In SPOM-NOM, we use nonlinear optical processes that are excited by a combination of two-color pulses. Such processes include nondegenerate TPEF, TPA, sum-frequency generation (SFG), FWM, SPE, CARS, and SRS. The spatial overlap between the two-color pulses is temporally modulated by beam pointing modulation as shown in
. As an example, we use nondegenerate TPEF microscopy to illustrate the principle of this technique. As shown in , the focal position of the excitation pulse at ω

_{1} is temporally modulated with displacement,

*δ*_{0}, at a modulation frequency,

*f*, whereas that of the other excitation pulse at ω

_{2} is fixed. We consider the TPEF intensity modulation at positions –

*δ*_{0}, 0, and +

*δ*_{0}. Because the TPEF intensity at ω

_{1} + ω

_{2} is proportional to the product of the intensities of the excitation pulses, the TPEF intensity,

*I*_{TPEF}(

*r*,

*t*), is temporally modulated according to

The TPEF intensity at

*r* = 0 is modulated at a frequency of 2

*f*, while those at –

*δ*_{0} and +

*δ*_{0} are modulated at a frequency of

*f* as shown in . Thus, the frequency dependence of the TPEF signal varies from the center to the edge of the irradiated region. Assuming Gaussian beam profiles, the distortion of beam profiles from purely sinusoidal waves appears as high harmonic components. From the even-multiple frequency components of the modulation frequency, we can acquire information from a smaller volume than the interaction volume. Therefore, the spatial resolution is enhanced by sensing the even-multiple frequency components of the modulation frequency. However, modulation in a straight line in the lateral (

*x*) direction does not improve the spatial resolution in the

*y* direction. This problem can be solved by moving the focal position of one pulse in a spiral locus in the

*xy* plane as expressed by

This spiral modulation enhances the 3D spatial resolution. We refer to these techniques as straight and spiral SPOM-NOM. Moreover, since the modulation depth of the spatial overlap in the focal region is much greater than those in out-of-focus regions, out-of-focus background signals, which limit the obtainable imaging depth, are well suppressed. Thus, SPOM-NOM can be used to extend the achievable imaging depth.

By using SPOM at a modulation frequency of *f* without intensity modulation of the excitation pulses (), the laser intensities are modulated at a frequency of 2*f* with the loss or gain from TPA () and SRS (). Thus, SPOM enables the loss or gain from TPA and SRS to be detected through a lock-in amplifier with a demodulation frequency of 2*mf* (*m* = 1, 2, 3,…).

We define the relation between signals in conventional nonlinear microscopy and SPOM-NOM. As an example, we consider nondegenerate TPEF signals. In conventional nondegenerate TPEF microscopy using two-color pulses at instantaneous intensities of *I*_{1}(*r*,*z*,*t*) and *I*_{2}(*r*,*z*,*t*), the TPEF signal is expressed by

where *T* is the pixel dwell time, σ^{(2)} is the two-photon absorption cross section, *τ* is the pulse duration, *f*_{L} is the pulse repetition rate, *I*^{(peak)} is the peak intensity, and *I*^{(ave)} is the time-averaged intensity. *η* is the proportionality coefficient including the detection efficiency, the fluorescence quantum efficiency, and so on. *r* and *z* are radial and axial coordinates, respectively, which denote the distances from the focal point. Since the SPOM-NOM signal for beam pointing modulation is detected through a lock-in amplifier with a demodulation frequency of 2*mf* (*m* = 1, 2, 3,…), the demodulated TPEF signal in SPOM-NOM can be written as

where *f* is the modulation frequency. In the beam-pointing modulation, *δ*(*z*) is expressed by

where

*δ*_{0} is the maximum displacement of the focal spot and

*f*_{OB} is the focal length of the objective lens. When

*m* =

*δ*_{0} = 0,

Eq. (2) corresponds to

Eq. (1) and expresses the number of total collected photons during SPOM-NOM at each point. According to

Eq. (2), the demodulated signal in SPOM-NOM can be negative. To estimate the difference between the number of required photons per pixel in SPOM-NOM and that in conventional TPEF microscopy for the same signal intensity and acquisition time, we simplify

Eq. (2) by making a few approximations. By employing a Taylor expansion,

*I*_{1} can be written as

Since

*δ*(

*z*) is much smaller and

*T* is much longer than 1/2

*mf*,

*I*_{1} components that cannot be neglected in the integration in

Eq. (2) are expressed by

Using

Eqs. (2) and

(5), the demodulated TPEF intensity in SPOM-NOM can be approximated by

When the intensity profile

*I*_{1}(

*r*,

*z*) is approximated by

the demodulated TPEF signal at 2

*f* can be written as

where

and

Here,

*w*_{0} is the focal spot radius (1/e

^{2}), λ

_{c} is the central wavelength, and

*n* is the refractive index.

Equation (7) indicates that the spatial resolution of SPOM-NOM at 2

*f* is enhanced by a factor of

relative to that of conventional TPEF microscopy.

According to

Eq. (7), the demodulated signal in SPOM is proportional to the square of the maximum displacement of the focal spot and is inversely proportional to the square of the focal spot size. Thus, when the focal spot size is spread by wavefront distortion at large depths, the demodulated signal in the focal region is reduced compared with those in out-of-focus regions near the sample surface. Consequently, the signal-to-background ratio is reduced. However, the reduced demodulated signal can be compensated by increasing the maximum displacement of the focal spot. Then, the out-of-focus background from the sample surface is hardly increased. Therefore, the signal-to-background ratio can be enhanced even when the focal spot size is spread by wavefront distortion. Of course, the wavefront distortion can be compensated by adaptive optics [

23].

The maximum magnitudes of

^{2}*I*_{1}/

*r*^{2},

^{4}*I*_{1}/

*r*^{4},

^{6}*I*_{1}/

*r*^{6}, and

^{8}*I*_{1}/

*r*^{8} can be written as 4

*I*^{1}/

*w*_{0}^{2}, 48

*I*_{1}/

*w*_{0}^{4}, 960

*I*_{1}/

*w*_{0}^{6}, and 2304

*I*_{1}/

*w*_{0}^{8}, respectively. Thus, the demodulated TPEF intensities in SPOM-NOM at demodulation frequencies of 2

*f*, 4

*f*, 6

*f*, and 8

*f* are respectively reduced by factors of

*δ*_{0}^{2}/2

*w*_{0}^{2},

*δ*_{0}^{4}/8

*w*_{0}^{4},

*δ*_{0}^{6}/48

*w*_{0}^{6}, and

*δ*_{0}^{8}/384

*w*_{0}^{8} compared with the conventional TPEF intensity. If

*δ*_{0} is set to

*w*_{0}/2, the number of the required photons per pixel in SPOM-NOM at a demodulation frequency of 2

*f* is eight times greater than that in conventional TPEF microscopy to obtain the same signal. Under this condition, we can achieve a sufficiently high enhancement of the spatial resolution. If the numbers of total corrected photons per pixel are equal both in microscopies, the demodulated TPEF intensity in SPOM-NOM at 2

*f* will be eight times lower than the TPEF intensity in conventional microscopy.