as nonlinear polarization rotation or based on Kerr mode-locking is provided by the fiber and the FBG section 14 for coupling resonator modes such that the laser pulses 1 represent a frequency comb as described below. The output coupler 16 is arranged for coupling laser pulses 1 light out of the resonator device (fiber) 11, e.g. for an application in a spectroscopy apparatus 20 (see FIG. 2).

(9) In the case of Talbot frequency comb, the mode-locking mechanism is effectively m times smaller than the conventional lasers. Therefore either strong nonlinear effects or matching the required dispersion (equation (6)) (see below) with higher fidelity (compared with conventional frequency combs) is provided to enforce mode-locking in the Talbot frequency comb. Intra-cavity dispersion is introduced through the FBG section 14 that can be designed with very large values for the group velocity dispersion and precise values for the higher order dispersions. To manufacture the FBG section 14, a photo sensitive fiber may be directly written with a UV laser. Up to the 6th order dispersion is commercially available.

(10) According to the embodiment of FIG. 2, the spectroscopy apparatus 20 includes the inventive laser pulse source apparatus 10, e.g. according to FIG. 1, a sample holder 21 for accommodating the sample 3, a detector device 22 for detecting the laser pulses 1 after an interaction with the sample, a spectrum analyzer device 23 for analyzing a detector signal of the detector device 22 and for obtaining beat signals created by the comb modes of the pulse train of laser pulses 1, and a calculation device 24 for determining the spectral response of the sample 3 from the beat signals. Analyzing the beat signals can be implemented as known from conventional dual-comb spectroscopy. The calculation device 24 can be included in the spectrum analyzer device 23 and/or a main control device (not shown) of the spectroscopy apparatus. The sample holder 21 comprises e.g. a gas cell for the investigation of gaseous samples, a cuvette for accommodating liquid samples or a support platform for solid samples. The spectrometer geometry can be designed for a transmission or reflection measurement. The detector device 22 comprises a photodiode.

(11) Between the laser pulse source apparatus 10 and the sample holder 21, a beam splitter 25 is optionally arranged. The beam splitter 25 directs a portion of the laser pulses 1 as reference light 4 to a reference detector device 26, which comprises another photodiode. Preferably, the detector devices 22, 26 have the same type of detectors and the same operations conditions. Detector signals from the reference detector device 26 are received by the spectrum analyzer device 23 for obtaining reference beat signals created by the comb modes of the pulse train of laser pulses 1 without an interaction with the sample 3. Comparing the reference beat signals with the beat signals in the detector signal from detector device 22 allow the correction of comb mode amplitude fluctuations.

(12) In the following, creating the Talbot frequency comb by mode-locking of the resonator modes of the laser pulse source apparatus 10 is described. Mode-locking is provided by adjusting an intra-cavity dispersion such that the modes of the resonator cavity have a mode spectrum e.g. according to

(13) $\begin{array}{cc}{}_n={}_0+\left(n+\frac{n^2}{m}\right){}_r& \left(1\right)\end{array}$

(14) Here, the modes are numbered around the optical carrier frequency .sub.0 with integers n=1, 2, 3, . . . and .sub.r is the mode spacing at the optical carrier frequency (.sub.o).

(15) Contrary to the conventional regularly spaced frequency comb [1], there is a quadratic term in n which leads to a non-equally spaced comb of RF modes (comb with linearly changing spacing) with mode spacing as follows:

(16) $\begin{array}{cc}{}_n={}_{n+1}-{}_n={}_r\left(1+\frac{2n+1}{m}\right)& \left(2\right)\end{array}$

(17) Since m is assumed to be much larger than 1 the mode spacings between the upper and lower modes relative to the optical carrier frequency .sub.o are approximately .sub.r (or an average of spacing to higher and lower mode). These RF modes are the result of beating between adjacent optical modes and can be seen in the power spectrum of the laser output. Higher order mode beatings like .sub.n+2.sub.n etc. are separated in the power spectrum by .sub.r.

(18) This is similar to the harmonics of the repetition rate in a conventional frequency comb. The mode spacing at the carrier frequency .sub.0 is given approximately by .sub.r for large m. It becomes the spacing for all modes for m.fwdarw. for which (1) turns into a conventional frequency comb [1].

(19) However, neither .sub.r nor .sub.0 are the usual repetition frequency and offset frequency. Nevertheless they can be measured in very much the same way as for a regular frequency comb (see self-referencing, described below). Besides representing an all new mode locking regime, the interesting aspect of (1) is that each mode beating uniquely belongs to one particular pair of modes. For example the RF signal at .sub.r (1+1/m) belongs to the beating between .sub.0 and .sub.1 and so on. Hence an RF spectrum recorded with a photo detector, e.g. 22 in FIG. 2, and a radio frequency spectrum analyzer, e.g. 23 in FIG. 2, directly displays a scaled down version of the optical spectrum of the laser pulse source apparatus 10.

(20) By placing a sample 3 between the laser pulse source apparatus 10 and the detector device 22 and recording the change of the RF spectrum one gets the absorption spectrum of the sample 3. This is similar to a dual frequency comb setting with a linearly increasing spacing between the modes of the two frequency combs [3, 4], albeit with a single laser avoiding problems due to the relative jitter of the combs.

(21) Depending on the magnitude of m, the mode spacing nominally becomes negative for n<m/2. Physically this means that the corresponding spectral region possesses a negative group velocity. While this is possible in principle, this is excluded in practical applications of the invention, in particular for obtaining a reasonable laser design. Accordingly, preferably, it is assumed that the active modes (modes contributing to the gain) of the laser are limited to n>m/2.

(22) To see how the laser spectrum are forced to the modes defined by equation (1), the electric field E(t), e.g. 2 in FIG. 1, at a fixed point inside the cavity is computed as follows. Assuming that the modes oscillate with some complex amplitudes a.sub.n, E(t) is obtained according to:

(23) $\begin{array}{cc}E\left(T\right)=E_o{e}^{-i{}_{0}t}\underset{n=-}{\overset{+}{.Math.}}{a}_n{e}^{-i\left(n+\frac{n^2}{m}\right){}_{r}t}& \left(3\right)\end{array}$

(24) Generally, this cannot represent a stable pulse in the time domain. However, assuming that m is an exact integer, the pulse will revive up to a phase factor after the time T

(25) $\begin{array}{cc}\begin{array}{c}E\left(t+T\right)=E_o{e}^{-i{}_0\left(t+T\right)}\underset{n}{.Math.}{a}_n{e}^{-i\left(n+\frac{n^2}{m}\right){}_{r}t-2i\left(\mathrm{mn}+n^2\right)}\\ =E_o{e}^{-i{}_0\left(t+T\right)}\underset{n}{.Math.}{a}_n{e}^{-i\left(n+\frac{n^2}{m}\right){}_{r}t}\\ =e^{-i2m{}_0/{}_r}E\left(t\right)\end{array}& \left(4\right)\end{array}$

(26) The revival time is the m multiple of the cavity round trip group delay measured at the mode with n=0, i.e. at .sub.0.

(27) FIG. 3 shows an example of the time dependency of the power |E(t)|.sup.2 at a fixed point inside the laser cavity according to equation (3) normalized to the time averaged power. The recurrence coefficient is m=10.sup.6, the carrier frequency .sub.0=310.sup.6.sub.r and the amplitudes follow a Gaussian distribution centered at .sub.0:a.sub.n=e.sup.(n/1000,000).sup.2. After m cavity round trips the pulse reassembles with a peak power of the initial pulse. With these parameters the mode beatings equation (2) are separated by 200 Hz. Accordingly, like in a conventional mode locked laser, the peak power enhancement over the time averaged power is roughly given by the number of active modes. However, in contrast to the latter, the large peak power occurs not every cavity round trip but only every m-th cavity round trip. The usual Kerr lens mode locking mechanism might be used to enforce an integer m by reduced loss of the high peak intensity pulse.

(28) In the case of the Talbot frequency comb, that large peak intensity occurs only every m-th round trip. Therefore, a strong self-amplitude modulation of the Kerr effect is used for successful mode locking.

(29) To find the required dispersion that results in the mode spectrum, equation (1) is resolved for n:

(30) $\begin{array}{cc}n=\frac{m}{2}\left(\sqrt{1+4\frac{{}_n-{}_0}{m{}_r}}-1\right)& \left(5\right)\end{array}$

(31) Like in any other laser with cavity length L, the round trip phase (w) at frequency has to fulfill the boundary condition:

(32) 0$\begin{array}{cc}\left(\right)=2n+\frac{L}{c}{}_0& \left(6\right)\\ =m\left(\sqrt{1+4\frac{-{}_0}{m{}_r}}-1\right)+\frac{L}{c}{}_0& \left(7\right)\end{array}$

(33) Since .sub.0 is the resonant mode with n=0, the last term has to be added to obtain the total round trip phase. Without loss of generality it is assumed in equation (1) that the parameter .sub.0 is the center of the emitted spectrum (see FIG. 5). Using equation (7) and computing the derivatives at .sub.0 the dispersion preferably provide for generating the mode spacing of equation (1) is obtained:

(34) $\begin{array}{cc}{}^{}\left({}_0\right)=-\frac{4}{m{}_r^2}& \left(8\right)\\ {}^{}\left({}_0\right)=+\frac{24}{m^2{}_r^3}& \left(9\right)\\ {}^{}\left({}_0\right)=-\frac{240}{m^3{}_r^4}\mathrm{.Math.}& \left(10\right)\\ {}_{{}_0}^{\left(k\right)}={\left(-1\right)}^{k+1}\frac{2^k\left(2k-3\right)!!}{m^{k-1}{}_r^k}& \left(11\right)\end{array}$

(35) For a practical laser pulse source apparatus 10, the requirements on the dispersion are quite extreme but not impossible (see FIGS. 1 and 5). These requirements are mitigated by large values of m, i.e. a long pulse revival time (for a given .sub.r). However this could lead to a reduced Kerr effect and hence weaken the mode locking mechanism. Once the laser is set up for a particular value of m, it will be reproduced every time it put in the mode locked condition.

(36) With the example parameters of FIGS. 3 and 5 equation (11) provides
(.sub.0)=3.210.sup.7fs.sup.2, (.sub.0)=3.010.sup.8fs.sup.3, .sup.4(.sub.0)=4.810.sup.9fs.sup.4
etc.

(37) Rather than this expansion one might use the first term in equation (7) directly to compute the required dispersion. Besides that the FBG section 14 also compensates the higher order dispersion of the remaining components. To estimate the required length of the FBG section 14 for obtaining the required dispersion, the difference of the round trip phase delay for the two ends of the spectrum using equation (7) is calculated and the same difference of a cavity without dispersion is subtracted
=(.sub.0+/2)(.sub.0/2)

(38) In this equation, is the spectral width of the spectral envelope of the laser pulses. This phase difference divided by 2 and multiplied by the carrier wavelength =2c/.sub.0=1 m gives the path length difference that needs to be added for the two colors by the FBG section 14. Since the light travels twice through the FBGs section 14, the actual length is half this value. With the values m=10.sup.6; .sub.r=2*100 MHz; .sub.0=2*300 THz; .sub.=2*10 THz, e.g. z=4.2 cm is obtained. The real length might then also depend on the requirements for the reflectivity. Fiber lasers generally come with a large optical gain so that it may be possible to compromise on that parameter. The design of the laser pulse source apparatus 10 could be a tradeoff between large m (=low dispersion) and small m (=stronger mode locking). To start operation at a very large value of m it may also conceivable to include an intracavity modulator that mimics all or parts of the temporal envelope shown in FIG. 3.

(39) FIG. 4 represents frequencies of the modes of the Talbot frequency comb according to equation (1) shown as the curve A. The dashed part belongs to the negative mode spacing section (i.e. negative group velocity) not considered here.

(40) Curve A defines the dispersion properties of the cavity whose expansion is shown in equation (11). The vertex of the curve at [m/2; .sub.0m.sub.r/4] and can be chosen without restriction by selecting m and .sub.r. The active modes, i.e. the laser spectrum, is assumed to be centered at .sub.0. It covers a certain range in .sub.n and n-space (rectangular area). The curvature of the parabola reflects the required group velocity dispersion which can be minimized by large values of m and .sub.r in accordance with equation (11).

(41) FIG. 5 shows comparing the exact round trip phase given by equations (5) and (6) with the expansion given by equation (11) for different order with m=10.sup.6 and .sub.r=2100 MHz. With dispersion compensation up to .sup.(6)(), the phase mismatch increases to 0.77 rad at the edges of the spectrum (width 10 THz), i.e. about 0.12 free spectral ranges. This mismatch is compensated by the Kerr effect like in a conventional mode locked laser.

(42) Self-Referencing of the Talbot Frequency Comb

(43) For self-referencing the two parameters of the Talbot frequency comb .sub.r and .sub.0 need to be measured and ideally stabilized. The recurrence index m is assumed to be known. One might get an estimate of it and then fix it to be an integer, by measuring the recurrence time and compare it to the cavity length. A more reliable method would be to measure a known optical frequency with a self-referenced Talbot frequency comb and then identify the proper m compatible with that measurement. This is the same method often applied with conventional frequency combs to determine the correct mode number.

(44) The parameter .sub.r can be determined by observing the mode beating dependence on n as expressed by equation (2), i.e. by the second order mode differences:

(45) $\begin{array}{cc}\frac{2}{m}{}_r={}_{n+1}-{}_n& \left(12\right)\end{array}$

(46) In practical terms this frequency is readily generated by mixing of the RF modes and can then be locked to a precise reference frequency such as an atomic clock by feeding back on the cavity length and presumably also on the pump laser power. In that sense .sub.r is determined almost as simple as the repetition rate of a regular frequency comb.

(47) The second parameters of the Talbot comb, .sub.0 can be measured in very much the same way as done with the carrier envelope offset frequency of a regular comb [1], i.e. with an f-2f interferometer. A part from the red side of the Talbot comb with mode number n.sub.1 is frequency doubled and superimposed the blue part of mode number n.sub.2. According to equation (1) the generated beat notes have the frequencies:

(48) $\begin{array}{cc}2{}_{n_1}-{}_{n_2}={}_0-\left(2n_2-n_1+\frac{2n_2^2-n_1^2}{m}\right){}_r& \left(13\right)\end{array}$

(49) The condition for a signal in the radio frequency domain is that out of the combinations of integers in the bracketed term there is one, that is large enough to multiply .sub.r all the way up to the optical frequency .sub.0. For m.fwdarw., this condition is identical of the comb spanning an optical octave. If the combs bandwidth is sufficient there many combinations of integers that fulfill this requirement, i.e. several radio frequency beat notes may be taken as .sub.0. Again this is very similar to regular frequency combs where the offset frequency is only determined modulo the repetition rate. Which of the beat notes is taken for .sub.0 does not matter as long as the mode numbering is adapted to that choice. Frequency doubling the Talbot comb however will generate even more frequencies as assumed in equation (13), as described with reference to FIG. 1.

(50) The features of the invention disclosed in the above description, the drawings and the claims can be of significance individually, in combination or sub-combination for the implementation of the invention in its different embodiments.